Alain Chenciner
email: chenciner@imcce.fr

¸ois Laudenbach
email: francois.laudenbach@univ-nantes.fr

MORSE 2-JET SPACE AND h-PRINCIPLE

Keywords: Mathematics Subject Classification. 58A20, 58E05, 57R45 Morse singularity, 2-jet, h-principle

A section in the 2-jet space of Morse functions is not always homotopic to a holonomic section. We give a necessary condition for being the case and we discuss the sufficiency.

Introduction

Given a submanifold Σ in an r-jet space (of smooth sections of a bundle over a manifold M), it is natural to look at the associated differential relation R(Σ) formed by the (r + 1)-jets transverse to Σ. For j r f being transverse to Σ at x ∈ M is detected by j r+1

x f . This is an open differential relation in the corresponding (r + 1)-jet space. One can ask whether the Gromov h-principle holds: is any section with value in R(Σ) homotopic to a holonomic section of R(Σ)? (We recall that a holonomic section of a (r + 1)-jet space is a section of the form j r+1 f .)

According to M. Gromov, the answer is yes when M is an open manifold and Σ is natural, that is, invariant by a lift of Dif f (M) to the considered jet space (see [START_REF] Gromov | Partial Differential Relations[END_REF] p. 79, [START_REF] Eliashberg | Introduction to the h-principle[END_REF] ch. [START_REF] Maller | Fitted diffeomorphisms of non-simply connected manifolds[END_REF]).

The answer is also yes when the codimension of Σ is higher than the dimension n of M; this case follows easily from Thom's transversality theorem in jet spaces (see [START_REF] Laudenbach | De la tranversalité de Thom au h-principe de Gromov[END_REF]). In the case of jet space of functions and when Σ is natural and codimΣ ≥ n + 1, it also can be seen as a baby case of a theorem of Vassiliev [9].

In this note we are interested in a codimension n case when M is a compact n-dimensional manifold. Let J r (M) denote the space of r-jets of real functions; when the boundary of M is not empty, it is meant that we speak of jets of functions which are locally constant on the boundary. We take Σ ⊂ J 1 (M) the set of critical 1-jets. Then R(Σ) ⊂ J 2 (M) is the open set of 2-jets of Morse functions. We shall analyze the obstructions preventing the h-principle to hold with this differential relation.

Index cocycles

It is more convenient to work with the reduced jet spaces J r (M), quotient of J r (M) by R which acts by translating the value of the jet. It is a vector bundle whose linear structure is induced by that of C ∞ (M). For instance, J 1 (M) is isomorphic to the cotangent space T * M. Let M denote the reduced 2-jets of Morse functions, that is the 2-jets which are transverse to the zero section 0 M of T * M (in the sequel, jet will mean reduced jet). Let π : J 2 (M) → J 1 (M) be the projection and π 0 : J 2 0 (M) → 0 M be its restriction over the zero section of the cotangent space. Since it is formed of critical 2-jets, it is a vector bundle whose fiber is the space of quadratic forms, S 2 (T * x M), x ∈ M. Let M 0 := M ∩ J 2 0 (M); it is a bundle over 0 M whose fiber consists of non-degenerate quadratic forms. Its complement in J 2 0 (M) is denoted D (like discriminant); it is formed of 2-jets which are not transverse to 0 M . When M is connected, M 0 has a connected component M i 0 for each index i ∈ {1, ..., n} of quadratic forms.

2.1. Tranverse orientation. Each M i 0 is a proper submanifold of codimension n in M. Moreover the differential dπ gives rise to an isomorphism of normal fiber bundles

ν(M i 0 , M) ∼ = π * (ν(0 M , T * M))|M i 0 .
Of course, ν(0 M , T * M) is canonically isomorphic to the cotangent bundle τ * M, whose total space is T * M. When M is oriented, so is the bundle τ * M. When M is not orientable, one has a local system of orientations of τ * M. Pulling it back by π yields a local system of orientations of ν(M i 0 , M) (that is, co-orientations of M i 0 ). Let us denote M even 0 (resp. M odd 0 ) the union of the M i 0 's for i even (resp. odd). We endow M even 0 with the above local system of co-orientations. For reasons which clearly appear below, it is more natural to equip M odd 0 with the opposite system of co-orientations. Lemma 2.2. Let s = j 2 f be a holonomic section of M meeting M 0 transversally. Then each intersection point of s(M) with M 0 is positive. The same statement holds when s is a local holonomic section only.

Proof. Let a be such an intersection point in s(M) ∩ M i 0 ; so i is the index of the corresponding critical 2-jet. We can calculate in local coordinates (x, y ′ , y ′′ ), where x = (x 1 , ..., x n ) are local coordinates of M,y ′ = (y ′ 1 , ..., y ′ n ) (resp. y ′′ = (y ′′ jk ) 1≤j≤k≤n ) are the associated coordinates of T *

x M (resp. S 2 T * x M). Since f is holonomic, we have y ′′ jk (a) = ∂y ′ j ∂x k (a). Finally, the sign of det y ′′ (a) (positive if i is even and negative if not) gives the sign of the Jacobian determinant at a of the map x → y ′ (x), that is the sign of the intersection point when M i 0 is co-oriented by the canonical orientation of the y ′ -space. As we have reversed this co-orientation when i is odd, the intersection point is positive whatever the index is.

Proposition 2.3. 1) Each M i 0 defines a degree n cocycle of M with coefficients in the local system Z or of integers twisted by the orientation of M. Let µ i be its cohomology class in

H n (M, Z or ); in particular, if s : M → M is a section, < µ i , [s] > is an integer. 2) When s is homotopic to a holonomic section j 2 f , then < µ i , [s]
> is positive and equals the number c i (f ) of critical points of the Morse function f . In particular the total number |Z| of zeroes of the section π • s (which, by construction, is transverse to the 0-section) satisfies:

|Z| ≥ n i=0 c i (f ) .
Proof. 1) Let σ be a singular n-cycle with twisted coefficients of M. It can be C 0 -approximated by σ ′ , an n-cycle which is transverse to M i 0 . As M i 0 is a proper submanifold, there are finitely many intersections points in σ ′ ∩ M i 0 , each one having a sign with respect to the local system of coefficients. The algebraic sum of these signs defines an integer c(σ ′ ). One easily checks that c(σ ′ ) = 0 if σ ′ is a boundary. As a consequence, if σ ′ 0 and σ ′ 1 are two approximations of σ, as σ ′ 1σ ′ 0 is a boundary, we have c(σ ′ 1 )c(σ ′ 0 ) = 0 which allows us to uniquely define c(σ) as the value of an n-cocycle on σ. Typically, the image of a section carries an n-cycle with twisted coefficients and this algebraic counting applies.

2) Since c defined in 1) is a cocycle, it takes the same value on s and on j 2 f . According to lemma 2.2, it counts +1 for each intersection point in j 2 f ∩ M i 0 , that is, for each index i critical point of f . Corollary 2.4. If s is a section of M which is homotopic to a holonomic section, the integers

m i :=< µ i , [s] > fulfill the Morse inequalities m 0 ≥ β 0 (F ) m 1 -m 0 ≥ β 1 (F ) -β 0 (F ) ... ... m 0 -m 1 + ... + (-1) n m n = β 0 (F ) -β 1 (F )... + (-1) n β n (F ) =: χ(M)
where F is a field of coefficients, β i (F ) = dim F H i (M, F or ) is the i-th Betti number with coeffcients in F or (F twisted by the orientation) and χ(M) is the Euler characteristic (independent of the field F ).

Corollary 2.5. The h-principle does not hold true for the sections of M.

Proof. It is sufficient to construct a section s of M which violates the Morse inequalities, for example a section which does not intersect M 0 0 . Leaving the case of the circle as an exercise, we may assume n > 1. One starts with a section s 1 of T * M tranverse to O M . Each zero of s 1 has a sign (if the local orientation of M is changed, so are both local orientations of s 1 and 0 M the sign of the zero in unchanged). For each zero a, one can construct a homotopy fixing a, with arbitrary small support, which makes s 1 linear in a small neighborhood of a. As GL(n, R) has exactly two connected components, one can even suppose that after the homotopy, s 1 is near a the derivative of a non degenerate quadratic fonction whose index can be chosen arbitrarily provided it is even (resp. odd) if a is a positive (resp. a negative) zero. Finally, one can achieve by homotopy that near each zero a, one has s 1 = df with a a non-degenerate critical point of f of index 2 (resp. 1) if a is a positive (resp. negative) zero.

Near the zeroes s 1 has a canonical lift to M by s 2 = j 2 f . Away from the zeroes, the lift s 2 extends as a lift of s 1 since the fibers of π are contractible over T * M \ 0 M . By construction, we have < µ 0 , [s 2 ] >= 0, violating the first Morse inequality.

Remark 2.6. Denote µ even = µ 0 + µ 2 + ... and µ odd = µ 1 + ... . The following statement holds true: µ even = µ odd if and only if the Euler characteristic vanishes.

Proof. Assume first µ even = µ odd . Proposition 2.3 yields for any holonomic section in M: m even = m odd , that is χ(M) = 0. Conversely, if χ(M) = 0, there exists a non-vanishing 1-form on M and hence, by lifting it to J 2 (M), a section v 0 in M avoiding M 0 . We form

W = {z ∈ J 2 (M) | z = z 0 + tv 0 , z 0 ∈ M 0 , t ≥ 0 or z 0 ∈ D, t > 0}.
It is a proper submanifold in M whose boundary (with orientation twisted coefficients) is M even 0 -M odd 0 . Therefore, every cycle c satisfies < µ even , c >=< µ odd , c >, which implies the wanted equality.

Are Morse inequalities sufficient?

This question is closely related to the problem of minimizing the number of critical points of a Morse function. This problem was solved by S. Smale in dimension higher than 5 for simply connected manifolds, as a consequence of the methods he developped for proving his famous h-cobordism theorem (see [START_REF] Smale | Notes on differentiable dynamical systems[END_REF] or chapter 2 in [START_REF] Franks | Homology and Dynamical Systems[END_REF]). Under the same topological assumptions we can answer our question positively. But there are other cases, discussed later, where the answer is negative. Proposition 3.1. Two sections s, t of M ⊂ J 2 (M) are homotopic as sections of M if and only if their algebraic intersection numbers m i with M i 0 are the same. Proof. According to proposition 2.3 1), the condition is necessary. Let us prove that it is sufficient. Leaving the 1-dimensional case to the reader, we assume dim M ≥ 2. Denote s 1 = π • s. Each zero of s 1 is given an index due to its lifting by s to a point of some

M i 0 . For each index i choose |m i | zeroes of s 1 , a 1 i , . . . , a |m i | i
, among its zeroes of index i; when m i > 0 (resp. m i < 0), we choose the a j i so that the corresponding intersection points of s(M) with M i 0 are positive (resp. negative). When m i = 0, no points are selected. In the same way,

|m i | zeros b 1 i , . . . , b |m i | i
of t 1 are chosen. The intersection signs being the same, one can find a homotopy of t in M, which brings the b j i to coincide with the a j i and makes the two sections s and t coincide in the neighborhood of these points.

The other zeroes of s 1 of index i can be matched into pairs of points {a j+ i , a j- i } of opposite sign. A Whitney type lemma allows us to cancel all these pairs by a suitable homotopy of s in M, reducing to the case when s 1 has no other zeroes than the a j i 's, j = 1, . . . , |m i |. A similar reduction may be assumed for t. Let us finish the proof in this case before stating and proving this lemma.

Both sections s 1 and t 1 of T * M are homotopic (among sections) by a homotopy which is stationary on a neighborhood N(a j i ). Making this homotopy h : M × [0, 1] → T * M transverse to the zero section, the preimage of 0 M consists of arcs {a j i } × [0, 1] and finitely many closed curves γ k . Each of these closed curves can be arbitrarily decorated with an index i. This choice allows us to lift h to J 2 (M) as a homotopy h from s to t; this h is the desired homotopy. More precisely, we proceed as follows for getting h. First h|γ k is lifted to M i 0 by using that the fiber of π : M i 0 → 0 M is connected. The transversality of h to 0 M allows us to extend this lifting to a neighborhood of γ k , making h transverse to M i 0 . Now it is easy to extend h to M × [0, 1], since the fiber of π over any point outside 0 M is contractible.

A Whitney type lemma. Let (b + , b -) be a pair of transverse intersection points of s with M i 0 having opposite sign when they are thought of as zeroes of s 1 in M. Let α be a simple path in M joining them avoiding the other zeroes of s 1 and let N be a neighborhood of α. Then there exists a homotopy S = (s u ) u∈[0,1] of s 0 = s into M, supported in N and cancelling the pair (b + , b -), that is, π • s 1 has no zeroes in N.

Proof. We choose an embedded 2-disk (with corners) ∆ in N × [0, 1[ meeting N × {0} transversally along α. We first construct the homotopy S 1 := π • S of s 1 among the sections of T * M, following the cancellation process of Whitney which we are going to recall. We require S 1 to be transverse to 0 M with (S 1 ) -1 (0 M ) = β, where α ∪ β = ∂∆. Using a trivialization of T * M|N, S 1 |N × [0, 1] reads S 1 (x, u) = (x, g(x, u)). The requirement is that g vanishes transversally along the arc β; it is possible exactly because dim M ≥ 2 and the end points have opposite signs. Let T be a small tubular neighborhood of β; its boundary traces an arc β ′ on ∆, "parallel" to β. Let α ′ be the subarc of α whose end points are those of β ′ . The restriction g|T is required to be a trivialization of T , but this latter may be chosen freely. We choose it so that the loop (g|β ′ ) ∪ (s 1 |α ′ ) be homotopic to 0 in (R n ) * \ {0}; of course, when n > 2 this condition is automatically fulfilled. Now g can be extended to the rest of ∆ as a non-vanishing map. As N × [0, 1] collapses onto N × {0} ∪ ∆ ∪ T , the extension of g can be completed without adding zeroes outside β, yielding the desired homotopy S 1 .

It remains to lift S 1 to M. The lifting is first performed along β with value in M i 0 . Then it is globally extended in the same way as in the above lifting process. Corollary 3.2. Let s be a section of M ⊂ J 2 (M) and m i be its algebraic intersection number with M i 0 . Let f : M → R be a Morse function whose number c i (f ) of critical points of index i satisfies c i (f ) = m i for all i ∈ {0, . . . , n}. Then s and j 2 (f ) are homotopic as sections of M. Corollary 3.3. We assume dim M ≥ 6 and π 1 (M) = 0. Let s be a section of M ⊂ J 2 (M) whose algebraic intersection numbers m i fulfills the Morse inequalities for every field of coefficients. In particular, they are non-negative. Then s is homotopic through sections in M to a holonomic section.

Proof. Under these topological assumptions the following result holds true: For any set of non-negative integers {c 0 , c 1 , • • • , c n } satisfying the Morse inequalities for any field of coefficients, there exists a Morse function on M with c i critical points of index i (see theorem 2.3 in [START_REF] Franks | Homology and Dynamical Systems[END_REF]). So we have a Morse function f : M → R with m i critical points of index i. According to corollary 3.2, s is homotopic in M to j 2 f . 3.4. We end this section by recalling that the Morse inequalities are not sharp for estimating the number of critical points of a Morse function on a non-simply connected closed manifold. Typically when π 1 (M) equals its subgroup of commutators (perfect group), some critical points of index 1 are required for generating the fundamental group, but the Morse inequalities allow c 1 = 0 (see [START_REF] Maller | Fitted diffeomorphisms of non-simply connected manifolds[END_REF] for more details). On the other hand, the only constraint for a section of M with intersection numbers m i is the Euler-Poincaré identity:

m 0 -m 1 + ... = χ(M).
So it is possible to find a section s whose intersection number m i is the minimal rank in degree i of a free complex whose homology is H * (M, Z), that is,

m i = β i + τ i + τ i-1 ,
where β i stands for the rank of the free quotient of H i (M, Z) and τ i denotes the minimal number of generators of its torsion subgroup ([2] p. 15). Such a set of integers satisfies the Morse inequalities but is far from being realizable by a Morse function. Finally this section s is not homotopic in M to a holonomic section.

Failure of the 1-parametric version of the h-principle

We thank Yasha Eliashberg who pointed out to us the failure of the h-principle in the 1parametric version of the problem under consideration.

Here M is assumed to be a product

] be the projection. When M is not 1-connected and dim M ≥ 6, according to Allen Hatcher the so-called pseudoisotopy problem has always a negative answer: there exists f without critical points which is not joinable to f 0 among the Morse functions (see [4]). But j 2 f can be joined to j 2 f 0 by a path γ in M. Indeed, take a generic homotopy γ 1 joining df to df 0 ; then arguing as in the proof of proposition 3.1 it is possible to lift it to M. When M is the n-torus T n , A. Douady showed very simply the stronger fact that the path γ 1 can be taken among the non-singular 1-forms (see appendix to [5]). This γ is not homotopic in M with end points fixed to a path of holonomic sections.


Cine-ASL -LLFAIRGE Myocardial perfusion mapping in rodents

A package of two fully non-invasive and robust ASL MRI methods for quantitative mapping of myocardial perfusion in

  rodents © CRMBM -Aix-Marseille University -Marseille, France contact: frank.kober at univ-amu.fr Package content Acquisition method cineASL for Bruker Paravision 4.0, 5.1, 6.0.1, 360 Features -Method implemented as described in Troalen et al. Magn Reson Med. 70. 2013: 1389-1398 -Allows fast high-resolution myocardial blood flow mapping of the myocardium Acquisition method LLFAIRGE for Bruker Paravision 4.0, 5.1, 6.0.1, 360 Features -Method implemented as described in Kober et al. Magn Reson Med. 51. 2004: 62-67 -Allows high-resolution T1-and myocardial blood flow mapping of the myocardium Features common to both methods -Supports GRAPPA acceleration (PV >= 6.0.1) -Interleaved and non-interleaved Tag/Control acquisition -Convenient setting of geometry and ASL timing IDL Post-processing tool Imdisp running in IDL Virtual Machine Python post-processing tool PymDisp (Beta Version) PymDisp post-processing Features Reading of Bruker native format Reading of DICOM format Two-click MBF map calculation Visualization by cardiac frame Summation of frames ROI analysis Plot of ROI result per cardiac frame ROIs saved to disk Maps saved to disk (Nifti format) One-click saving of animated GIF Software package released for distribution to partners under non-disclosure agreement (NDA)