Claire David

Michel L Lapidus

Weierstrass

Weierstrass Fractal Drums -II Towards a Fractal Cohomology

Keywords: MSC Classification: 28A80-35R02 Weierstrass Curve, prefractal approximations, Complex Dimensions of a fractal drum, discrete differential operators, differentials, fractal cohomology, cohomology groups, topological invariants, fractal expansions, cohomology infinitesimal, fractal complex, Poincaré duality

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Introduction

Thus far, the set up of a fractal cohomology theory appears as a very little explored field, where almost everything remains to be built. Many unanswered questions are involved, among which one may first retain the following one: what are the topological invariants of a fractal object?

In [START_REF] David | h-Laplacians on singular sets[END_REF], by using the concept of h-differentiation, where h denotes a very small real parameter, the first author and Gilles Lebeau have connected the associated h-cohomology to the classical de Rham cohomology. The results of that work go further than the existing ones on small scale parameter based cohomology (see the paper by Alain Connes and Henri Moscovici [START_REF] Connes | Cyclic cohomology, the Novikov conjecture and hyperbolic groups[END_REF], then, the articles of Laurent Bartholdi, Thomas Schick, Stephen Smale, and Nathan Smale [START_REF] Bartholdi | Hodge theory on metric spaces[END_REF], along with the complement written by Stephen and Nathan Smale in [START_REF] Smale | Abstract and classical Hodge-de Rham theory[END_REF]), in so far as, for the first time, we pass to the limit of the resolvent of the h-scale Laplacian, when the scale parameter h tends towards zero, along with the ad hoc Hodge decomposition. Among the many difficulties encountered in order to obtain the connections between the h-Laplacian and the Laplacian from fractal analysis, one had to define a proper differential, suitably adapted to fractals. This was done by means of discrete paths across prefractals (a sequence of finite graphs that converge towards the fractal).

Note that this approach is completely different from the one that can be found in the works of Fabio Cipriani and Jean-Luc Sauvageot in [START_REF] Cipriani | Fredholm modules on P.C.F. self-similar fractals and their conformal geometry[END_REF], followed up by that of Marius Ionescu, Luke G. Rogers and Alexander Teplyaev in [START_REF] Ionescu | Derivations and Dirichlet forms on fractals[END_REF], based on the construction of Fredholm modules. We refer to the introduction of [START_REF] David | h-Laplacians on singular sets[END_REF] for detailed comments.

In [START_REF] Michel | Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] and [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], the second author and Machiel van Frankenhuisjen have suggested that there should exist a fractal cohomology having direct links with the theory of Complex Dimensions, developed for many years now by Michel L. Lapidus and his collaborators, for example in [START_REF] Michel | Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture[END_REF], [START_REF] Michel | Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function[END_REF], [START_REF] Michel | Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl-Berry conjecture[END_REF], [START_REF] Michel | The Riemann zeta-function and the onedimensional Weyl-Berry conjecture for fractal drums[END_REF], [START_REF] Michel | The Riemann hypothesis and inverse spectral problems for fractal strings[END_REF], [START_REF] Michel | A tube formula for the Koch snowflake curve, with applications to complex dimensions[END_REF], [START_REF] Michel | Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes[END_REF], [START_REF] Michel | Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [LR Ž17a], [LR Ž18], [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF], [START_REF] Herichi | Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality[END_REF], [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF]. Recall that this theory of Fractal Complex Dimensions makes the connection between the geometry of an object and its differentiability properties, which is done by means of geometric (or fractal) zeta functions. Those zeta functions stand for the trace of a differential operator at a complex order s. The Complex Dimensions are obtained as the poles of the meromorphic continuations of those fractal zeta functions. The hint is that to each Complex Dimension ω associated to a fractal, one could associate a cohomology group H ω . See also [START_REF] Michel | Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes[END_REF] (and the relevant references herein) for further discussion of the expected connections between this potential fractal cohomology theory and number theory, along with arithmetic geometry.

Substantial results on fractal cohomology have later been obtained by the second author and Tim Cobler in [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF], where are studied the properties of the derivative operator D = d dz acting on a particular family of weighted Bergman space consisting of entire functions on C. This operator has to be understood as a suitable substitute for the Frobenius operator, and -via the spectrum of its action on the corresponding fractal cohomology spaces -the associated (generalized) Polya-Hilbert operator encodes all of the key information about a given meromorphic function, including especially its zeros and poles. The Frobenius operator -or a suitable analog -appears as being perfectly suited to fractals, when iterated function systems (ifs), and the associated prefractal approximations are involved. Somehow, one intuitively understands that points -vertices -at the m th step of the approximation process -necessarily carry the information coming from the previous (m -1) th step. In light of the prefractal sequence, composed of finite graphs where neighboring points are connected by edges, a shift operator also appears to be very natural. We note that the work in [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF] is pursued in the second author's book [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF], dedicated in large part to the subject of fractal cohomology and its motivations via the theory of Complex Dimensions and a suitably complexified version of quantized number theory [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Herichi | Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality[END_REF]; see also the epilog of [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF] for a brief overview.

But how could one go further, and obtain the underlying connections between Complex Dimensions and cohomology groups? The idea that came to our minds was to study a fractal object which could also be associated with a function. A test case, in a way. Our choice fell on the (nowhere differentiable) Weierstrass Curve.

We first tackled the problem in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], by determining, at first, the Complex Dimensions of the Curve -and establishing, in the process, a fractal tube formula expressing the volume of a small εneighborhood of the Weierstrass Curve as an infinite sum indexed by the corresponding Complex Codimensions. As previously evoked, those Complex Dimensions are obtained as the poles of the tube (or, equivalently, the distance) zeta function associated to the Curve. As could have been expected, the Minkowski dimension (or box dimension) D W is the Complex Dimension with maximal real part, and zero imaginary part. In [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], we also proved new geometric properties of the Curve and of the associated function, in relation with its discrete local Hölder and reverse Hölder continuity, with explicit estimates that had not been obtained before. In particular, we showed that the Codimension 2 -D W is the optimal Hölder exponent for the Weierstrass function W. This, naturally, goes in the direction of the existence of fractional derivatives, of orders expressed in function of the corresponding Dimensions.

As a corollary of these results, we concluded, in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF] and [START_REF] David | Polyhedral neighborhoods vs tubular neighborhoods: New insights for fractal zeta functions[END_REF], that the Weierstrass Curve Γ W is fractal , in the sense of the theory of Complex Dimensions ([LvF00], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [LR Ž17a], [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF]), because it admits nonreal Complex Dimensions -and hence also, intrinsic oscillations in its geometry. See Theorem 3.1, on page 18 and Remark 4.7, on page 44 below for a brief overview of some of the relevant main results of [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], including a fractal tube formula for the expression of the volume of the ε-neighborhoods of the Weierstrass Curve, as a sum over its Complex Codimensions. Now, a very interesting new question and feature arise: can one rewrite the series expansion of the Weierstrass function, in a form that directly yields a second one, indexed by the Complex Dimensions? This echoes the Hölder properties of the function, with a direct connection between the value of the Hölder exponent and the Complex Dimensions. We will show here that it is, indeed, the case.

The natural question which then arises is the following one: are those Complex Dimensions connected to the cohomological properties of the Curve? The answer is yes, even though it was rather difficult to prove it and to properly formulate it. In order to solve this problem, we naturally used the same kind of differentiation (from a necessary generalization) as the one obtained in [START_REF] David | h-Laplacians on singular sets[END_REF], by means of discrete paths along prefractals; see Section 4.3. In the process, we also came accross a very interesting and useful feature, in the context of our study, where the fractal object -the Weierstrass Curve Γ W -is directly connected with a function: in addition to its discrete local Hölder and reverse-Hölder properties (see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]), the Weierstrass function satisfies a new geometric condition which amounts to a uniform continuity condition in its argument part. In this light, by considering classes of functions which satisfy the same kind of properties, we prove that one can obtain, by induction, a complete characterization of all the elements of the cohomology groups associated to the Curve. And, as expected, we show that these cohomology groups consist of continuous functions (on the Weierstrass Curve) which are expressed in terms of (finite, or countably infinite) sums indexed by the Complex Dimensions. More precisely, as obtained in Theorem 4.11, on page 50, one has an exact correspondance between the m th cohomology group and the real terms in the sums. As for the complex terms, they can be understood as modes of vibrations of the Weierstrass fractal Curve.

As could perhaps also be expected, the sum that enables us to express any given function belonging to the m th cohomology group, at a given vertex of the associated m th prefractal graph, is in accordance with the expression of the Complexified Weierstrass function at the same point. In fact, we show, in Property 3.2, on page 18, that the Complexified Weierstrass function can exactly be expressed as an infinite sum indexed by the Complex Dimensions. Going further, this might be interpreted as a kind of generalized Taylor expansion, with fractional derivatives of orders k (2 -D W ) + i k p, where p denotes the oscillatory period of the Weierstrass Curve, for k ∈ N arbitrary, and where k ∈ Z satisfies a specific condition governed by the underlying geometry of the Curve; see Theorem 4.3, on page 31.

The Weierstrass Curve has a natural symmetry with respect to the vertical line x = 1 2 . It is then natural to define the associated left and right side differentials and complexes, as well as a Complex Conjugation with respect to this vertical line. One then naturally obtains a suitable version of Poincaré Duality for the fractal cohomology of the Weierstrass Curve, in relation with our notion of paths (from Section 4.4) accross the prefractal graphs; see Proposition 4.10, on page 49.

We point out that the fractal cohomology groups H m associated with each of the m th prefractal approximations of the Weierstrass Curve Γ W (where m ∈ N is arbitrary) are characterized in Theorem 4.3, on page 31, while the total fractal cohomology group of Γ W , H ⋆ -interpreted and defined as an appropriate inductive limit of the groups H m (where m ∈ N ) -is fully described in Theorem 4.11, on page 50.

The results of this paper and of [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF] are announced in the survey article [START_REF] David | Fractal complex dimensions and cohomology of the Weierstrass curve[END_REF], where the main results are presented in a summarized form. As is commented upon in several places throughout the present paper (and [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]), our results shed new light on the theory and interpretation of Complex Dimensions -particularly in connection with the vibrations of fractal drums (or strings), as well with the key notion of scales (including transition scales) in mathematical and theoretical physics.

Geometric Framework

Henceforth, we place ourselves in the Euclidean plane, equipped with a direct orthonormal frame. The usual Cartesian coordinates are denoted by (x, y). The horizontal and vertical axes will be respectively referred to as (x ′ x) and (y ′ y).

Notation 1 (Set of all Natural Numbers, and Intervals).

As in Bourbaki [START_REF] Bourbaki | Theory of Sets[END_REF] (Appendix E. 143), we denote by N = {0, 1, 2, . . .} the set of all natural numbers, and set Given two positive-valued functions f and g, defined on a subset I of R, we use the following notation, for all x ∈ I: f (x) ≲ g(x) when there exists a strictly positive constant C such that, for all x ∈ I, f (x) ⩽ C g(x).

N ⋆ = N \ {0}. Given a, b with -∞ ⩽ a ⩽ b ⩽ ∞, ]a, b[ = (a

Notation 3 (Weierstrass Parameters).

In the sequel, λ and N b are two real numbers such that

0 < λ < 1 , N b ∈ N ⋆ and λ N b > 1 . (♣)
Note that this implies that N b > 1 (i.e., N b ⩾ 2).

As explained in [START_REF] David | On fractal properties of Weierstrass-type functions[END_REF], we deliberately made the choice to introduce the notation N b which replaces the initial number b, in so far as, in Hardy's paper [START_REF] Harold | Weierstrass's Non-Differentiable Function[END_REF] (in contrast to Weierstrass's original article [START_REF] Weierstrass | Über continuirliche Funktionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differential quotienten besitzen[END_REF]), b is any positive real number satisfying λ b > 1, whereas we deal here with the specific case of a natural integer, which accounts for the natural notation N b .

Definition 2.1 (Weierstrass Function, Weierstrass Curve).

We consider the Weierstrass function W (also called, in short, the W-function) defined, for any real number x, by

W(x) = ∞ n=0 λ n cos 2 π N n b x . (1)
We call the associated graph the Weierstrass Curve, and denote it by Γ W .

Due to the one-periodicity of the W-function, from now on, and without loss of generality, we restrict our study to the interval [0, 1[= [0, 1).

Definition 2.2 (Weierstrass Complexified Function).

We introduce the Weierstrass Complexified function W comp , defined, for any real number x, by

W comp (x) = ∞ n=0 λ n e 2 i π N n b x .
Notation 4 (Logarithm).

Given y > 0, ln y denotes the natural logarithm of y, while, given a > 1, ln a y = ln y ln a denotes the logarithm of y in base a; so that, in particular, ln = ln e .

Notation 5 (Minkowski Dimension and Hölder Exponent).

For the parameters λ and N b satisfying condition (♣) (see Notation 3, on page 5), we denote by

D W = 2 + ln λ ln N b = 2 -ln N b 1 λ ∈ ]1, 2[
the box-counting dimension (or Minkowski dimension) of the Weierstrass Curve Γ W , which happens to be equal to its Hausdorff dimension [START_REF] Kaplan | The Lyapunov dimension of a nowhere differentiable attracting torus[END_REF], [START_REF] Barańsky | On the dimension of the graph of the classical Weierstrass function[END_REF], [START_REF] Shen | Hausdorff dimension of the graphs of the classical functions[END_REF], [START_REF] Keller | A simpler proof for the dimension of the graph of the classical Weierstrass function[END_REF]. We point out that the results in our previous paper [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF] also provide a direct geometric proof of the fact that D W , the Minkowski dimension (or box-counting dimension) of Γ W , exists and takes the above values, as well as of the fact that W is Hölder continuous with optimal Hölder exponent

2 -D W = - ln λ ln N b = ln N b 1 λ .
Convention (The Weierstrass Curve as a Cyclic Curve).

In the sequel, we identify the points (0, W(0)) and (1, W(1)) = (1, W(0)). This is justified by the fact that the Weierstrass function W is 1-periodic, since N b is an integer.

Remark 2.1. The above convention makes sense, because, in addition to the periodicity property of the W-function, the points (0, W(0)) and (1, W(1)) have the same vertical coordinate.

Property 2.1 (Symmetry with Respect to the Vertical Line x = 1 2 ).

Since, for any x ∈ [0, 1],

W(1 -x) = ∞ n=0 λ n cos 2 π N n b -2 π N n b x = W(x) ,
the Weierstrass Curve is symmetric with respect to the vertical straight line x = 1 2 .

Proposition 2.2 (Nonlinear and Noncontractive Iterated Function System (IFS)).

Following our previous work [START_REF] David | Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function[END_REF], we approximate the restriction Γ W to [0, 1[×R, of the Weierstrass Curve, by a sequence of graphs, built via an iterative process. For this purpose, we use the nonlinear iterated function system (IFS) of the family of C ∞ maps from R 2 to R 2 denoted by

T W = T 0 , . . . , T N b -1 ,
where, for any integer i belonging to {0, . . . , N b -1} and any point (x, y) of R 2 ,

T i (x, y) = x + i N b , λ y + cos 2 π x + i N b .
Property 2.3 (Attractor of the IFS [START_REF] David | Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function[END_REF], [START_REF] David | On fractal properties of Weierstrass-type functions[END_REF]).

The Weierstrass Curve Γ W is the attractor of the IFS T W , and hence, is the unique nonempty

compact subset K of R 2 satisfying K = N b -1 ⋃ i=0 T i (K); in particular, we have that Γ W = N b -1 ⋃ i=0 T i (Γ W ).
Notation 6 (Fixed Points).

For any integer i belonging to {0, . . . , N b -1}, we denote by

P i = (x i , y i ) = i N b -1 , 1 1 -λ cos 2 π i N b -1
the unique fixed point of the map T i ; see [START_REF] David | On fractal properties of Weierstrass-type functions[END_REF].

Definition 2.3 (Sets of Vertices, Prefractals).

We denote by V 0 the ordered set (according to increasing abscissae) of the points P 0 , . . . , P N b -1 .

The set of points V 0 -where, for any integer i in {0, . . . , N b -2}, the point P i is linked to the point P i+1 -constitutes an oriented finite graph, ordered according to increasing abscissae, which we will denote by Γ W 0 . Then, V 0 is called the set of vertices of the graph Γ W 0 .

For any positive integer m, i.e., for m ∈ N ⋆ , we set

V m = N b -1 ⋃ i=0 T i (V m-1 ).
The set of points V m , where two consecutive points are linked, is an oriented finite graph, ordered according to increasing abscissa, called the m th -order W-prefractal. Then, V m is called the set of vertices of the prefractal Γ W m ; see Figure 2, on page 10.

Definition 2.4 (Adjacent Vertices, Edge Relation).

For any m ∈ N, the prefractal graph Γ W m is equipped with an edge relation ∼ m , as follows: two vertices X and Y of Γ W m (i.e. two points belonging to V m ) are said to be adjacent (i.e., neighboring or junction points) if and only if the line segment [X, Y ] is an edge of Γ W m ; we then write X ∼ m Y . Note that this edge relation depends on m, which means that points adjacent in V m might not remain adjacent in V m+1 .

Property 2.4. [START_REF] David | Bypassing dynamical systems: A simple way to get the box-counting dimension of the graph of the Weierstrass function[END_REF] For any m ∈ N, the following statements hold :

i. V m ⊂ V m+1 . ii. #V m = (N b -1) N m b + 1, where #V m
(N b -1) k (N b -1) N m b , W (N b -1) k (N b -1) N m b , 1 ⩽ k ⩽ N m b -1 .
Hence, the total number of junction points is N m b -1. For instance, in the case N b = 3, the polygons are all triangles; see Figure 1.

In the sequel, we will denote by P 0 the initial polygon, whose vertices are the fixed points of the maps T i , 0 ⩽ i ⩽ N b -1, introduced in Notation 6 and Definition 2.3, on page 7, i.e., P 0 , . . . , P N b -1 ; see, again, Figure 1, on page 9.

M j,m = j (N b -1) N m b , W j (N b -1) N m b .
We also introduce, for any integer j in 0, . . . , (N b -1) N m b -1 : i . the elementary horizontal lengths:

L m = 1 (N b -1) N m b ;
ii . the elementary lengths:

l j,j+1,m = d M j,m , M j+1,m = L 2 m + h 2 j,j+1,m ,
where h j,j+1,m is defined in iii. just below.

iii . the elementary heights:

h j-1,j,m = W j (N b -1) N m b -W j -1 (N b -1) N m b , h j,j+1,m = W j + 1 (N b -1) N m b -W j (N b -1) N m b ; 1 x -1 1 y (a)
W 0 , Γ W 1 , Γ W 2 , Γ W 3 , Γ W 4 , Γ W 5 , in the case when λ = 1 2 and N b = 3.
iv . the geometric angles:

θ j-1,j,m = ̂ (y ′ y), M j-1,m M j,m , θ j,j+1,m = ̂ (y ′ y), M j,m M j+1,m ,
where (y ′ y) denotes the vertical axis, which yield the following value of the geometric angle between consecutive edges, namely, M j-1,m M j,m , M j,m M j+1,m , with arctan = tan -1 :

θ j-1,j,m + θ j,j+1,m = arctan L m h j-1,j,m + arctan L m h j,j+1,m .
(Note that, of course, θ j-1,j,m = arctan L m h j-1,j,m and θ j,j+1,m = arctan L m h j,j+1,m .)

Property 2.5 (Density of the Set V ⋆ in the Weierstrass Curve [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]).

The set V ⋆ = ⋃ n∈N V n is dense in the Weierstrass Curve Γ W .
Proof.

Since the function W is continuous, it suffices to remark that the set of the abscissae of the vertices is dense in [0, 1].

Formally, as is given in Definition 2.5, on page 9, for all m ∈ N , the set of consecutive vertices of V m is given by

M j,m 0⩽j⩽#V m -1 = j (N b -1) N m b , W j (N b -1) N m b 0⩽j⩽#V m -1 , with [0, 1] = #V m -2 ⋃ j=0 j (N b -1) N m b , j + 1 (N b -1) N m b .
(2)

Let us now consider a point M = (x, W(x)) of the Weierstrass Curve Γ W , along with a strictly postive number ε. Since the Weierstrass function is continuous, there exists a nonnegative integer m 0 ∈ N ⋆ such that, for all m ⩾ m 0 , we have that

∀ x ′ ∈ [0, 1] ∶ |x -x ′ | ⩽ 1 (N b -1) N m b ⟹ W(x) -W x ′ ) ⩽ ε
By using relation (2), we obtain the existence of an integer m 1 ⩾ m 0 such that, for all m ⩾ m 1 , there exist j ∈ {0, . . . , #V m -2}

j (N b -1) N m b ⩽ x ⩽ j + 1 (N b -1) N m b , from which we deduce that W(x) -W j + 1 (N b -1) N m b ⩽ ε ,
which yields the expected density result.

Property 2.6. For the geometric angle θ j-1,j,m , 0 ⩽ j ⩽ (N b -1) N m b , m ∈ N, we have the following relation:

tan θ j-1,j,m = h j-1,j,m L m .
At this point, it is interesting to use the scaling properties of the Weierstrass function, as obtained in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF].

Property 2.7 (Scaling Properties of the Weierstrass Function, and Consequences [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]).

Since, for any real number x,

W(x) = ∞ n=0 λ n cos 2 π N n b
x , one also has

W(N b x) = ∞ n=0 λ n cos 2 π N n+1 b x = 1 λ ∞ n=1 λ n cos 2 π N n b x = 1 λ (W(x) -cos (2 π x)) ,
which yields, for any strictly positive integer m and any j in {0, . . . , #V m },

W j (N b -1) N m b = λ W j (N b -1) N m-1 b + cos 2 π j (N b -1) N m b .
By induction, one then obtains that

W j (N b -1) N m b = λ m W j (N b -1) + m-1 k=0 λ k cos 2 π N k b j (N b -1) N m b .
Property 2.8 (A Consequence of the Symmetry with Respect to the Vertical Line x = 1 2 ).

For any strictly positive integer m and any j in {0, . . . , #V m }, we have that

W j (N b -1) N m b = W (N b -1) N m b -j (N b -1) N m b ,
which means that the points

(N b -1) N m b -j (N b -1) N m b , W (N b -1) N m b -j (N b -1) N m b and j (N b -1) N m b , W j (N b -1) N m b
are symmetric with respect to the vertical line x = 1 2 ; see Figure 3, on page 13. Property 2.9 ([DL22]).

Since, for any natural integer n,

N n b = (1 + N b -1) n = n k=0 n k (N b -1) k ≡ 1 mod N b -1 ,
one obtains, for any integer j in {0, . . . , N b -1}:

W j N b -1 = ∞ n=0 λ n cos 2 π N n b j (N b -1) = ∞ n=0 λ n cos 2 π j N b -1 = 1 1 -λ cos 2 π j N b -1 .
We observe that the point

j N b -1 , W j N b -1 = j N b -1 , 1 1 -λ cos 2 π j N b -1
is also the fixed point of the map T j introduced in Proposition 2.2, on page 7, and Notation 6, on page 7 following it.

Property 2.10 ([DL22]).

For 0 ⩽ j ⩽ (N b -1) 2 (resp., for (N b -1) 2 ⩽ j ⩽ N b -1), we have that W j + 1 N b -1 -W j N b -1 ⩽ 0 resp., W j + 1 N b -1 -W j N b -1 ⩾ 0 .
Notation 7 (Signum Function).

The signum function of a real number x is defined by

sgn (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -1, if x < 0, 0, if x = 0, +1, if x > 0 . Property 2.11 ([DL22]).
Given any strictly positive integer m, we have the following properties: i. For any j in {0, . . . , #V m }, the point

j (N b -1) N m b , W j (N b -1) N m b is the image of the point j (N b -1) N m-1 b -i, W j (N b -1) N m-1 b -i = j -i (N b -1) N m-1 b (N b -1) N m-1 b , W j -i (N b -1) N m-1 b (N b -1) N m-1 b under the map T i , 0 ⩽ i ⩽ N b -1.
Consequently, for 0 ⩽ j ⩽ N b -1, the j th vertex of the polygon P m,k , 0 ⩽ k ⩽ N m b -1, i.e., the point

(N b -1) k + j (N b -1) N m b , W (N b -1) k + j (N b -1) N m b ,
is the image of the point

⎛ ⎜ ⎝ (N b -1) k -i (N b -1) N m-1 b + j (N b -1) N m-1 b , W ⎛ ⎜ ⎝ (N b -1) k -i (N b -1) N m-1 b + j (N b -1) N m-1 b ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ,
which is also the j th vertex of the polygon

P m-1,k-i (N b -1) N m-1 b
. Therefore, there is an exact correspondance between vertices of the polygons at consecutive steps m -1, m.

ii. Given j in {0, . . . , N b -2} and k in 0, . . . , N

m b -1 , we have that sgn W k (N b -1) + j + 1 (N b -1) N m b -W k (N b -1) + j (N b -1) N m b = sgn W j + 1 N b -1 -W j N b -1 .
Property 2.12 (Lower Bound and Upper Bound for the Elementary Heights [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]).

For any strictly positive integer m and any j in 0, . . . , (N b -1) N m b , we have the following estimates:

C inf L 2-D W m ⩽ |W ((j + 1) L m ) -W (j L m )| h j,j+1,m ⩽ C sup L 2-D W m , m ∈ N, 0 ⩽ j ⩽ (N b -1) N m b , (✠)
where the finite and positive constants C inf and C sup are given by

C inf = (N b -1) 2-D W min 0⩽j⩽N b -1, W j+1 N b -1 ≠W j N b -1 W j + 1 N b -1 -W j N b -1 and C sup = (N b -1) 2-D W max 0⩽j⩽N b -1 W j + 1 N b -1 -W j N b -1 + 2 π (N b -1) (λ N b -1)
.

One should note, in addition, that these constants C inf and C sup depend on the initial polygon P 0 .

Theorem 2.13 (Sharp Local Discrete Reverse Hölder Properties of the Weierstrass Function [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]).

For any natural integer m, let us consider a pair of real numbers (x, x ′ ) such that

x = (N b -1) k + j (N b -1) N m b = ((N b -1) k + j) L m , x ′ = (N b -1) k + j + (N b -1) N m b = ((N b -1) k + j + ) L m , where 0 ⩽ k ⩽ N m b -1, and i. if the integer N b is odd, 0 ⩽ j < N b -1 2 and 0 < j + ⩽ N b -1 2 or N b -1 2 ⩽ j < N b -1 and N b -1 2 < j + ⩽ N b -1 ;
ii. if the integer N b is even,

0 ⩽ j < N b 2 and 0 < j + ⩽ N b 2 or N b 2 + 1 ⩽ j < N b -1 and N b 2 + 1 < j + ⩽ N b -1 .
This means that the points (x, W(x)) and x ′ , W(x ′ ) are vertices of the polygon P m,k (see Property 2.4, on page 8 above), both located on the left-side of the polygon, or both located on the right-side; see Figure 4, on page 13.

Then, one has the following (discrete, local ) reverse-Hölder inequality, with sharp Hölder expo-

nent - ln λ ln N b = 2 -D W , C inf |x ′ -x| 2-D W ⩽ W(x ′ ) -W(x) ,
where (x, W(x)) and x ′ , W(x ′ ) are adjacent vertices of the same m th prefractal approximation, Γ W m , with m ∈ N arbitrary. Here, C inf is given as in Property 2.12, on page 15 just above.

Remark 2.2. It is clear that, for any m ∈ N, and any pair (x, W(x)) , x ′ , W(x ′ ) of adjacent vertices of the finite prefractal graph Γ W m , the same following (discrete, local) Hölder and reverse-Hölder inequality, with sharp Hölder exponentln λ ln N b = 2 -D W , holds (see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]); i.e., still with C inf and C sup given as in Property 2.12, on page 15 above, we have that

C inf |x ′ -x| 2-D W ⩽ W(x ′ ) -W(x) ⩽ C sup |x ′ -x| 2-D W .
Corollary 2.14 (Optimal Hölder Exponent for the Weierstrass Function (see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF])).

The local reverse Hölder property of Theorem 2.13, on page 15 just above -in conjunction with the Hölder condition satisfied by the Weierstrass function (see also [START_REF] Zygmund | Trigonometric Series[END_REF], Chapter II, Theorem 4.9, page 47) -shows that the Codimension 2 -D W = -ln λ ln N b ∈ ]0, 1[ is the best (i.e., optimal ) Hölder exponent for the Weierstrass function (as was originally shown, by a completely different method, by G. H. Hardy in [START_REF] Harold | Weierstrass's Non-Differentiable Function[END_REF]).

Note that, as a consequence, since the Hölder exponent is strictly smaller than one, it follows that the Weierstrass function W is nowhere differentiable.

Corollary 2.15 (of Property 2.12, given on page 15 (see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF])).

Thanks to Property 2.12, on page 15, one may now write, for any strictly positive integer m and any integer j in 0, . . . , (N b -1) N m b -1 : i. for the elementary heights:

h j-1,j,m = L 2-D W m O (1) ;
ii. for the elementary quotients:

h j-1,j,m L m = L 1-D W m O (1) ,
and where

0 < C inf ⩽ O (1) ⩽ C sup < ∞ .
Corollary 2.16 (Nonincreasing Sequence of Geometric Angles (Coming from Property 2.11; see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]) ).

For the geometric angles θ j-1,j,m , 0 ⩽ j ⩽ (N b -1) N m b , m ∈ N, we have the following result:

tan θ j-1,j,m = L m h j-1,j,m (N b -1) > tan θ j-1,j,m+1 ,
which yields

θ j-1,j,m > θ j-1,j,m+1 and θ j-1,j,m+1 ≲ L D W -1 m .
Corollary 2.17 (Local Extrema (Coming from Property 2.11, given on page 14; see [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]) ).

i. The set of local maxima of the Weierstrass function on the interval [0, 1] is given by

(N b -1) k N m b , W (N b -1) k N m b ∶ 0 ⩽ k ⩽ N m b -1, m ∈ N ,
and corresponds to the extreme vertices of the polygons at a given step m (i.e., they are the vertices connecting consecutive polygons).

ii. For odd values of N b , the set of local minima of the Weierstrass function on the interval [0, 1] is given by

(N b -1) k + N b -1 2 (N b -1) N m b , W (N b -1) k + N b -1 2 (N b -1) N m b ∶ 0 ⩽ k ⩽ N m b -1, m ∈ N ,
and corresponds to the bottom vertices of the polygons at a given step m.

Complex Dimensions, and Associated Series Expansion of the Complexified Weierstrass Function

The following results were obtained in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], as a consequence of the very precise fractal tube formula also obtained in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]. (See Remark 4.7, on page 44 below for an abbreviated version of this fractal tube formula.)

Theorem 3.1 (Complex Dimensions of the Weierstrass Curve [DL22]).
The possible Complex Dimensions of the Weierstrass Curve are all simple, and given as follows: The next result is new and will play a key role in the rest of this paper (see, especially, Theorem 4.3, on page 31 below), as well as in one of its sequels, [START_REF] David | Fractional Taylor series: Conditions of existence, and explicit formulas[END_REF]. We refer to Definition 2.2, on page 6 above for the definition of W comp .

D W -k (2 -D W ) + i m p , with k ∈ N , m ∈ Z , 1 -2 k + i m p , with k ∈ N , m ∈ Z,

Property 3.2 (Complex Dimensions Series Expansion of the Weierstrass Complexified Function W comp ).

We have, for any sufficiently large positive integer m and any j in {0, . . . , #V m -1}, the following exact expansion, indexed by the Complex Codimensions k i. 0,j,m ∈ Z is arbitrary and

(D W -2) + i k k,j,m p, with 0 ⩽ k ⩽ m, W comp j ε m m = (N b -1) 2-D W ε m (2-D W ) m W comp j N b -1 + (N b -1) 2-D W m-1 k=0 c k,j,m ε k (2-D W ) k ε i k k,j,m p k , (3)
c 0,j,m = exp 2 i π N b -1 j ε m m . i. For 1 ⩽ k ⩽ m -1, k,j,m ∈ Z and c k,j,m ∈ C are respectively given by k,j,m = -j (N b -1) k-1 k N m-k b , (⋄) (4)
and

c k,j,m = (N b -1) 2-D W exp ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 i π j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ k (N b -1) k-1 k N m-k b - (N b -1) k-1 k N m-k b ln(N b -1) ln N b ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (⋄⋄) (5)
Since relation (6) is valid for any m ∈ N ⋆ , we note that the associated Complex Dimensions (i.e., in fact, the Complex Dimensions associated with the Weierstrass function) are

D W -k (2 -D W ) + i k,j,m p where 0 ⩽ j ⩽ #V m -1 and 0 ⩽ k ⩽ m -1.
This immediately ensures, for the Weierstrass function (i.e., the real part of the Weierstrass complexified function W comp ), that, for any strictly positive integer m and any j in {0, . . . , #V m },

W j ε m m = (N b -1) 2-D W ε m (2-D W ) m W comp j N b -1 + m-1 k=0 ε k (2-D W ) k Re c k,j,m ε i k k,j,m p k = (N b -1) 2-D W ε m (2-D W ) m W comp j N b -1 + 1 2 m-1 k=0 ε k (2-D W ) k c k,j,m ε i k k,j,m p k + c k,j,m ε -i k k,j,m p k . (6)
Proof. Thanks to Property 2.7, on page 12 (the Scaling Property of the Weierstrass function W), for any strictly positive integer m and any j in {0, . . . , #V m -1}, we then have that

W j (N b -1) N m b = λ m W j N b -1 + m-1 k=0 λ k cos 2 π N k b j (N b -1) N m b .
By using the Minkowski Dimension D W introduced in Notation 5, on page 6, this latter expression can be rewritten as

W j ε m m = N m (D W -2) b W j N b -1 + m-1 k=0 N k (D W -2) b cos 2 π N k (D W -2) b j ε m m = (N b -1) 2-D W ε m (2-D W ) m W j N b -1 + (N b -1) 2-D W m-1 k=0 ε k (2-D W ) k cos 2 π N b -1 j ε m-k m = (N b -1) 2-D W ε m (2-D W ) m W j N b -1 + (N b -1) 2-D W m-1 k=0 ε k (2-D W ) k Re exp 2 i π N b -1 j ε m-k m .
We then aim at determining wether there exist integers 0,j,m , . . . , m-1,j,m and complex numbers c 0,j,m , . . . , c m-1,j,m such that, for any integer k in {0, . . . , m -1},

(N b -1) 2-D W exp 2 i π N b -1 j ε m-k m = c k,j,m ε i k k,j,m p k = c k,j,m exp i k,j,m p ln ε k k .
We note that, when k = 0, we have that

ε i k k,j,m p k = 1 ,
which directly ensures that 0,j,m ∈ Z is arbitrary, and that

c 0,j,m = exp 2 i π N b -1 j ε m m .
We then have that, for 1

⩽ k ⩽ m -1, exp i k,j,m p ln ε k k = exp 2 i k,j,m π ln N b ln 1 (N b -1) N k b = exp - 2 i π k,j,m ln N b ln (N b -1) N k b = exp - 2 i π k,j,m ln N b (ln(N b -1) + k ln N b ) = exp - 2 i π k,j,m ln(N b -1) ln N b exp -2 i π k k,j,m , along with exp 2 i π N b -1 j ε m-k m = exp 2 i π j k k (N b -1) 1 (N b -1) -k N m-k b = exp 2 i π j k N b -1) k-1 k N m-k b = exp 2 i π j k (N b -1) k-1 k N m-k b + (N b -1) k-1 k N m-k b .
where [y] denotes the integer part of the real number y, and {y} denotes the fractional part of the real number y.

By letting

k,j,m = -j

(N b -1) k-1 k N m-k b and (N b -1) 2-D W exp 2 i π j k (N b -1) k-1 k N m-k b = c k,j,m exp - 2 i π k,j,m ln(N b -1) ln N b = c k,j,m e -i k,j,m p ln(N b -1) ,
i.e., equivalently,

c k,j,m = (N b -1) 2-D W exp 2 i π j k (N b -1) k-1 k N m-k b + i k,j,m p ln(N b -1) = (N b -1) 2-D W exp 2 i π j k (N b -1) k-1 k N m-k b e -i j (N b -1) k-1 k N m-k b p ln(N b -1) ,
we obtain the desired result.

Remark 3.1. One may note that only the Complex Dimensions

D W -k (2 -D W ) + i k,j,m p,
where m ∈ N is arbitrary, are involved in the expansion obtained in Property 3.2, on page 18. This calls for the existence, among the set of Complex Dimensions of the Weierstrass Curve, of a subset of Complex Dimensions only associated with the Weierstrass function W (as opposed to the Curve Γ W ). Such a result seems natural, in light of the possible interpretation of the Complex Dimensions given in [DL22]: the vertical line x = 0 (i.e., the Complex Dimensions of Γ W with real part 0) corresponds to oscillations coming from points -indeed, at any step m of the prefractal graph approximation, the prefractal graph Γ W m is, at first, constituted of points; as for the vertical line x = 1 (i.e., the Complex Dimensions of Γ W with real part 1), it corresponds to oscillations coming from lines (or, rather, line segments): indeed, at any step m of the prefractal graph approximation, the polygonal curve Γ W m is constituted of line segments, in an Euclidean space of dimension two.

Fractal Cohomology

Review and Preliminaries

For the benefit of the reader who may not be familiar with mathematical notions pertaining to cohomology, we shall first recall several relevant definitions.

Definition 4.1 (A-Module of p-Fermions on a General Topological Space [DL23c]).

Given a ring A of characteristic different from 2, a general topological space X, along with a nonnegative integer p, we call p-fermion on X, with values in A, any antisymmetric map f from X p+1 to A, i.e., such that, for any transposition τ of {0, . . . , p} and any x 0 , . . . , x p in X p+1 ,

f x 0 , . . . , x p = -f x τ (0) , . . . , x τ (p) .
A 0-fermion on X is simply a map f from X to A. We will henceforth adopt the convention according to which a 0-fermion on X is a 0-antisymmetric map on X.

We then denote by F p (X, A) the A-module of p-fermions on X with values in A, which makes it an abelian group with respect to the addition, with an external law from

A × F p (X, A) to F p (X, A),
where Given a ring A of characteristic different from 2, a general topological space X and a nonnegative integer p, we define the p-differentials δ p from F p (X, A) to F p+1 (X, A), for any f in F p (X, A), by

∀ (a, b) ∈ A 2 , ∀ (f, g) ∈ F p (X, A) 2 ∶ a (f + g) = a f + a g , (a + b) f = a f + b f .
∀ x 0 , . . . , x p+1 ∈ X p+2 ∶ δ p (f ) x 0 , . . . , x p+1 = c p p q=0
(-1) q f . . . , x q-1 , x q+1 , . . . , where c p ∈ A denotes a constant, the value of which is defined only when necessary.

Note that the 0-differential δ

0 , from F 0 (X, A) to F 1 (X, A), is defined, for any f in F 0 (X, A), by ∀ (x 0 , x 1 ) ∈ X 2 ∶ δ 0 (f ) (x 0 , x 1 ) = c 0 (f (x 1 ) -f (x 0 )) .
A key property is that

∀ p ∈ N ∶ δ p+1 • δ p = 0 . The so-called (cochain) Complex F • (X, A) , δ
• is the following algebraic structure, which consists in a sequence of abelian groups (or, here, A-modules) F p (X, A) p ∈ N , where, for each inte-

ger p ∈ N, the group F p (X, A) is connected to the group F p+1 (X, A) by means of the p-differential δ p ; namely, F 0 δ 0 ⟶ . . . F p δ p ⟶ F p+1 δ p+1 ⟶ . . . .
We then set

F • (X, A) = ∞ ⨁ p=0 F p (X, A) .
The associated group cohomology, i.e., the set consisting of ker δ 0 and of the p-cohomology

groups ker δ p+1 /Im δ p , p ∈ N ⋆ , is denoted by H • F • (X, A) , δ • .

General Considerations: h-Cohomology and Complex Dimensions

In [START_REF] David | h-Laplacians on singular sets[END_REF], we evoked a prefractal cohomology where the cohomology groups were of the form

ker δ #V m /Im δ #V m-1 , m ∈ N ⋆ .
In doing so, we joined the approach developed by Michel L. Lapidus and Machiel van Frankenhuisjen in [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] and [START_REF] Michel | Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions[END_REF], where the authors suggest that there should exist a fractal cohomology having direct links with the theory of Complex Dimensions. Recall from the introduction that further results along these lines have been obtained by Michel L. Lapidus and Tim Cobler in [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF]; see also [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF] and, especially, the second author's forthcoming book, [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF].

Undoubtedly, in our present setting, such a cohomology exists, as will be shown in this section. However, it is interesting to look at Figure 5 not vertically, but horizontally, i.e., along a line y = m p, m ∈ Z. And then, things look rather different: this horizontal line travels across all the vertical lines

x = D W -k (2 -D W ) , for k ∈ N .
As explained just above, one may infer that topological invariants, as the intrinsic objects of any cohomology theory, have to be searched along the aforementioned horizontal line, y = m p (m ∈ N); which means, for a given purely imaginary mode, their conservation when switching from the m th prefractal graph, to the (m + 1) th one. From a physical point of view, this makes sense. So, one could think of a cohomology as being indexed by the purely imaginary modes.

With regard to the real parts of the Complex Dimensions, one must bear in mind that they are nothing but the values of the maximal orders of differentiation. Differentiation, through the Dirac operator, is directly connected to the maximal length between adjacent vertices of a given m th prefractal graph approximation, which decreases proportionnaly to

N -m (2-D W ) b
when one switches from the m th prefractal graph, to the (m + 1) th one. This point will next be taken into account through our concept of generalized fermions. Henceforth, the ring A discussed aboved is choosen to be A = C, the field of complex numbers.

We point out that, in the present paper (and by contrast with [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]), the aforementioned vertical lines will be replaced by the vertical lines By analogy with particle physics, given a pair of integers (m, p), with m ∈ N and p ∈ N ⋆ , we will call (m, p)-fermion on V m , with values in C, any antisymmetric map f from V p+1 m to C. Note that these maps are not assumed to be multilinear (which would be meaningless here, anyway).

x = -k (2 -D W ) , for k ∈ N ,
For any m ∈ N, an (m, 0)-fermion on V m (or a 0-fermion, in short) is simply a map f from V m to C. As in Definition 4.1, on page 22, we adopt the convention according to which a 0-fermion on V m is a 0-antisymmetric map on V m .

In the sequel, for any (m, p) ∈ N 2 , we will denote by F p (V m , C) the C-module (i.e., the complex vector space) of (m, p)-fermions on V m , with values in C, which makes it an abelian group with respect to the addition, with an external law from

C × F p (V m , C) to F p (V m , C),

Differentials

One now has to define a proper notion of differential. The problem is a bit more complicated than in our previous work [START_REF] David | h-Laplacians on singular sets[END_REF], since one does not deal with a fixed space: at stake are the prefractals Γ W m , m ∈ N.

In order to understand how things go, one may look at the initial polygon P 0 : this polygon has exactly N b vertices, which mean, in terms of potential topological invariants (under the form of a complex-valued function f defined on the whole set of vertices M j,m , 0 ⩽ j ⩽ #V m -1, m ∈ N of the m th prefractal approximation Γ W m ), a number which will -or not -be conserved when switching to Γ W 1 .

This can be achieved by examining the following quantities (see Figure 6, on page 25):

f P j -f M 1,1 + f M 2,1 . . . + (-1) N b -1 f P j+1 , with 0 ⩽ j ⩽ N b ,
where the points P j , for 0 ⩽ j ⩽ N b -1, are the vertices of P 0 ; namely, they are the fixed points of the underlying maps T j in the i.f.s. T W = T 0 , . . . , T N b -1 .

One thus deals with alternate (i.e., antisymmetric) expressions with N b + 1 terms.

When reaching the m th prefractal graph, with m ⩾ 1, one will also have to examine quantities of the following form (see Figure 7, on page 26):

f M i,m-1 -f M j,m + f M j+1,m . . . + (-1) N b -1 f M i+1,m-1 , for 0 ⩽ i ⩽ #V m-1 -1
, and where (with the notation of Definition 2.4, on page 8 above)

M i,m-1 ∼ m M i+1,m-1 ,
while, at the same time,

M i,m-1 ∼ m-1 M j,m , M j,m ∼ m-1 M j+1,m , . . . ,
which means that the points M i,m-1 and M i+1,m-1 are consecutive vertices of V m-1 , and that the set of consecutive vertices of V m located strictly between M i,m-1 M i+1,m-1 consists of the points M j,m , M j+1,m ; and so on. Given a strictly positive integer m, and two adjacent vertices

P j+1 V 0 P j V 0 M 1,1 V 1 \V 0 M 2,1 V 1 \V 0 M 1,N b-1 V 1 \V 0
M j,m ϵ V m \V mE1 M i,mE1 ϵ V mE1 ⊂ V m M i+1,m-1 ϵ V m-1 ⊂ V m M j+1,m ϵ V m \V m-1
X m-1,k , X m-1,k+1 in V m , for 0 ⩽ k ⩽ #V m-1 -1, we call (m -1, m)-path between X m-1,k , X m-1,k+1 the ordered set of vertices P m-1,m X m-1,k , X m-1,k+1 = X m, , 0 ⩽ ⩽ N b , where X m, = X m-1,k and X m, +N b = X m-1,k+1 ;
see Figure 7, on page 26. Given a strictly positive integer m, we define the

(m -1, m)-differential δ m-1,m from F 0 (V m , C) to F N b +1 (V m , C), for any f in F 0 (Γ W , C) and any M i,m-1 , M i+1,m-1 , M j+1,m , . . . , M j+N b -2,m ∈ V N b +1 m such that M i,m-1 = M j,m and M i+1,m-1 = M j+N b ,m , by δ m-1,m (f ) M i,m-1 , M i+1,m-1 , M j+1,m , . . . , M j+N b -1,m = c m-1,m N b q=0 (-1) q f M j+q,m ,
where c m-1,m denotes a suitable positive constant. Note that as one deals with these differentials in the present paper, one does not need to know -or fix -the value of this constant. It becomes important, however, when operators involving the differentials, such as the Laplacian, are involved; see, for instance, Section 6 of [START_REF] David | h-Laplacians on singular sets[END_REF].

For k < m, we define the (m -k, m)-differential δ m-k,m from F 0 (V m , C) to F N k b +1 (V m , C), for any f in F 0 (V m , C) and any M i,m-k , M i+1,m-k , M j+1,m , . . . , M j+k (N b -1)-1,m ∈ V N k b +1 m such that M i,m-k = M j,m and M i+1,m-k = M j+N k b ,m , by δ m-k,m (f ) M i,m-k , M i+1,m-k , M j+1,m , . . . , M j+N k b -1,m = c m-k,m ⎛ ⎜ ⎜ ⎝ N k b q=0 (-1) q f M j+q,m ⎞ ⎟ ⎟ ⎠
, where c m-k,m denotes a suitable positive constant. Again, we will not need to specify its value here.

By induction, we can, equivalently, consider the (0, 1)-differential δ 0,1 from F

0 (V 0 , C) to F N b +1 (V 1 , C), then, the (1, 2)-differential δ 1,2 from F N b +1 (V 1 , C) to F N 2 b +1 (V 2 , C); and so on, which means that the (m -1, m)-differential δ m-1,m is defined from F N m-1 b +1 (V m-1 , C) to F N m b +1 (V m , C), for any m ∈ N ⋆ .
In fact, at a given step m ⩾ 0, between two adjacent vertices of V m , there are

N b -1 consecutive ver- tices of V m+1 \ V m . Hence, this results in N b -1 + 2 = N b + 1 consecutive vertices of V m+1 ⊃ V m . Among those N b + 1 vertices, there are then N b pairs of consecutive vertices of V m+1 ⊃ V m , which themselves involve N 2 b pairs of consecutive vertices of V m+2 ⊃ V m+1 , i.e., N 2 b + 1 consecutive vertices of V m+2 ⊃ V m+1 ⊃ V m ;
and so on, by induction.

Remark 4.1. Note that because of the density of the set

V ⋆ = ⋃ n∈N V n in the Weierstrass Curve Γ W
(see Property 2.5, on page 11), and since Γ W is compact, a continuous -and hence, also, uniformly continuous -function defined on the Weierstrass Curve is obviously uniquely defined by its restriction to each set V m , for m ∈ N (i.e., by its restriction to V ⋆ ). This topological property will be crucial in the forthcoming prefractal cohomology of Section 4.4 below.

Conversely, any (uniformly) continuous function on V

⋆ can be uniquely extended to a necessarily uniformly continuous function on the compact set Γ W ⊂ R 2 .

Fractal Complex, and Cohomology Groups

Proposition 4.1 (Fractal Complex).

Hereafter, the (cochain) Complex involved, denoted by F

• (Γ W , C) , δ
• is the algebraic structure, which consists in the sequence of abelian groups (of (m, p) fermions, see Definition 4.3, on page 24)

F N m b +1 (V m , C) m ∈ N
, where, for each integer m ⩾ 2, the group

F N m-1 b +1 (V m-1 , C) is connected to the group F N m b +1 (V m , C) by means of the (m -1, m)-differentials δ m-1,
m introduced in Definition 4.5, on page 26, and viewed as group homomorphisms; namely,

F 0 (V 0 , C) δ 0,1 ⟶ . . . F N m b +1 (V m , C) δ m-1,m ⟶ F N m+1 b +1 (V m+1 , C) δ m,m+1 ⟶ . . . .

Because of the density of the set V

⋆ = ⋃ n∈N V n in the Weierstrass Curve Γ W (see Remark 4.1 just
above, on page 27), and due to the definition of the differentials in Definition 4.5, on page 26, this complex can also be written in the following form: In our present setting, with the differential introduced in Definition 4.5, on page 26, the cohomology groups are the quotient groups

F 0 (Γ W , C) δ 0,1 ⟶ . . . F N m b +1 (Γ W , C) δ m-1,m ⟶ F N m+1 b +1 (Γ W , C) δ m,m+1 ⟶ . . . .
H m = ker δ m-1,m /Im δ m-2,m-1 , for m ⩾ 0 ,
with the additional convention that δ -2,-1 = 0 and δ -1,0 = 0, which ensures that H 0 = {0}.

Remark 4.2. For any k ∈ N and m ∈ N ⋆ , the quotient of the Complex Dimensions

k (2 -D W ) m p
is also a quotient of frequencies, for oscillatory maps defined on the countable set of vertices

V ⋆ = ⋃ n ∈ N V n .
The fundamental frequencies

2 -D W p , 1 p ,
respectively associated to oscillations in the vertical and horizontal directions, happen to be proportional to each other.

Proof. The search for invariants amounts in finding oscillatory maps f that are preserved either when switching from the m th prefractal graph to the (m + 1) th one, or, alternatively, from the initial prefractal graph to the m th prefractal graph.

Also, because applying the map T j , for any j ∈ {0, . it makes sense to observe that, given a complex number s = r + i t (with (r, t) ∈ R 2 ), we have that

λ N b s = e m (r+i t) (ln λ-ln N b ) = λ r N r b e -i t (2-D W ) ln N b e -i t ln N b .
Note that the frequencies

1 p = ln N b 2 π and 2 -D W p
are proportional to each other.

As a search for topological invariants, a fractal cohomological approach must take into account the specific geometric properties of an object. For our present Fractal Curve Γ W , the symmetry with respect to the vertical line x = 1 2 plays an important role in this setting. Indeed, not only is it present at the initial step of the prefractal graph construction, but it also continues to hold, at a given integer step m ⩾ 1, with the associated polygons P m,k , for 0 ⩽ k ⩽ N

M (N b -1) k+j,m = M (N b -1) k ′ +j ′ ,m-1 and M (N b -1) (k+1),m = M (N b -1) k ′ +j ′ +1,m-1 , for 0 ⩽ k ′ ⩽ N m-1 b -1 and 0 ⩽ j ′ ⩽ N b -1.
Hence, the consecutive edges of P m,k are nothing but a path between M (N b -1) k+j,m and M (N b -1) (k+1)+j,m . This deformed path, in a sense, is the image of the direct one

M (N b -1) k ′ +j ′ ,m-1 ↝ M (N b -1) k ′ +j ′ +1,m-1 .
One can go as far as saying that, in some sense, these two paths are homotopic.

In this light, it thus seems natural to encode this symmetry in the cohomological approach. From now on, given any m ∈ N, we will call m th cohomology infinitesimal the number ε m m > 0 which also corresponds to the elementary horizontal length introduced in part i. in Definition 2.5, on page 9; i.e., ε

m m = (ε m ) m = 1 N b -1 1 N m b .
Observe that, clearly, ε m itself -and not just ε for each m ∈ N -an infinitesimal , called the cohomology infinitesimal . Note that this cohomology infinitesimal is the one naturally associated to the scaling relation of Proposition 2.7, on page 12.

In the sequel, it is also useful to keep in mind that the sequence of positive numbers (ε m )

∞ m=0 itself satisfies ε m ∼ 1 N b , as m → ∞ ; i.e., ε m → 1 N b , as m → ∞.
In particular, ε m / → 0, as m → ∞, but, instead, ε m tends to a strictly positive and finite limit.

We also introduce, given any m ∈ N, the m th intrinsic cohomology infinitesimal , denoted by ε m > 0, such that

ε m = 1 N m b , where ε = 1 N b .
We call ε the intrinsic scale, or intrinsic subdivision scale. Given a nonzero complex number z, we denote by arg (z) the argument of z; i.e., the angle between the positive real axis and the line joining the origin and the point M with affix z. Implicitly, we always choose the same convention (e.g., arg (z) ∈ ]-π, π]) whenever evaluating arg (z) -and hence also, arg

Note that

(z) -arg (z ′ ), for (z, z ′ ) ∈ C ⋆ × C ⋆ .
Definition 4.7 (Set of Functions of the Same Nature as the Weierstrass Function W).

i . We say that a continuous, complex-valued function f , defined on

Γ W ⊃ V ⋆ ,
-z| 2-D W ⩽ f (z ′ ) -f (z) ⩽ Csup |z ′ -z| 2-D W ,
where Cinf and Csup denote positive and finite constants (but not necessarily the same ones as for the Weierstrass function W itself, in Property 2.12, on page 15). This can be written, equivalently, as

z -z ′ 2-D W ≲ f (z) -f (z ′ ) ≲ z -z ′ 2-D W . (♦) (7)
Hereafter, we will denote by Höld (Γ W ) the set consisting of the continuous, complex-valued functions f , defined on Γ W ⊃ V ⋆ and satisfying relation (7); see the discussion just above.

ii . Moreover, we will denote by Höld geom (Γ W ) ⊂ Höld (Γ W ) the subset of Höld (Γ W ) consisting of the functions f of Höld (Γ W ) which satisfy the following additional geometric condition (8), again, for any pair of adjacent vertices (M, M ′ ) with respective affixes (z, z ′ ) ∈ C 2 of the prefractal graph V m , with m ∈ N arbitrary; we have that,

arg (f (z)) -arg f (z ′ ) ≲ |z -z ′ | D W -1 . (♠) (8)
We can now state the following key result.

f M j,m = m k=0 c k f, M j,m ε k (2-D W ) k , M ⋆,m ∈ V m , c k f, M ⋆,m ∈ C , (♦♦) (9)
where, for each integer k such that 0 ⩽ k ⩽ m, the number ε k k > 0 is the k th cohomology infinitesimal introduced in Definition 4.6, on page 29 above.

The coefficients c k (f, M ⋆ ) are complex quantities, which only depend on the function f involved, and on the point M ⋆ at which they are evaluated. As will follow from an obvious modification in Definition 4.8, on page 41, and Proposition 4.5, on page 42 below, these coefficients c k (f, M ⋆ ) are the residues (at the possible Complex Dimensions -k (2 -D W )) of a suitable global scaling zeta function.

Note that, insofar as the functions f involved are, in a sense, determined, at any vertex M j,m ∈ V m , by the expansion given in relation (9), it is natural to identify the cohomology groups H m with those functions.

ii. If the functions f of part i. belong to Höld geom (Γ W ) (see part ii. of Definition 4.7, on page 30 above), then, for any strictly positive integer m, and again with the convention H 0 = Im δ -1,0 = {0}, the cohomology groups

H m = ker δ m-1,m /Im δ m-2,m
j,m ∈ V m , f M j,m = m k=0 c k f, M j,m ε k (2-D W ) k ε i k k,j,m p k = m k=0 c k f, M j,m ε k (2-D W )+i k k,j,m p k , M ⋆,m ∈ V m (♠♠) (10)
where p denotes the oscillatory period introduced in [DL22],

p = 2 π ln N b ,
and where the coefficients c k (⋆, ⋆) are complex numbers which still depend on the function f involved, and on the point at which they are evaluated. Here, in relation (10), for each integer k such that 0 ⩽ k ⩽ m, k,j,m denotes an integer (in Z) satisfying the estimate

k k,j,m ln ε k k ln N b ≲ ε k (D W -1) k 2 π . ( ) (11)
The proof of Theorem 4.3, on page 31, will be given on page 34, just after Remark 4.5.

Remark 4.4 (A Few Important Comments About Theorem 4.3).

i. Note that since, the fractional part map is one-periodic, this results in a kind of periodicity with respect to the integers k,j,m : the set k,j,m , k ∈ N is infinite and equal to Z. In particular, | k,j,m | → ∞ as k → ∞. We thus recover completely analogous results to those obtained in Theorem 3.1, on page 18 above (from [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]).

ii. Much as in part i. of Theorem 4.3, the coefficients c k (f, M ⋆ ), for 0 ⩽ k ⩽ m, depend only on the function f and on the point M ⋆ of V m at which they are evaluated. Note that, obviously, the values of the constants c k in part ii. are different from those obtained in part i. The specific topic of their possible extension to all M ⋆ ∈ Γ W and their possible continuity in M ⋆ will be studied in more detail in a forthcoming work, [START_REF] David | Fractional Taylor series: Conditions of existence, and explicit formulas[END_REF].

Furthermore, as will follow from Definition 4.8, on page 41, and Proposition 4.5, on page 42, these coefficients c k (f, M ⋆ ) are the residues (at the possible Complex Dimensionsk (2 -D W ) + i k,j,m p ) of a suitable global scaling zeta function; see relation (13) of Proposition 4.5, on page 42.

iii. In both expansions (9) and (10), the coefficients c k (f, M ⋆ ), for 0 ⩽ k ⩽ m, reflect the dependence of the value taken by the map f at the vertex M j,m on the values taken by f at previous steps -vertices -of the m th prefractal graph approximation, in conjunction with values taken by f at neighboring vertices of M j,m at the same level (m) of the prefractal sequence and with vertices which, in addition, strictly belong to the same polygon P m,k introduced in Property 2.4, on page 8, with 1 ⩽ k ⩽ N m b -1 (by "strictly" here, we mean that the junction vertices are not included).

iv. The expansion in part ii. (namely, (10)) might be interpreted as a kind of generalized Taylor expansion with corresponding complex derivatives of orders -ω k = k (2 -D W ) + i k,j,m p, where k ∈ N is arbitrary, the coefficients c k (f, M ⋆ ) can thus be interpreted as (discrete) derivatives of complex order -ω k of the function f , evaluated at the point M ⋆ of V ⋆ ⊂ Γ W . A similar comment can be made about the expansion in part i. (namely, (9)), but now with the above value of -ω k replaced by k (2 -D W ). Again, as in part i., it is natural to identify the cohomology groups H m with the sets of functions satisfying, at any vertex M j,m ∈ V m , the expansion given in relation (10).

v. Note that the result given in part ii. above is significantly stronger than the one given in part i. of this theorem. Indeed, the expansion (9) in part i. only involves (modulo a translation by -D W ) the real parts of some of the Complex Dimensions listed in Theorem 3.1, on page 18, and Remark 3.1, on page 22; namely, the Complex Dimensions with real parts different from 0 and 1. By contrast, in part ii., we can also recover the imaginary parts of those same Complex Dimensions; note that this is possible only because the functions involved satisfy the geometric condition (8) introduced in Definition 4.7 above, on page 30.

vi. It immediately follows from the expansion given in (10) of part ii. above, as well as from the fact that the sequence of sets of vertices (V m ) m ∈ N is increasing (see part i. of Property 2.4), that the sequence of cohomology groups (H m ) m ∈ N is increasing; i.e.,

∀ m ∈ N ∶ H m ⊂ H m+1 .
vii. Finally, in both expansions (9) and (10), the number ε is not an arbitrary one. It is directly connected to the scaling properties of the Weierstrass Curve, and enables us to express the exact form of the associated scaling relationship (see Property 2.7, along with Property 3.2, on page 18). In other words, the number ε -or, rather, the corresponding m th cohomology infinitesimal ε m m , in the sense of Definition 4.6, on page 29 above -depends on the geometry of the Curve. In light of our expansions, it can be interpreted as a (geometric) coefficient connecting the Weierstrass Curve and the complex-valued functions defined on the set

V ⋆ = ⋃ n ∈ N V n .
Corollary 4.4 (The Cohomology Groups are Nontrivial).

Let us set, for any real number t in [0, 1],

γ W (t) = (t, W(t)) .
For any positive integer m, we then have that the identity function on the Weierstrass Curve, 1 Γ W , restricted to the set of vertices V m , namely, 1 Γ W |V m , obviously belongs to H m . Hence, H m ≠ {0}, and is nontrivial.

Similarly, it will follow from Theorem 4.11, on page 50 below that the identity function on Γ W , namely, 1 Γ W , belongs to the total cohomology group H ⋆ , a fact which we condense as follows: the Weierstrass function itself belongs to its own fractal cohomology.

Remark 4.5. The above geometric condition (8), introduced in Definition 4.7, on page 30, and used in part ii . of Theorem 4.3, on page 31, in an essential manner -a kind of Hölder (or even Lipschitz) condition in the argument of the functions -is satisfied by the most natural function on the Weierstrass Curve, namely, the Weierstrass function itself. We came across this property in our previous work [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], where we proved that, for the geometric angles θ j-1,j,m , with 0 ⩽ j ⩽ (N b -1) N k b , m ∈ N (see part iv. of Definition 2.5, on page 9):

θ j-1,j,m+1 ≲ ε m (D W -1) m . One then has, for vertices M j m+1 ,m+1 ∈ V m+1 and M j m ,m ∈ V m ⊂ V m+1 such that M j m+1 ,m+1 ∼ m+1 M j m ,m , |arg M j m+1 ,m+1 -arg M j m ,m | ≲ ε m (D W -1) m , or, equivalently, arg M j m ,m = arg M j m-1 ,m-1 + C m ε m (D W -1) m
, for some nonnegative constant C m ∈ R .

Let us now denote by

k m an integer (in Z) such that k m ln ε ln N b = C m ε m (D W -1) m 2 π , with {y} ∈ [0, 1[ denoting the fractional part of y ∈ R. It then follows that ε i m k m p m = e i m k m p ln ε m = e i 2 π m k m ln ε m ln N b = e i 2 π m k m ln ε m ln N b +2 π i m k m ln ε m ln N b = e i 2 π m k m ln ε m ln N b = e i C m ε m (D W -1) m .
Note that, for practical reasons, it is more convenient to write

m k m ln ε m ln N b ≲ ε m (D W -1) m 2 π .
This key condition is the one that enables us to obtain the expansion (10) in part ii. of Theorem 4.3 above: indeed, without this condition, it is impossible to obtain the terms

ε i k k,j,m p k
, for some suitable values of k,j,m ∈ Z.

More precisely, this condition results, for the argument of the involved function f , in an arithmetic growth proportional to ε m (D W -1) m when switching from the m th prefractal graph to the (m + 1) th one, which is in perfect agreement with the interpretation given in [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF]. At the same time, it goes further than [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF] with regard to the imaginary parts of the Complex Dimensions, since it enables us to show that the m th prefractal graph bears, in a sense, the oscillations of its predecessors.

We are now ready to provide the proof of Theorem 4.3, on page 4.3, along with some preliminary discussion.

Proof. (of Theorem 4.3, given on page 31)

Notation 9. In this proof, for the sake of simplicity, we will often write M ⋆,m , m ∈ N, for the points M j,m , with 0 ⩽ j ⩽ #V m -1, of the prefractal graph Γ W m . In the same way, we will also use ⋆ as a dummy expression in order to denote the relevant quantities involved at the different levels of our forthcoming induction.

We will also make use of complex coefficients of the form c ⋆ (⋆). When c 0 (⋆) coefficients are involved, they denote constants depending on the argument; for instance, a coefficient c 0 (f, ⋆) refers to values of the function f on vertices ⋆ of the Curve.

Premilinaries. We hereafter place ourselves within the set Höld (Γ W ) introduced in part i. of Definition 4.7, on page 30.

i . Henceforth, we aim at proving, by induction, that, for any integer m ⩾ 1, and with the convention H 0 = Im δ -1,0 = {0}, the cohomology groups H m = ker δ m-1,m /Im δ m-2,m-1 consist of the restrictions to V m of (m, N m b + 1)-fermions, involving the restrictions of continuous, complex-valued functions f defined on the set Γ

W ⊃ V ⋆ = ⋃ n ∈ N V n , such that, for any vertex M j,m ∈ V m , f M j,m = m k=0 c k f, M j,m,k ε k (2-D W ) k , c k (f, M ⋆ , k) ∈ C ,
where ε k k > 0 is the k th cohomology infinitesimal introduced in Definition 4.6, on page 29, namely

(asymptotically as m → ∞), ε m m ∼ 1 N m b ,
and where, for 0

⩽ k ⩽ m, the complex coefficients c k (f, M ⋆ , k) are such that 0 ⩽ Cinf ≲ |c k (f, M ⋆ , k)| ≲ Csup < ∞ . (⋆⋆) (12)
Note that, accordingly, some of the coefficients c k could vanish.

↝ Initialization:

Let us thus consider a continuous, complex-valued function f in Höld (Γ W ). The values of the function on the set of vertices V 0 can then be considered as constant ones (i.e., independent of ε). For any vertex M j,0 ∈ V 0 , 0 ⩽ j ⩽ #V 0 -1, we therefore set f M j,0 = c 0 f, M j,0 , c 0 f, M j,0 ∈ C .

Let us suppose that

f ∈ ker δ 0,1 /Im δ -1,0 ; i.e., f ∈ ker δ 0,1 .

One thus has, for any

M i,0 , M i+1,0 , M j+1,1 , . . . , M j+N b -1,1 ∈ V N b +1 1 such that M i,0 = M j,1 and M i+1,0 = M j+N b ,1 , δ 0,1 (f ) M i,0 , M i+1,0 , M j+1,1 , . . . , M j+N b -1,1 = = c 0,1 c 0 f, M i,0 + (-1) N b c 0 f, M i+1,0 + N b -1 q=1 (-1) q f M j+q,1 = 0 ,
where the positive constant c 0,1 is the one associated to the differential δ 0,1 (see Definition 4.5, on page 26).

Each value f M j+1,1 can then be expressed as

f M j+1,1 = 0⩽q⩽N b -2 (-1) q+1 f M j+q+1,1 + c 1 (f, M i,0 ) + c 0 (f, M i+1,0 ) ,
where the coefficients c ⋆ (f, ⋆) belong to C and depend on the points M ⋆,0 .

The discrete local Hölder and reverse-Hölder property of the function then yields, for 0

⩽ q ⩽ N b -2, ε 2-D W 1 ≲ |f M j+q,1 -f M j+q+1,1 | ≲ ε 2-D W 1
, which enables us to express the sum

0⩽q⩽N b -2 (-1) q+1 f M j+q+1,1 in the form 0⩽k⩽N b -2 (-1) q+1 f M j+q+1,1 = c 0 f, M ⋆,1 + c 1 f, M ⋆,1 ε 2-D W 1
, where the coefficients c ⋆ (f, ⋆) belong to C, and with

(N b -1) Cinf ⩽ |c 1 (f, ⋆)| ⩽ (N b -1) Csup .
We thus obtain the sought for expression, i.e.,

f M j+1,1 = 0⩽k⩽1 c k f, M j+1,1 ε k (2-D W ) k , c k f, M ⋆,1 ∈ C .
↝ At a given strictly positive step m:

Our induction hypothesis consists in the following assumptions:

M j,m ∈ V m , f M j,m = m k=0 c k f, M j,m ε k (2-D W ) k , c k f, M ⋆,k ∈ C . b. Given δ m-2,m-1 (ϕ m-2 ) in Im δ m-2,m-1
, and involving the restrictions to V m-1 of functions

g f ∈ Höld (Γ W ) such that, for any M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m-1 b -1,m-1 ∈ V N m-1 b +1 m-1 sat- isfying M i,0 = M j,-1 and M i+1,0 = M j+N m-1 b -1,m , we have that δ m-2,m-1 (ϕ m-2 ) M i,0 , M i+1,0 , M j+1,m-1 , . . . , M j+N m-1 b -1,m-1 = m-1 k=0 c k g f , M ⋆,m-1 ε k (2-D W ) k , where the coefficients c ⋆ g f , ⋆ belong to C.

Let us then consider an (m, N m+1 b

+ 1)-fermion, denoted by ϕ m+1 , belonging to the (m + 1) th cohomology group H m+1 . It involves the restrictions to V m+1 of functions f ∈ Höld (Γ W ), such that, for any

M i,0 , M i+1,0 , M j+1,m , . . . , M j+N b -2,m ∈ V N b +1 m+1 satisfying M i,0 = M j,m+1 and M i+1,0 = M j+N m+1 b ,m+1 , we have that δ m,m+1 (ϕ m+1 ) M i,0 , M i+1,0 , M j+1,m+1 , . . . , M j+N m+1 b -1,m+1 = 0 , with δ m,m+1 (ϕ m+1 ) M i,0 , M i+1,0 , M j+1,m+1 , . . . , M j+N m+1 b -1,m+1 = c m,m+1 ⎛ ⎜ ⎜ ⎝ N m+1 b q=0 (-1) q f M j+q,m+1 ⎞ ⎟ ⎟ ⎠ = c m,m+1 ⎛ ⎜ ⎜ ⎝ f M i,0 + (-1) N m+1 b f M i+1,0 + N m+1 b -1 q=1 (-1) q f M j+q,m+1 ⎞ ⎟ ⎟ ⎠ . ( )
At the same time, given δ m-1,m ϕ m 1 in Im δ m-1,m , and involving the restrictions to V m of continuous functions g f on Γ W , such that, for any

M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m ∈ V N m b +1 m satisfying M i,0 = M j,m and M i+1,0 = M j+N m b ,m , we have that δ m-1,m (ϕ m-1 ) M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m = = c 0,1 ⎛ ⎜ ⎝ c 0 g f , M i,0 + (-1) N m b c 0 g f , M i+1,0 + N m b -1 q=1 (-1) q g f M j+q,m ⎞ ⎟ ⎠ .
Since the sequence of sets of vertices (V m ) m ∈ N is increasing, the sum

N m b -1 q=1 (-1) q g f M j+q,m
can also be written in the form

N m-1 b -1 q=1 (-1) q g f M j ′ +q,m-1 + N m b -N m-1 b q=1 (-1) q g f M j+q,m ,
where the vertices M j ′ +q,m-1 ∈ V m-1 are the intersection points between V m-1 and V m , for 1

⩽ j ′ + q ⩽ #V m-1 -1.
We can then make use of part b. of the induction hypothesis, and obtain that

δ m-1,m (ϕ m-1 ) M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m = = c m-1,m ⎛ ⎜ ⎜ ⎝ m-1 k=0 c k g f , M ⋆,m-1 ε k (2-D W ) k + N m b -N m-1 b q=1 (-1) q g f M j+q,m ⎞ ⎟ ⎟ ⎠ ,
where the coefficients c ⋆ g f , ⋆ belong to C.

The discrete, local, Hölder and reverse-Hölder property of the function g f then yields, for 1

⩽ j + q ⩽ N m b -N m-1 b , ε m (2-D W ) m ≲ |g f M j+q,m -g f M j+q+1,m | ≲ ε m (2-D W ) m
, which enables us to express the sum

N m b -N m-1 b q=1 (-1)
q g f M j+q,m in the following form:

N m b -N m-1 b q=1 (-1) q g f M j+q,m = c 0 g f , M ⋆,m + c m g f , M ⋆,m ε m (2-D W ) m
, where the coefficients c ⋆ g f , ⋆ belong to C. Bearing in mind that the sequence of sets of vertices (V m ) m ∈ N is increasing, we thus obtain that

δ m-1,m (ϕ m-1 ) M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m = = m-1 k=0 c k g f , M ⋆,m-1 ε k (2-D W ) k + c m g f , M ⋆,m ε m (2-D W ) m = m k=0 c k g f , M ⋆,m ε k (2-D W ) k
, where the coefficients c ⋆ (f, ⋆) belong to C.

Back to relation ( ), i.e., the relation translating the fact that the (m, N m+1 b

+ 1)-fermion ϕ m+1 belongs to the (m + 1)

th cohomology group H m+1 , we can then deduce that each value f M j+1,m+1 can be expressed in the following form:

f M j+1,m+1 = c 0 M j+1,m+1 + N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1 + m k=0 c k g f , M ⋆,m ε k (2-D W ) k .
In light of the discrete local Hölder and reverse-Hölder properties of the function f , we have that,

for 2 ⩽ q ⩽ N m+1 b -2, ε (m+1) (2-D W ) m+1 ≲ |f M j+q,m+1 -f M j+q+1,m+1 | ≲ ε (m+1) (2-D W ) m+1
, which enables us to express the sum

N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1 as N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1 = c0 M j+q,m+1 + c m+1 f, M j,m+1 ε (m+1) (2-D W ) m+1
, where the coefficients c0 (f, ⋆) and c ⋆ (f, ⋆) belong to C.

In the end, and because, implicitly, the function g f itself depends on f , we obtain the sought for expression,

f M j+1,m+1 = 0⩽k⩽m+1 c k f, M j+1,m+1 ε k (2-D W ) k , c k f, M ⋆,k ∈ C .
This completes the proof by induction of part i. of the theorem.

ii . If the functions f of part i . belong to Höld geom (Γ W ); i.e., if they satisfy the following additional geometric condition (8) in Definition 4.7, again, for any pair of adjacent vertices (M, M ′ ) of respective affixes (z, z ′ ) ∈ C 2 of the prefractal graph Γ W m , with m ∈ N arbitrary, we have that

arg (f (z)) -arg f (z ′ ) ≲ |z -z ′ | ; (♠)
then the expected result is obtained in the manner explained in Remark 4.5, on page 33 above. For the sake of simplicity, we will only rewrite the additional points in the induction.

↝ Initialization:

It is the same as in i .. We simply set 0 = 0. As a result, we do not repeat the argument here.

↝ At a given strictly positive step m:

Our induction hypothesis now consists in the following assumptions:

j,m ∈ V m , f M j,m = m k=0 c k f, M j,m ε k (2-D W ) k ε i k k,j,m p k , c k f, M ⋆,k ∈ C ,
k,j,m ln ε k k ln N b ≲ ε k (D W -1) k 2 π . b ′ . Given δ m-2,m-1 (ϕ m-2 ) in Im δ m-2,m-1
, and involving the restrictions to V m-1 of functions

g f ∈ Höld geom (Γ W ) such that, for any M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m-1 b -1,m-1 ∈ V N m-1 b +1 m-1 satisfying M i,0 = M j,-1 and M i+1,0 = M j+N m-1 b ,-1m , we have that δ m-2,m-1 (ϕ m-2 ) M i,0 , M i+1,0 , M j+1,m-1 , . . . , M j+N m-1 b -1,m-1 = m-1 k=0 c k g f , M ⋆,m-1 ε k (2-D W ) k ε i k k,⋆,m-1 p k ,
where the coefficients c ⋆ g f , ⋆ belong to C, and where, for 0 ⩽ k ⩽ m -1, k,j,m denotes an integer (in Z) satisfying the same estimate as in a., i.e.,

k,j,m ln ε k k ln N b ≲ ε k (D W -1) k 2 π .
Let us then consider an (m, N m+1 b

+ 1)-fermion, denoted by ϕ m+1 , belonging to the (m + 1) th cohomology group H m+1 , which involves the restrictions to V m+1 of functions f ∈ Höld geom (Γ W ).

We then proceed as in b. above. For the sake of simplicity, we will only give details for the additionnal changes. We start by considering δ m-1,m ϕ m 1 in Im δ m-1,m , and involving the restrictions to V m of functions g f ∈ Höld geom (Γ W ) such that, for any

M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m ∈ V N m b +1 m , satisfying M i,0 = M j,m and M i+1,0 = M j+N m b ,m ,
we have that

δ m-1,m (ϕ m-1 ) M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m = = c m-1,m ⎛ ⎜ ⎝ c 0 g f , M i,0 + (-1) N m b c 0 g f , M i+1,1 + N m b -1 q=1 (-1) q g f M j+q,m ⎞ ⎟ ⎠ .
As was explained in Remark 4.5, on page 33, by reasoning through a chain of adjacent vertices (i.e., a path in the corresponding, oriented finite graph), the geometric condition yields, for 1

⩽ q ⩽ N m b -N m-1 b -1, arg M j+q,m = arg M j ′ ,m + C m ε m (D W -1) m , 0 ⩽ j + q ⩽ #V m -1 , C m ∈ R .
We thus obtain that

δ m-1,m (ϕ m-1 ) M i,0 , M i+1,0 , M j+1,m , . . . , M j+N m b -1,m = = c m-1,m m-1 k=0 c k g f , M ⋆,m-1 ε k (2-D W ) k ε i k k,⋆,m-1 p k + c m g f , M j+q,m ε m (2-D W ) m ε i m m,j+q,m p m
, where the coefficients c ⋆ (g, ⋆) belong to C, and where, for 0 ⩽ k ⩽ m, k,j,m denotes an integer (in Z) satisfying the estimate

k,j,m ln ε k k ln N b ≲ ε k (D W -1) k 2 π .
Back to the fact that the (m, N m+1 b

+ 1)-fermion ϕ m+1 belongs to the (m + 1) th cohomology group H m+1 , we can then deduce that each value f M j+1,m+1 can then be expressed in the following form:

f M j+1,m+1 = c 0 M j+1,m+1 + N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1 + m k=0 c k g f , M ⋆,m ε k (2-D W ) k ε i k k,⋆,m p k .
The geometric condition (8), on page 31, in conjunction with the discrete local Hölder and reverse-Hölder properties of the function f , enables us this time to express the sum

N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1
as

N m+1 b -1 q=2 (-1) q+1 f M j+q,m+1 = c0 f, M j+q,m+1 + c m+1 f, M j,m+1 ε (m+1) (2-D W ) m+1 ε i (m+1) m+1,j,m+1 p m+1
, where the coefficients c0 (f, ⋆) and c ⋆ (f, ⋆) belong to C, and where m+1,j,m+1 denotes an integer (in Z) satisfying the estimate m+1,j,m+1

ln ε m+1 m+1 ln N b ≲ ε (m+1) (D W -1) m+1 2 π .
We therefore obtain the desired result and complete the (strong) induction.

Note that, given m ∈ N, this proof thus gives the value of f at any vertex M j,m ∈ V m , for 1 ⩽ j ⩽ #V m -1. In fact, the expressions associated to the extreme vertices M 0,m and M #V m , which are, also, the extreme vertices (i.e., the endpoints) of the Curve Γ W , are still valid, with zero coefficients in factor of the terms ε

k (2-D W ) k ε i k k,⋆,m p k , for k ⩾ 1.
Two other comments on our proof are warranted: ↝ In both parts i . and ii ., the first induction hypothesis a. simply states that the desired resultnamely, the Taylor-like expansion -is true at the considered step m ∈ N.

↝ In both parts i . and ii ., the second induction hypothesis b. (resp. b' .) is the one that is required in order to carry out the strong induction.

This completes the proof of Theorem 4.3, given on page 31.

We note that, in light (and with the notation) of Theorem 4.3, on page 31 above, and Theorem 4.11, on page 50 below, Definition 4.8, on page 41 and Proposition 4.5, on page 42 just below can be applied to any f ∈ H m , for m ∈ N, and to any f ∈ H ⋆ , respectively. Given m ∈ N, and a function f in Höld geom (Γ W ) (see part ii. of Definition 4.7, on page 30 above), possessing, by Theorem 4.3, on page 31, for any vertex M j,m ∈ V m , an expansion of the form (♠♠) obtained in part ii. of Theorem 4.3, on page 31; i.e.,

f M j,m = m k=0 c k f, M j,m ε k (2-D W )+i k k,j,m p k , M ⋆,k ∈ V k , c k f, M j,m ∈ C ,
where, for 0 ⩽ k ⩽ m, ε k k has been introduced in Definition 4.6, on page 29, and where the coefficients c k f, M ⋆,m are uniformly bounded complex numbers, we define the local scaling zeta function ζ f,M j,m ,loc associated to f at the vertex M j,m , as the following truncated Mellin transform, initially given, for any complex number s such that Re(s) > 0, by

ζ f,M j,m ,loc (s) = m k=0 c k f, M ⋆,m ε k k 0 t s-1 t k (2-D W )+i k,j,m p dt .
By meromorphic continuation to all of C, one then obtains the local scaling zeta function ζ f,M j,m ,loc for all s ∈ C, as given by the equality just above. More precisely, we have that, for all s ∈ C,

ζ f,M j,m ,loc (s) = m k=0 c k f, M j,m ε s+k (2-D W )+i k,j,m p s + k (2 -D W ) + i k,j,m p .
Note that, similarly, for any k ∈ N, ε k = (ε k ) k denotes the k th component of the k th cohomology infinitesimal introduced in Definition 4.6, on page 29; the same comment applies to Proposition 4.5, on page 42 just below.

Proposition 4.5 (Global Scaling Zeta Function).

Given a point M ⋆,m ∈ V m , for some arbitrary m ∈ N, and a function f in Höld geom (Γ W ), possessing, by Theorem 4.3, on page 31, for any vertex M j,m ∈ V m , an expansion of the form (10) in part ii. of Theorem 4.3 (see also Definition 4.8 ust above, on page 41), we introduce the global scaling zeta function ζ f,M ⋆,m ,gl associated to f at the point M ⋆,m , as the following convergent (and, even, absolutely convergent) series. Namely, ζ f,M ⋆,m ,gl is the meromorphic function on C given, for all s ∈ C, by

ζ f,M ⋆,m ,gl (s) = ∞ k=0 c k f, M ⋆,m ε s+k (2-D W )+i k,j,m p s + k (2 -D W ) + i k,j,m p . ( 13
)
Accordingly, in light of Definition 4.8, on page 41 just above, ζ f,M ⋆,m ,gl is a countably infinite linear combination (with coefficients c k f, M ⋆,m , for all k ∈ N) of truncated Mellin transforms determined by the k th cohomology infinitesimal ε k (for all k ∈ N).

Note that the set of local Complex Dimensions (i.e., the set of poles of ζ f,M ⋆,m ,gl ),

CD f,M ⋆,m = -k (2 -D W ) -i k,j,m p , c k f, M ⋆,m ≠ 0 , k ∈ N ,
depends on the function f and on the point M ⋆,m .

By contrast, naturally, the set of possible local Complex Dimensions,

CD possible f,M ⋆,m = -k (2 -D W ) -i k,j,m p , k ∈ N , does not depend on f or M ⋆,m .
More precisely, the poles of ζ f,M ⋆,m ,gl are all simple, and are the complex numbers

ω k = -k (2 -D W ) -i k,j,m p, for each k ∈ N such that c k f, M ⋆,m ≠ 0.
Furthermore, for all k ∈ N, the residue of ζ f,M ⋆,m ,gl at ω k is given by

res ζ f,M ⋆,m ,gl (s), ω k = c k f, M ⋆,m .
Proof. The convergence, for all s ∈ C, of the series involved just above in the expression of the global scaling zeta function ζ f,M ⋆,m ,gl is ensured by the fact that the coefficients c k f, M ⋆,m are uniformly bounded complex numbers (with respect to k ∈ N); see the inequalities in relation (12) of the proof of Theorem 4.3, on page 31 above.

Indeed, for each k ∈ N, c k f, M ⋆,m ε s+k (2-D W )+i k,j,m p s + k (2 -D W ) + i k,j,m p ≲ ε Re(s)+k (2-D W ) (Re(s) + k (2 -D W )) 2 + Im(s) + k,j,m p 2 ≲ ε Re(s)+k (2-D W ) k (2 -D W ) = ε Re(s) k (2 -D W ) ε 2-D W ) k ≲ 1 k (2 -D W ) 1 N Re(s) b 1 N 2-D W b k , ( )
where k ∈ N 1 k 1 N 2-D W b k < ∞,
which can be checked, for example, via the ratio test for series.

Weierstrass' theorem, in conjunction with the uniform estimate ( ) just above, then enables us to conclude that, for all s

∈ C \ -k (2 -D W ) -i k,j,m p , k ∈ N , the series of meromorphic -and even, holomorphic -functions s ↦ c k f, M ⋆,m ε s+k (2-D W )+i k,j,m p s + k (2 -D W ) + i k,j,m p , for k ∈ N, is itself holo- morphic.
For k 0 ∈ N, and s = ω k 0 = -k 0 (2 -D W ) -i k 0 p, the same computation as above, in conjunction with the local uniform convergence (in s ∈ C, on every compact subset of C) of the series of

functions s ↦ c k f, M ⋆,m ε s+k (2-D W )+i k,j,m p s + k (2 -D W ) + i k,j,m p , shows that lim s→ω k 0 s -ω k 0 ζ f,M ⋆,m ,gl (s) = k ∈ N lim s→ω k 0 s -ω k 0 ζ f,M ⋆,m ,gl (s) = c k 0 f, M ⋆,m , + k ∈ N, k≠k 0 lim s→ω k 0 s -ω k 0 s -ω k c k f, M ⋆,m ε s-ω k = c k 0 f, M ⋆,m , ε 0 = c k 0 f, M ⋆,m , ;
i.e., the residue of ζ f,M ⋆,m ,gl exists and is given (still for any

k 0 ∈ N) by res ζ f,M ⋆,m ,gl (s), ω k 0 = c k 0 f, M ⋆,m .
Note that ζ f,M ⋆,m ,gl is initially defined by an infinite sum of truncated Mellin transforms -exactly as in Definition 4.8, on page 41, for ζ f,M j,m ,loc , except for m now replaced by ∞ in the summation. Also as in Definition 4.8, on page 41, the resulting computation is initially valid for Re(s) > 0, but by meromorphic continuation to all of C, the corresponding (convergent) series representation of ζ f,M ⋆,m ,gl is seen to remain valid for all s ∈ C.

In the above discussion, we can consider that our global zeta function ζ f,M ⋆,m ,gl takes its values in the Riemann sphere P 1 (C), or C = C ∪ ∞; i.e., the extended complex plane, or the complex projective line, equipped with the chordal metric, as is done in Chapter 3 and Appendix C of [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], or in Chapter 6 of [START_REF] Herichi | Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality[END_REF]. We then have that the global scaling zeta function ζ f,M ⋆,m ,gl is meromorphic in all of C (and hence, is a continuous function with values in P 1 (C), taking the value ∞ ∈ P 1 (C) at each pole), with all of its poles being simple poles.

Remark 4.6. The result of Theorem 4.3, on page 31, is in perfect agreement with the one obtained, in the specific case of the Complexified Weierstrass function W comp , studied in detail in Proposition 3.2, on page 18 above.

We may also note that those results (in Proposition 3.2, on page 18, and Proposition 4.5, on page 42, as well as in Theorem 4.3 above and 4.11 below) enable us to distinguish between the set of Complex Dimensions associated to the Weierstrass function W, and the set of Complex Dimensions associated to the Weierstrass Curve Γ W , restricted here to their imaginary parts in {0, . . . , m p}. We will further discuss this issue in a later work.

Remark 4.7 (A Consequence for the Interpretation of Complex Dimensions).

In [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], the Complex Dimensions of the Weierstrass Curve Γ W have been obtained by means of a fractal tube formula; i.e., in our case, for any ε > 0 sufficiently small, the area of a two-sided εneighborhood of the Curve, which (apart from two terms associated with the Complex Dimensions 0 and -2) consists of an expansion of the form

α real part of a Complex Dimension ε 2-α G α ln N b 1 ε , (⋆)
where, for any real part α of a Complex Dimension, G α denotes a continuous and one-periodic function -and where, for α = D W -k (2 -D W ), with k ∈ N arbitrary, so that if k = 0, 2 -D W is the optimal Hölder exponent for the Weierstrass function, as shown in Corollary 2.14, on page 16; in this case, the function G α is nonconstant, for any k ∈ N -as well as, if k = 0, bounded away from zero and infinity. It is, therefore, important to note that, by contrast, the expansions of Theorem 4.3, on page 31 above (as well as those of Theorem 4.11, on page 50 below) are only valid for the value of the cohomology infinitesimal introduced in Definition 4.6, on page 29, i.e., the infinitesimal ε associated to the scaling law of the Weierstrass Curve Γ W ; see, however, Corollary 4.12, on page 52 below for an extension to smaller infinitesimals.

As a consequence, it makes it possible to express, in an extremely precise way, the relations satisfied by the functions which belong to the cohomology groups.

Remark 4.8 (The Case of Junction Vertices).

In the case of junction vertices, i.e., points M j,m = M j ′ ,m+1 belonging to consecutive prefractal graphs Γ W m and Γ W m+1 , for m ∈ N, 0 ⩽ j ⩽ #V m , 0 ⩽ j ′ ⩽ #V m+1 -1, the expansion (10) given in part ii . of Theorem 4.3, on page 31, is of the form

f M j,m = f M j ′ ,m+1 = m+1 k=0 c k f, M j ′ ,m+1 ε k (2-D W ) k ε i k k,j,m+1 p k , M ⋆,k ∈ V k , c k f, M j,m,k ∈ C , with c m+1 f, M j ′ ,m+1 = 0 and ∀ k ∈ {0, . . . , m} ∶ c k f, M j ′ ,m+1 = c k f, M j,m ,
as well as for some suitable sequence ( k ) 0⩽k⩽m+1 ∈ Z m+2 , as specified in the statement of part ii. of Theorem 4.3, on page 31.

Note that it is in perfect agreement with the fact that the sequence of cohomology groups (H m ) m ∈ N is increasing, as seen at the end of Theorem 4.3, on page 31. Given a function f ∈ Höld geom (Γ W ) (see Definition 4.7, on page 30), a natural question consists in asking how one can determine, for any integer m ⩾ 0 and any integer j ∈ {0, . . . , #V m -1}, the explicit expression of the coefficients c k f, M ⋆,k involved in the expansion (10).

In fact, things present themselves as follows:

↝ At step m = 0:
One simply determines the values of the complex coefficients c 0 f, M j,0 , for 0 ⩽ j ⩽ #V 0 -1, such that

∀ j ∈ {0, . . . , #V 0 -1} ∶ f M j,0 = c 1 f, M j,0 . ↝ At step m = 1:
The point is to determine the values of the complex coefficients c k f, M j,2 , for 1

⩽ j ⩽ #V 1 -1, such that f M j,1 = 1 k=0 c k f, M j,2 ε k (2-D W ) k ε i k k,j,1 p k . ↝ At step m = 2:
The point is to determine the values of the complex coefficients c k f, M j,2 , for 1

⩽ j ⩽ #V 2 -1, such that f M j,2 = 2 k=0 c k f, M j,2 ε k (2-D W ) k ε i k k,j,2 p k .
As is shown in the proof of Theorem 4.3, given on page 31, we have that

0⩽k⩽N b -2 (-1) k+1 f M j+k+1,2 = c 2 f, M j+1,2 ε 2 (2-D W ) 2 ε i 2 2,j+1,2 p 2 .
Consequently, for all integers j in {0, . . . , #V 2 -1} such that f M j+1,2 ≠ 0, one immediately obtains that

c 2 f, M j+1,2 = 0⩽k⩽N b -2 (-1) k+1 f M j+k+1,2 ε -2 (2-D W ) 2 ε -i 2 2,j+1,2 p 2 .
One could proceed likewise at every step m. Therefore, the difficulty seems to determine, at a given step m ⩾ 1, the values of the coefficients c k f, M j+1,m , for 0 ⩽ k ⩽ m -1.

The result obtained in Proposition 3.2, on page 18, yields the exact expression for those coefficients, in the case of the Complexified Weierstrass function W comp . One possible approach is to test wether analoguous expressions, adapted to functions satisfying the hypotheses of Theorem 4.3, on page 31, are suitable and can be obtained (i.e., they belong to the cohomology groups). This will be the object of our forthcoming work [START_REF] David | Fractional Taylor series: Conditions of existence, and explicit formulas[END_REF].

Proposition 4.7 (Generators of the Cohomology Groups).

For any integer m ⩾ 1, and with the convention H 0 = Im δ -1,0 = {0}, the generators of the (additive) cohomology groups

H m = ker δ m-1,m /Im δ m-2,m-1
are to be understood in the sense of the k th cohomology infinitesimals ε k k , for 0 ⩽ k ⩽ m (see Definition 4.6, on page 29). Note that by generators we do not refer to the notion of group generators in group theory, but in a broader sense, instead. In our present context, indeed, we express the values, at the vertices of V m , of the continuous, complex-valued maps defined on the Weierstrass Curve

Γ W ⊃ ⋃ n ∈ N V n ,
which constitute the quotient groups H m , by means of Taylor-like expansions involving fractional powers of the k th cohomology infinitesimals ε k k , for 0 ⩽ k ⩽ m. So, in a sense, one could think of those generators as generalized (fractional) polynomial variables. Those generators are thus of the following form:

ε k (2-D W ) k ε i k k,j,m p k = ε k (2-D W )+i k,j,m p k , with 0 ⩽ k ⩽ m and k,j,m ∈ Z such that k ln ε k k ln N b ≲ ε k (D W -1) k
2 π ; see also Proposition 4.5, on page 42 above.

Proof. This follows at once from the statement of part ii. of Theorem 4.3, on page 31.

Poincaré Duality

The goal of this section is to obtain an appropriate analog, in the present context, of Poincaré Duality (from algebraic topology and geometry). Very useful references on Poincaré Duality can be found in the book of Jean Gallier and Jocelyn Quaintance [GQ22] (Chapter 7), and in the Lecture notes of Weiyi Zhang [START_REF] Zhang | Cohomology and Poincaré Duality[END_REF].

In our context, where we lay the foundations of prefractal cohomology, we also have in mind to obtain, in a future work, an appropriate fractal homology theory of which the present existing fractal cohomology theory (or a suitable modification therein) would be, in some sense, the dual theory. At the same time, we have at our disposal this natural symmetry (with respect to the vertical line x = 1 2 (see Property 2.1, on page 6). In this light, we could not bypass defining the associated left and right side differentials and complexes, as well as a Complex Conjugation with respect to this vertical line, as is done below. Also, our prefractal graphs enable us to obtain a polyhedral neighborhood of the Weierstrass Curve Γ W , see [START_REF] David | Iterated fractal drums ∼ Some new perspectives: Polyhedral measures, atomic decompositions and Morse theory[END_REF], [START_REF] David | Polyhedral neighborhoods vs tubular neighborhoods: New insights for fractal zeta functions[END_REF]. As is discussed in loc. cit., an immediate consequence of Property 4.8 below, on page 47, is that we have a correspondance between the polygons located from either side of the x = 1 2 , directly connected to Poincaré Duality.

Proposition 4.8 (Conjugation).

In light of the symmetry of Γ W with respect to the vertical line x = 1 2 (see Property 2.1, on page 6), it seems natural to consider the following change of variables from real to complex coordinates (with an obvious relabelling of the axes):

Z = y + i x - 1 2
, where (x, y) ∈ R 2 .

We then define the corresponding complex conjugation, which, to any point M of the complex plane, associates its symmetric point M ′ with respect to the vertical line x = 1 2 ,

M = M ′ ,
with affix the complex number Z defined by Z = yi x -1 2 .

This yields, for any strictly natural integer m and any j in 0, . . . ,

(N b -1) N m b 2 , M (N b -1) N m b -j,m = M j,m , or j (N b -1) N m b , W j (N b -1) N m b = (N b -1) N m b -j (N b -1) N m b , W (N b -1) N m b -j (N b -1) N m b .
In the following definitions and results (i.e., Definition 4.9, on page 47, and Definition 4.10, on page 48, along with Proposition 4.10, on page 49), it is helpful to remember that, in light of part ii . of Property 2.4, on page 8, we have that, for any integer m ⩾ 0,

#V m -1 = (N b -1) N m b .
Definition 4.9 (Left and Right Side of the Prefractal Sequence and Curve).

We define the Left Side of the Prefractal Sequence Γ W m m∈N as the following sequence of sets:

(L m ) m ∈ N = M j,m , 0, ⩽ j ⩽ (N b -1) N m b 2 m ∈ N .
In the same way, we define the Right Side of the Prefractal Sequence Γ W m m∈N as the following sequence of sets:

(R m ) m ∈ N = M j,m , (N b -1) N m b 2 ⩽ j ⩽ N b -1) N m b m ∈ N = L m m ∈ N .
Thus, for any m ∈ N, L m represents the Left Side of the prefractal graph Γ W m , while R m stands for its Right Side. Then, the Complex Conjugation defined in Proposition 4.8, on page 47, exactly yields the following identity:

δ Z,m-1,m = δ Z,m-1,m .
We can similarly define the left-side and right-side complexes, in the obvious manner. Given a strictly positive integer m, we respectively define: i . the Left and Right Side cohomology groups, for each integer m ⩾ 1, as

H m = ker δ Z,m-1,m /Im δ Z,m-2,m-1 , H m = ker δ Z,m-1,m /Im δ Z,m-2,m-1 ,
with the additional convention

H 0 = Im δ Z,-1,0 = {0} , H 0 = Im δ Z,-1,0 = {0} .
ii . For each integer m ⩾ 1 and for 0 ⩽ j ⩽

(N b -1) N m b

2

, the j th Left and Right Side cohomology groups, respectively denoted by H m,j and H m,#V m-1 -j , as consisting of the respective restrictions to V m of continuous, complex-valued functions f on the Weierstrass Curve Γ W such that, for any pair of conjugate vertices

M j,m , M #V m-1 -j,m ∈ V 2 m , satisfying f M j,m = m k=0 c k f, M j,m ε k (2-D W ) k ε i k k,j,m p k , M ⋆,k ∈ V k , c k f, M j,m ∈ C , and
f M #V m-1 -j,m = m k=0 c k f, M #V m-1 -j,m ε k (2-D W ) k ε i k k,j,m p k , M ⋆,k ∈ V k , c k f, M #V m-1 -j,m ∈ C , we have that f M j,m = f M #V m-1 -j,m .
Proposition 4.10 (Poincaré Duality).

Given a strictly positive integer m, the conjugation property obtained in Proposition 4.9, on page 48, which results in the isomorphism of the kernels

ker δ Z,m-1,m ≃ ker δ Z,m-1,m ,

2

, to the following Poincaré duality (expressed as a group isomorphism):

H m,j ≃ H m,#V m-1 -j .
Remark 4.9. The above duality in Proposition 4.10, on page 49, simply states the natural isomorphism between a path connecting two vertices located on the Left Side and the path connecting the corresponding two vertices located on the Right Side of the m th prefractal graph, symmetrically. In terms of the functions f , which comprise the left and right-side cohomology groups, it simply translates into the idempotent map M ↦ f M .

Total Fractal Cohomology of the Weierstrass Curve

Regarding our final and main theorem just below, we refer the reader to Section 4.4 above for the definition of the fractal complex and of the cohomology groups associated to the Weierstrass Curve Γ W ; we also refer to Remark 4.1, on page 27, and Remark 4.3, on page 30. Within the set Höld geom (Γ W ) of continuous, complex-valued functions f , defined on the Weier-

strass Curve Γ W ⊃ V ⋆ = ⋃ m ∈ N V m (
H ⋆ = H • F • (Γ W , C) , δ • = ∞ ⨁ m=0 H m ,
where, for any integer m ⩾ 1, and with the convention H 0 = Im δ -1,0 = {0}, H m is the cohomology group

H m = ker δ m-1,m /Im δ m-2,m-1 .
Then, H ⋆ is the set consisting of functions f on Γ W , viewed as 0-fermions (in the sense of Definition 4.3, on page 24), and, for any integer m ⩾ 1, of the restrictions to V m of (m, N m b + 1)fermions, i.e., the restrictions to (the Cartesian product space) V N m b +1 m of antisymmetric maps on Γ W , with N m b + 1 variables (corresponding to the vertices of V m ), involving the restrictions to V m of the continuous, complex-valued functions f on Γ W -as, naturally, the aforementioned 0-fermions -satisfying the following convergent (and even, absolutely convergent)Taylor-like expansions (with

V ⋆ = ⋃ n∈N V n ), ∀ M ⋆,⋆ ∈ V ⋆ ∶ f M ⋆,⋆ = ∞ k=0 c k f, M ⋆,⋆ ε k (2-D W ) k ε i k k,j,m p k = ∞ k=0 c k f, M ⋆,⋆ ε k (2-D W )+i k k,j,m p k , (
k k ln N b ≲ ε k (D W -1) k 2 π .
Note that since the functions f involved are uniformly continuous on the Weierstrass Curve Γ W ⊃ V ⋆ , and since the set V ⋆ is dense in Γ W , they are uniquely determined by their restriction to V ⋆ , as given by (14). We caution the reader, however, that at this stage of our investigations, we do not know wether f (M ) is given by an expansion analogous to the one in (14), for every M ∈ Γ W , rather than just for all M ∈ V ⋆ .

The convergence (or even, the absolute convergence) of the series

∞ k=0 c k f, M ⋆,⋆ ε k (2-D W )+i k,j,m p k
directly comes from the fact that the coefficients c k (⋆, ⋆) are uniformly bounded (see the inequalities (12) in the proof of Theorem 4.3, on page 31), and that, for any k

∈ N ⋆ , ε k (2-D W )+i k,j,m p k = ε k (2-D W ) k = ε 2-D W k , with 0 < ε < 1 and 2 -D W > 0 .
Finally, for each M ⋆ = M ⋆,m ∈ V ⋆ , as was shown in Proposition 4.5, on page 42 (of Section 4.4), the coefficients c k (⋆, ⋆) (for any k ∈ N) are the residues at the possible Complex Dimensions

-k (2 -D W ) + i k,j,m p ) of
), z -z ′ 2-D W ≲ f (z) -f (z ′ ) ≲ z -z ′ 2-D W , (♦) (15)
along with the geometric condition

arg (f (z)) -arg f (z ′ ) ≲ |z -z ′ | . (♠) (16) Now, given a point M ⋆,⋆ ∈ V ⋆ = ⋃ n∈N V n , there exists a (smallest) nonnegative integer m 0 such that M ⋆,⋆ = M j m 0 ,m 0 ∈ V m 0 , with 0 ⩽ j m 0 ⩽ #V m 0 -1.
Then, since the sequence of sets of vertices (V m ) m ∈ N is increasing (see part i. of Property 2.4, on page 8), we must have that

∀ m ⩾ m 0 ∶ M ⋆,⋆ ∈ V m . In fact, this amounts, for each m ⩾ m 0 , to M ⋆,⋆ = M j m ,m ∈ V m , with 0 ⩽ j m ⩽ #V m -1.
For the sake of simplicity, we will write M ⋆,m , m ∈ N, for such points M j m ,m , with 0 ⩽ j m ⩽ #V m -1.

In conjunction with the series expansion (10) obtained in part ii. of Theorem 4.3, on page 31, we have that

f M ⋆,⋆ = lim m→∞ f M ⋆,m = lim m→∞ m k=0 c k f, M ⋆,m ε k (2-D W ) k ε i k k,j,m p k = ∞ k=0 c k f, M ⋆,⋆ ε k (2-D W )+i k,j,m p k ,
where, for each integer k ⩾ 0, c k f, M ⋆,⋆ denotes a complex coefficient which depends on the function f and on the vertex M ⋆,⋆ involved, and where, for k ⩾ 0, k,j,m denotes an integer such that

k,j,m ln ε k k ln N b ≲ ε k (D W -1) k 2 π .
In our present setting, where we deal with an inductive limit, this immediately yields the desired result.

Corollary 4.12 ((of Theorem 4.11) Extension of the Fractal Expansions to Smaller Infinitesimals).

Upon a first reading, the explicit formulas (14) of Theorem 4.11, on page 50, are satisfied if for each integer k ⩾ 0, the number ε k k > 0 is the k th cohomology infinitesimal introduced in Definition 4.6, on page 29. This is a translation of the fact that, for each integer k ⩾ 0, the number ε k > 0 corresponds to the elementary length between two neighboring points of the k th prefractal graph (much like the small increments in classical Taylor formulas).

In order to extend those explicit formulas (14) to a smaller value of ε (corresponding to closer points), since one cannot choose the same value for each scale n ∈ N (all the scales are involved at the same time), we can interpret this infinitesimal as a sequence n n n ∈ N of positive real numbers (necessarily tending to zero) such that its n th component n = ( n ) n is less than or equal to ε n , for all n ⩾ 0.

Indeed, we observe that for any n 0 ∈ N, and any m ∈ N, the set of vertices V m N

∀ x ∈ R ∶ W n 0 (x) = ∞ n=0 λ n cos 2 π N n 0 b n x , is contained in the set of vertices V m (N b ) associated to the value N b , i.e., ∀ x ∈ R ∶ W(x) = ∞ n=0 λ n cos 2 π N n 0 b n x .
So, we simply have to take n 0 large enough in order to obtain the desired result.

Note that, much as in Definition 4.6, and in contrast to the infinitesimal n = ( n ) n n ∈ N itself, the sequence of positive numbers ( n ) n ∈ N need not tend to zero. Remark 4.10. Naturally, an entirely analogous result remains valid for the explicit formulas (or extensions) (9) and (10)obtained, respectively, in part i. and ii. of Theorem 4.3, on page 31. The cohomology infinitesimal (or, equivalently, the elementary length) -which obviously depends on the magnification scale (i.e., the chosen prefractal approximation) -can be seen as a transition scale between the fractal domain and the classical one. In fact, we could say that the system is fractal below this scale, and classical above (for the level of magnification considered). In the limit when the integer m associated with the prefractal approximation tends to infinity, the system is fractal below the cohomological infinitesimal (which is really an infinitesimal, in this case), i.e., at small scales, and is classical beyond, i.e., on a large scale. This is in perfect agreement with what is described by the French physicist Laurent Nottale in [START_REF] Nottale | La relativité dans tous ses états[END_REF] about scale-relativity, and fractal-type objects in nature, pages 306-307: "La plupart des objets fractals qu'on rencontre dans la nature (...) sont également caractérisés géométriquement par l'existence d'une échelle de transition inférieure et/ou supérieure au-delà de laquelle ils redeviennent des objets standards" -"Most fractal objects encountered in nature (...) are also characterized geometrically by the existence of a lower and/or upper transition scale beyond which they become standard objects again". In fact, and as explained again by Nottale, "the transition between fractal (at small relative scales) and non-fractal (at larger scales) behavior is identified with the quantum/classical transition". This latter point appears to us to be of the greatest interest in the present context, and warrants much further exploration in the future.

Recall that in the standard mathematical theory of Complex Dimensions (see, e.g., [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [LR Ž17b], [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF]), a geometric object is said to be fractal if it admits at least one nonreal Complex Dimension (defined as a pole of the associated geometric or fractal zeta function). In our present context, indeed, the Weierstrass Curve Γ W is fractal, since it has infinitely (and countably) many Complex Dimensions; see [START_REF] David | Fractal complex dimensions and cohomology of the Weierstrass curve[END_REF], [START_REF] David | Polyhedral neighborhoods vs tubular neighborhoods: New insights for fractal zeta functions[END_REF]. In particular, it is both principally fractal (since it has infinitely many Complex Dimensions with real part D W , the Minkowski dimension of Γ W ) and is fractal in infinitely (and countably) many values of d (namely, d = D W -k (2 -D W ), with k ∈ N arbitrary).

At the same time, as is also proved in [START_REF] David | Polyhedral neighborhoods vs tubular neighborhoods: New insights for fractal zeta functions[END_REF], for all sufficiently large positive integers m (i.e., for all m ⩾ m 0 , for some m 0 ∈ N), the m th prefractal approximation Γ W m to the Weierstrass Curve Γ W When the resolution (in L. Nottale's words) improves (i.e., when m increases), fractality arises. Then, classical tools, for instance, usual differential operators, cannot be applied. We are therefore looking for their fractal counterpart, which make it possible to express, in a natural way, the evolution equations associated with the morpho-biological and physical phenomena involved, taking into account the underlying fractal nature.

We can also interpret the cohomology H ⋆ of the Weierstrass Curve Γ W (or of W) by means of the following rules: i . Rule 1 -Self-Awareness: The geometry itself belongs to the (total) cohomology; i.e., the Weierstrass function W belongs to the total cohomology space H ⋆ ; see Corollary 4.4, on page 33 above and Remark 4.12, on page 54 just below.

ii . Rule 2 -Optimality: The "restriction" (or rather, the truncation involving the expansion in terms of the allowed Complex Dimensions, up to order m ∈ N ⋆ ) of the geometry (say, the given function f ) to H m (the m th cohomology quotient group) satisfies a certain optimality condition.

In fact, it coincides with the projection of the geometry from the total cohomology space H ⋆ onto H m .

iii . Rule 3 -Naturality: The total cohomology space H ⋆ is the space of objects "of the same nature" (see Definition 4.7, on page 30), whose Complex Dimensions form a subset -or rather, are in direct correspondence with a subset -of the set of Complex Dimensions of the Weierstrass Curve Γ W ; see Theorem 3.1 from [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF] and Remark 3.1, on page 22 (now revisited in the light of Theorems 4.3, on page 31, and 4.11, on page 50).

Note that, contrary to the classical cases of arithmetic or differentiable varieties, for which the decomposition of the total cohomology is indexed by integers, the total (fractal) cohomology H

⋆ is a sum of spaces indexed by the complex dimensions (as is expected in [START_REF] Michel | Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] and [START_REF] Michel | Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes[END_REF], and discussed in detail in [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF], [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF] and [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF]), with an underlying quasiperiodicity property, induced by the estimate (on the imaginary parts of the complex dimensions) (11) in part ii. of Theorem 4.3, on page 31. This quasiperiodic geometric property (reminiscent of, but not identical to, that established in Chapter 3 of [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] for nonlattice self-similar strings) can possibly be connected to the structure of a (generalized) quasicrystal (see [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], Problem 3.22, page 89, and [START_REF] Michel | Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes[END_REF], especially, Chapter 5 and Appendix F), especially in the case not considered in the present paper when N b is not an integer. The aforementioned quasiperiodic geometric property suggests the existence of a natural transfer operator (Frobenius operator), under the action of which the the Weierstrass Curve Γ W remains invariant. More precisely, given , ∈ N, this transfer operator enables us to switch from each m th graph approximation Γ W m to the (m + 1)

th graph approximation Γ W m+1 . This transfer operator also acts on the (total) cohomology. This operator, that we have now identified, will be described in detail in our forthcoming work [START_REF] David | From Weierstrass to Riemann: The Frobenius Pass -When Fractal Flows come into Play with Zeta Functions[END_REF].

Of course, this goes hand in hand with Nottale's perception concerning a characteristic feature of fractality in [START_REF] Nottale | La relativité dans tous ses états[END_REF]: "l'apparition de nouvelles structures quand la résolution s'améliore" -"the appearance of new structures when the resolution improves." Remark 4.12 (About the Functions Defined on the Weierstrass Curve).

Continuous, complex-valued maps f defined on the Weierstrass Curve Γ W can also be viewed as continuous parametrized plane curves γ. As was done in Corollary 4.4, on pag 33, if, for any real number t in [0, 1], γ W (t) = (t, W(t)) , we then have that

γ = f • γ W ; i.e., ∀ t ∈ [0, 1] ∶ γ(t) = f (γ W (t)) = f (t, W(t)) .
Hence, in particular, when we say that the Weierstrass function W belongs to H ⋆ , we really mean that the identity function on the Weierstrass Curve, 1 Γ W , itself belongs to H ⋆ , which is obviously true.

Therefore, because of the properties of the Weierstrass function W, the choice of f = 1 Γ W (or, equivalently, γ = γ W ) is valid, and proves that H ⋆ ≠ {0}, which is nontrivial, as was pointed out in Corollary 4.4, on page 33.

Concluding Comments

Our above results aim at providing a better understanding of Fractal Cohomology. As was mentioned in the introduction, the Weierstrass Curve stands as a test case, since it constitutes a fractal object which is, also, associated to a function. For the sake of consistency, in addition to the original function, we also choose to consider the Complexified Weierstrass function, in the spirit of the fractal explicit formulas obtained in [START_REF] Michel | Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF] and [LR Ž17b]. This function possesses an exact expansion, consisting in a countably infinite sum indexed by the Complex Dimensions. This feature is all the more interesting that the generators of the cohomology groups can be expressed by means of the same kind of fractal expansions; see Proposition 4.7, on page 46.

Going further, those expansions might be interpreted as generalized Taylor expansions, with fractional derivatives of underlying orders the corresponding Complex Dimensions. The conditions of existence of such Taylor-like expansions, as well as the explicit expressions of the corresponding (local) fractional derivatives, will be the object of a future work, [START_REF] David | Fractional Taylor series: Conditions of existence, and explicit formulas[END_REF].

A natural conjecture is that the coefficients c k (f, M ), for k ∈ N, involved in the expansions of Theorem 4.11, on page 50, can not only be interpreted as (local) fractional derivatives of the function f (in the cohomology space of the Weierstrass Curve) in question, evaluated at the point M and expressed using the derivative of the corresponding complex, but also -at least in the case when the function f is the Weierstrass function W -as local curvatures, evaluated at a given point M of the curve, and depending, of course, on the Complex Dimensionk (2 -D W ) + i k,j,m p . Moreover, when f is not necessarily equal to the Weierstrass function W, the coefficients c k (f, M ) could again be interpreted as local curvatures, still depending on the function f , but now seen by means of an ε-microscope, still depending on the Complex Dimensionk (2 -D W ) + i k,j,m p mentioned above. When ε = ε m m ∞ m=0 → 0, we could thus obtain the transition microscope; i.e., the transition of optimal scales determined only by the geometry of the underlying fractal curve (here, the elementary lengths L k ). In the case of a fractal surface, for each scale k ∈ N, we would have two elementary parameters -degrees of freedom -perhaps two radii of curvature in the sense of Gauss, and for a fractal manifold of dimension n ⩾ 3, we would have n such values, associated for example with the underlying local curvature tensor, in the sense of Riemann, again for each scale k ∈ N and evaluated at a given point M of the fractal surface (or, more generally, of the fractal manifold). This is not all. Indeed, in the case of the Weierstrass Curve, our differentials δ and δ (see Section 4.5) enable us to define the associated Laplacian δ δ + δ δ, which needs to be further explored in this context -especially, in conjunction with a corresponding version of Hodge theory. Moreover, naturally, we intend to establish general results -concerning, in particular, both the Complex Dimensions and the cohomology groups -valid for a large class of (typically nowhere differentiable) fractal curves and functions, which will also be the subject of a future work, [START_REF] David | Complex Dimensions of nowhere differentiable Weierstrass-type functions (tentative title)[END_REF]. Building on the present fractal cohomology theory, we also intend to develop an appropriate extension of Morse theory dedicated to fractal sets, where, in addition to the height function, Complex Dimensions should enable us to essentially characterize them, [START_REF] David | Iterated fractal drums ∼ Some new perspectives: Polyhedral measures, atomic decompositions and Morse theory[END_REF].

At a later stage, we envision extending the present theory to fractal surfaces and higher-dimensional fractal manifolds, as was alluded to just above. Moreover, an interesting and challenging problem (see [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF], [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF]) consists in constructing an appropriate fractal homology theory of which the present fractal cohomology theory (or a suitable modification therein) would be, in some sense, the dual theory.

Finally, in the long term, we would like to obtain a fruitful synthesis between the pure analytical and number-theoretic approach (and program) developed in ([LvF00], [START_REF] Michel | Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Michel | Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings[END_REF], [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF], [START_REF] Michel | An overview of complex fractal dimensions: From fractal strings to fractal drums, and back[END_REF], [START_REF] Michel | From Complex Fractal Dimensions and Quantized Number Theory To Fractal Cohomology: A Tale of Oscillations, Unreality and Fractality[END_REF]) and the present geometric approach to fractal cohomology, relying, in part, on [START_REF] Cobler | Towards a fractal cohomology: Spectra of Polya-Hilbert operators, regularized determinants and Riemann zeros[END_REF], [START_REF] David | h-Laplacians on singular sets[END_REF] and [START_REF] David | Weierstrass fractal drums -I -A glimpse of complex dimensions[END_REF], in addition to the aforementioned work. Some progress in this direction has recently been made by the authors in [START_REF] David | From Weierstrass to Riemann: The Frobenius Pass -When Fractal Flows come into Play with Zeta Functions[END_REF].

  , b) denotes an open interval, while, for example, ]a, b] = (a, b] denotes a half-open, half-closed interval. Notation 2 (Wave Inequality Symbol).

  denotes the number of elements in the finite set V m . iii. The prefractal graph Γ W m has exactly (N b -1) N m b edges. iv. The consecutive vertices of the prefractal graph Γ W m are the vertices of N m b simple nonregular polygons P m,k with N b sides. For any strictly positive integer m, the junction point between two consecutive polygons is the point

Figure 1 :

 1 Figure1: The initial polygon P 0 , and the polygons P 1,0 , P 1,1 , P 1,2 , in the case when λ = 1 2 and N b = 3. (See also Figure2, on page 10.)

  The prefractal graph Γ W 0 , in the case when λ = 1 2 and N b = 3. The prefractal graph Γ W 5 , in the case when λ = 1 2 and N b = 3.

Figure 2 :

 2 Figure 2: The prefractal graphs Γ W 0 , Γ W 1 , Γ W 2 , Γ W 3 , Γ W 4 , Γ W 5 , in the case when λ = 1 2 and N b = 3.

Figure 3 :Figure 4 :

 34 Figure 3: Symmetric points with respect to the vertical line x = 1 2 .

  along with -2 and 0 , where p = 2 π ln N b is the oscillatory period of the Weierstrass Curve; see Figure 5, on page 19. Consequently, since it admits nonreal Complex Dimensions, the Weierstrass Curve is fractal, in the sense of the theory of Complex Dimensions developed in [LvF00], [LvF06], [LvF13], [LR Ž17b] and [Lap19].

Figure 5 :

 5 Figure 5: The Complex Dimensions of the Weierstrass Curve. The nonzero Complex Dimensions are periodically distributed (with the same period p = 2 π ln N b , the oscillatory period of Γ W ) along countably many vertical lines, with abscissae D W -k (2 -D W ) and 1 -2 k, where k ∈ N is arbitrary. In addition, 0 and -2 are Complex Dimensions of Γ W .

Definition 4. 2

 2 (p-differential, Complex of Fermions, and Associated Cohomology[START_REF] David | h-Laplacians on singular sets[END_REF]).

  while the above horizontal lines will be replaced by the horizontal lines y = m p, for m ∈ N and some suitable choice of m ∈ Z; see, especially, Theorem 4.3, on page 31 and Theorem 4.11, on page 50, along with Definition 4.8, on page 41, and Proposition 4.5, on page 42 below. Definition 4.3 ((m, p)-Fermion (Generalization of Definition 4.1, given on page 22)).

  given at the end of Definition 4.1, on page 22 (with A = C).

Figure 6 :

 6 Figure 6: In search of invariants, when switching from the initial prefractal graph, to the first one.

Figure 7 :

 7 Figure 7: In search of invariants, when switching from the (m -1) th prefractal graph, to the m th one.

  Definition 4.5 ((m -1, m) and (mk, m)-differentials).

Proposition 4. 2 (

 2 Prefractal Cohomology Groups).

  . . , N b -1}, corresponds, in the argument of f , to a multiplication by 1 N b in the horizontal direction, and by λ = 1 N b 2-D W in the vertical direction,

.

  m b -1, in the following precise sense: i . When the integer N b is odd, for 0 ⩽ k ⩽ N m b -1 2 , the polygons P m,k and P m,N m b -k are symmetrical ones. The line x = 1 2 is also a symmetry axis for the middle polygon P ii . When the integer N b is even, for 0 ⩽ k ⩽ N m b 2 , the polygons P m,k and P m,N m b -k are symmetrical ones. Another interesting point is that, for 0 ⩽ k ⩽ N m b -1, the extreme vertices of P m,k , i.e., the points M (N b -1) k,m and M (N b -1) (k+1),m , are also adjacent points of the prefractal graph of level m -1:

  Our forthcoming results (see Theorems 4.3, on page 31, and Theorem 4.11, on page 50 below) highlight the part played, at each step m ∈ N of the prefractal graph approximation, by a characteristic number called the m th Cohomology Infinitesimal (see Definition 4.6, on page 29 just below), in relation with local scaling zeta functions; see Definition 4.8, on page 41, and Proposition 4.5, on page 42 below. Definition 4.6 (m th Cohomology Infinitesimal and m th Intrinsic Cohomology Infinitesimal [DL23b]).

mm

  -depends on m. In addition, since N b > 1, ε m m satisfies the following asymptotic behavior, ε m m → 0 , as m → ∞, which, naturally, results in the fact that the larger m, the smaller ε m m . It is for this reason that we call ε m m -or rather, the sequence ε m m ∞ m=0 of positive numbers tending to zero as m → ∞, with ε m m = (ε m ) m ,

  m th intrinsic cohomology infinitesimal ε m is asymptotic (when m tends to ∞) to the m th cohomology infinitesimal ε m m . Remark 4.3. Henceforth, it is useful to view V ⋆ = ⋃ n ∈ N V n as being densely embedded into the Weierstrass Curve Γ W ⊃ V ⋆ ; see Remark 4.1, on page 27 above for the relevance of this comment to Theorem 4.3, on page 31 just below, as well as to Theorem 4.11, on page 50 further on. Notation 8 (Argument of a Complex Number).

  Theorem 4.3 (Complex Dimensions Series Expansion and Characterization of the Prefractal Cohomology Groups H m ). Let m ∈ N be arbitrary. Then: i. Within the set Höld (Γ W ) (see part i. of Definition 4.7, on page 30 just above), then, for any integer m ⩾ 1, and with the convention H 0 = Im δ -1,0 = {0}, the cohomology groups H m = ker δ m-1,m /Im δ m-2,m-1 are comprised of the restrictions to V m of (m, N m b + 1)-fermions, i.e., the restrictions to (the Cartesian product space) V N m b +1 m of antisymmetric maps on Γ W , with N m b + 1 variables (corresponding to the vertices of V m ), involving the restrictions to V m of continuous functions f on Γ W , such that, for any vertex M j,m ∈ V m , the following Taylor-like expansion is satisfied,

  -1 are comprised of the restrictions to V m of (m, N m b + 1)-fermions, i.e., the restrictions to V N m b +1 m of antisymmetric maps on Γ W , with N m b + 1 variables (corresponding to the vertices of V m ), involving the restrictions to V m of continuous functions f on Γ W , such that, for any vertex M

  a. the (m, N m b + 1)-fermions, denoted by ϕ m , belonging to H m (or, equivalently, the restrictions to V N m b +1 m of antisymmetric maps, with N m b + 1 variables, belonging to H m ), involve the restrictions to V m of functions f ∈ Höld (Γ W ) such that, for any vertex

  a. the (m, N m b + 1)-fermions, denoted by ϕ m , belonging to H m (or, equivalently, the restrictions to V N m b +1 m of antisymmetric maps, with N m b + 1 variables, belonging to H m ), involve the restrictions to V m of functions f ∈ Höld geom (Γ W ) such that, for any vertex M

  and where, for 0 ⩽ k ⩽ m, k,j,m denotes an integer (in Z) satisfying the estimate (still with ε k denoting the k th component of the k th cohomology infinitesimal in Definition 4.6, on page 29 above)

Definition 4. 8 (

 8 Local Scaling Zeta Function).

  Corollary 4.6 ((of Theorem 4.3, on page 31) Recursive Method for Obtaining the Coefficients of the Fractal Expansions).

  Proposition 4.9 (Complex Differentials). Under the prism of the symmetry with respect to the vertical line x = 1 2 (see Property 2.1, on page 6), it seems natural to consider, for any strictly positive integer m, instead of the differentials introduced in Definition 4.5, on page 26: i. The Left Side Differential, δ z,m-1,m as the restriction of the differential δ m-1,m to the Left Side of the Prefractal graph L m ; i.e., δ Z,m-1,m = δ m-1,m |L m . ii. The Right Side Differential, δ Z,m-1,m , as the restriction of the differential δ m-1,m to the Right Side of the Prefractal graph R m ; i.e., δ Z,m-1,m = δ m-1,m |R m .

  Definition 4.10 (Left and Right Side Cohomology Groups).

  as well as of the images,Im δ Z,m-2,m-1 ≃ Im δ Z,m-2,m-1 ,immediately leads to the Dualityker δ m-1,m |L m /Im δ m-2,m-1 |L m ≃ ker δ m-1,m |R m /Im δ m-2,m-1 |R m and, for 0 ⩽ j ⩽ (N b -1) N m b

  Theorem 4.11 (Fractal Cohomology of the Weierstrass Curve).

  14) where, for each integer k ⩾ 0, the coefficient c k (⋆, ⋆) = c k (f, ⋆) ∈ C is the same as in part ii. of Theorem 4.3, on page 31, the number ε k k > 0 is the k th k th cohomology infinitesimal introduced in Definition 4.6, on page 29, and where k,j,m denotes an integer (in Z) such that k,j,m ln ε

b

  for the Weierstrass function, namely,

  Remark 4.11 (Interpretation).

  is fractal and even, principally fractal. More specifically, it is fractal in dimension d, for precisely the following values of d:d = D W -k (2 -D W ) , with the integer k satisfying 0 ⩽ k ⩽ m.This latter point -as surprising as it may seem, at first sight -therefore corresponds to a transition when the integer m reaches m 0 . Our m th cohomology infinitesimal ε m m (see Definition 4.6, on page 29), is directly connected to the intrinsic (subdivision) scale ε = 1 N b , also introduced in Definition 4.6, since ε

  is of the same nature as the Weierstrass function W, if it satisfies local Hölder and reverse-Hölder properties analogous to those satisfied by the Weierstrass function W; i.e., for any pair of adjacent vertices (M, M

′ ) of respective affixes (z, z ′ ) ∈ C 2 of the prefractal graph Γ W m , with m ∈ N arbitrary (see Remark 2.2, on page 16), Cinf |z ′

  see part ii. of Definition 4.7 above, on page 30), let us consider the Complex (which can be called the Fractal Complex of Γ W ),

  a suitable global scaling zeta function ζ f,M ⋆,m ,gl , also introduced in Proposition 4.5, on page 42. is called the total fractal cohomology group of the Weierstrass Curve Γ W (or else, of the Weierstrass function W). is to be understood in the sense of the inductive limit of the sequence of cohomology groups (H m ) m ∈ N ; namely, for each fermion ϕ ∈ H ⋆ , and each m ∈ N, the restriction ϕ |V m of ϕ to the set of vertices V m belongs to H m ; the restriction ϕ |V m+1 |V m to V m of the restriction ϕ |V m+1 of ϕ to the set of vertices V m+1 (which is itself in H m+1 ), coïncides with the restriction ϕ |V m of ϕ to V m ; i.e., ∀ m ∈ N ∶ ϕ |V m ∈ H m and ϕ |V m+1 |V m = ϕ |V m .

	The group H	⋆ =
	Note that H
	This amounts, for each ϕ ∈ H

∞ ⨁ m=0 H m ⋆ ⋆ , to ϕ = (ϕ m ) m ∈ N ,

where, for each m ∈ N, ϕ m ∈ H m , while, if we denote by π ∶ H m+1 → H m the projection from H m+1 onto H m , we have that π (ϕ m+1 ) ∈ H m coincides with ϕ m .

Proof. In light of Theorem 4.3, on page 31, and Proposition 4.7, on page 46, let us consider a complexvalued function f , which satisfies the same discrete local Hölder and reverse-Hölder property as the Weierstrass function W, i.e., for any pair of adjacent vertices (M, M ′ ), of respective affixes (z, z ′

Acknowledgements

The authors would like to thank the anonymous referee for his judicious remarks about the first version of this paper, which contributed to significantly improve it. We would also like to thank the editor, Professor Camil Muscalu.

The research of M. L. L. was supported by the Burton Jones Endowed Chair in Pure Mathematics, as well as by grants from the U. S. National Science Foundation.

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