Thomas Le Gall

Nathalie Godin

Thomas Monnier

Claudio Fusco

Zeina Hamam

Acoustic Emission modeling from the source to the detected signal: model validation and identification of relevant descriptors

INTRODUCTION

Acoustic Emission (AE) is a method for non-destructive evaluation allowing real time detection of damage mechanisms. As a consequence, it appears as a well-suited technique to retrace damage evolution in metals and composite materials. In fact, microscopic events occurring in the material during damage are at the origin of ultrasonic waves, which propagate up to the surface. Using AE sensors, it is possible to measure the surface vibrations due to the propagated waves. The analysis of the recorded signal gives some information about the source mechanisms, which are responsible for the wave generation. The common practice consists in describing the measured signal through some appropriate parameters (rise time, peak amplitude, centroid frequency, peak frequency, etc.), and regrouping signals into clusters thanks to classification algorithms [START_REF] Sibil | Optimization of acoustic emission data clustering by a genetic algorithm method[END_REF]. The objective is to identify separate clusters corresponding each corresponding to a different source mechanism. However, the wave originating from the source is altered during propagation. Hence, the main limitation of the AE techniques is that changes due to the acquisition chain are not well known. First, it is modified by the propagation medium itself (phase displacement, reflections, dispersion) [START_REF] Aggelis | Effect of plate wave dispersion on the acoustic emission parameters in metals[END_REF] and then, by the recording system. All these transformations make the interpretation of the signals very difficult. In other words, we would like to discern the role of the source from the effect of the transformations due to the propagation medium and the recording system on the measured signal. An experimental study alone cannot provide a satisfying answer. Therefore, the aim of this work is to build a quantitative relationship between the AE sources and the detected signal by unravelling the effect of each stage of the acquisition chain, namely the source, the propagation medium and the detection system. For this purpose, an AE modelling is carried out using the finite element method (FEM) code ABAQUS®. In this case each stage of the acquisition chain is considered individually. In this paper, once we validate our FE Model for wave propagation, we study the effects of the geometry, of the type of sources and finally of the sensor.

MODEL VALIDATION

For obvious reasons, we first perform validation tests of the FE model by comparing experimental and numerical results. In the experimental setup, pencil lead breaks are introduced onto the surface of aluminium specimens having three different shapes shown in Figure 1, the thickness for all specimens being 3.7 mm. The EPL16 specimen has the exact shape of the dumbbell test pieces typically used in the laboratory for material characterisation in general and AE tests in particular. Other shouldered test bars EPL3 and EPL80 were especially designed to mimic the behaviour of a thin rod and a plate respectively, for which analytical solutions of guided propagating modes are easily calculable. The signals are detected by means of a laser vibrometer, which measures the out-of-plane velocity on a surface point. Here we have not used a piezoelectric sensor in order to avoid its effect on the signal waveform, in particular the so-called aperture effect, so that we can isolate the signal recorded from the transformations due to the sensor.

EFFECT OF GEOMETRY

In the three geometries (EPL3, EPL16 and EPL80), the same displacement history (with a chirp signal showed in Figure 3) is introduced on the surface in order to rather equally stimulate all frequencies up to 1.2 MHz. This simulation allows us to identify the different modes of propagation by presenting the 2D Fast Fourier Transform (FFT2D). Those simulated modes are compared to theoretical modes calculated for plates and beams in the case of EPL80 and EPL3 specimen respectively. The results are presented as 2D Fast Fourier Transform shown in Figure 4; this way to present the results allows us to identify the excited modes so that we can compare the simulated results to theoretical calculation. Here the results show that the propagation medium geometry affects the propagated signals. Knowing that, in these geometries, the signal propagates only following specific modes, we deduce that for the large specimen (EPL80) the plate modes are stimulated, while for the thin one (EPL3) the beam modes are stimulated. In the intermediate specimen (EPL16), for which the FFT2D is more complex, and the identification of modes is practically impossible: this geometry cannot be assigned neither to a plate nor to a beam. This study shows us that AE tests done on a classical specimen e.g. EPL16 (Figure 1) in the laboratory cannot replace an industrial case from a geometry perspective.

In the following part, we will study the effect of the type of source on the recorded signal.

EFFECT OF TYPE OF SOURCES

In this part, we are interested in simulating physical sources. Three sources corresponding to three different damage mechanisms have been modelled as buried point sources, according to [START_REF] Scruby | Quantitative acoustic emission techniques[END_REF], in three identical specimens (of equal thickness = 3.7 mm). The sources are located in the gauge section, in the median plan and the centre of the specimen width. They are modelled as dipole forces as shown in Figure 5, representing the expansion, fracture in-plane shear mode and the fracture-opening mode. The sources rise time is 1µs. the particle velocity is calculated on the section of the specimen from the epicentre to its end. The results are presented in the form of FFT2D (wave number in terms of frequency). All studies above are done without considering the piezoelectric AE sensor. This latter, depending on its type, is known by having resonant frequencies and limited bandwidth. So, in the last part we will take into account the sensor effect on the recorded signals.

The last study in this paper concerns the effects of the sensor on the signals. The strategy is to study the AE signals without considering the sensor, then adding the sensor effect and compare results. The results without sensor are based on simulation results. The sensor is taken into account by its transfer function experimentally determined by the reciprocity method [START_REF] Dia | Primary calibration of Acoustic Emission sensors by the method of Reciprocity: theoretical and experimental considerations[END_REF]. This transfer function is added in the post-processing phase in the Fourier domain, following the equation 2: -Sources (Figure 7): the fracture opening mode (S1) and fracture in-plan shear mode (S2) -Geometries (Figure 1) : EPL3, EL16 and EPL80

) ( ) ( ) ( f S f M f S surf sensor   (2)
Figure 7: Sources S1: fracture opening mode and S2: fracture in-plan shear mode applied to the three geometries

In order to have a good EA descriptors analysis, we selected relevant descriptors. The criteria for this selection are:

-the descriptor is constant regardless of the distance from the source -it is independent of specimen geometry -its value depends on the characteristics of the source Based on those criteria, the pertinent descriptors selected are the zero-crossing rate (temporal descriptor) and frequency centroid.

The left panel in Figure 8 left shows the results of classification in terms of the pertinent descriptors without sensor. We can clearly identify two classes. Each class is assigned to a different source. However the right panel of the same figure shows the results of classification after considering the sensor (in this case: a typical commercial PACμ80 sensor from Euro Physical Acoustics). We observe an overlapping of classes, which highlights the transformation of classification results due to the piezoelectric sensor.

CONCLUSIONS

The aim of using the modeling of AE is to understand the transformations of the signal from the source to the acquisition system. Firstly, comparing simulated results to experimental ones validates the propagation medium model. Then the model is used to study the effect of the geometry, the effect of sources and the effect of sensors. Each of these parameters has its own intervention on the signal and this paper showed how its affect the recorded signals. This paper basically treats the case of isotropic homogenous material. It is considered as a first step in the modeling of Acoustic Emission, before addressing the problem of anisotropic materials. The modeling of AE in composite material is ongoing in view of its significance to understand the different type of sources generally observed during the degradation of such more complex materials.

Figure 1 :Figure 2 :

 12 Figure 1: The three Aluminum geometries (EPL3, EPL16, and EPL80) used in all studies of this paperConcerning the FE modelling, loads by the pencil lead breaks are modelled as displacement sources according to[START_REF] Sause | Investigation of pencil-lead breaks as acoustic emission sources[END_REF]. A 3D geometry has been built for each specimen (EPL3, EPL16 and EPL80), where the material is considered homogeneous, isotropic and perfectly elastic (Young modulus = 180GPa, Poisson ratio = 0.49 and density = 2170 kg/m 3 ), the damping is taken into account by Rayleigh parameters. The dumbbells have been meshed with 3D hexagonal C3D8R elements. The average size of one element respects the following criterion:

Figure 3 :

 3 Figure 3: the Chirp source in terms of time and frequency

Figure 4 :

 4 Figure 4: FFT2D for the three geometries excited by the same source simulated on the surface, a) the large specimen (width = 80mm), b) the fine specimen (width=3mm) and c) the medium specimen (width=16mm)

Figure 5 :

 5 Figure 5: The modeled sources: expansion, fracture in-plane shear mode and the fracture-opening modeThe Figure6shows the FFT2D results for the different types of damage. The stimulated modes are not the same in the three cases neither the stimulated frequency range. These results highlight an acoustic signature for each kind of source.

Figure 6 :

 6 Figure 6: FFT2D of the same geometry stimulated by different type of sources; a) expansion, b) fracture in-plan shear mode and c) fracture opening mode

Where

  of the simulated signal retrieved from the surface representing the sensor's sole at the sensor position.In this part, the analysis is based on a traditional study of AE signals i.e. descriptor analysis. To obtain several AE signatures as results, different kind of sources and different geometries are considered.

Figure 8 :

 8 Figure 8: Classification result without considering sensor (left panel) and after considering the µ80 sensor effect (right panel) Finally, this last result presents what we see in our screen as recorded signal when using sensor but the result without sensor presents what kind of improvement in terms of clustering could actually be obtained when deconvolving the sensor effect on the signals.


Hans-Werner Henn

A MINI-COURSE ON MORAVA STABILIZER GROUPS AND THEIR COHOMOLOGY

Introduction

The Morava stabilizer groups play a dominating role in chromatic stable homotopy theory. In fact, for suitable spectra X, for example all finite spectra, the chromatic homotopy type of X at chromatic level n > 0 and a given prime p is largely controlled by the continuous cohomology of a certain p-adic Lie group G n , in stable homotopy theory known under the name of Morava stabilizer group of level n at p, with coefficients in the corresponding Morava module (E n ) * X.

These notes notes are slightly edited notes of a mini-course of 4 lectures delivered at the Vietnam Institute for Advanced Study in Mathematics in August 2013. The aim of the course was to introduce participants to joint work of the author with Goerss, Karamanov, Mahowald and Rezk which uses group cohomology in a crucial way to give a new approach to previous work by Miller, Ravenel, Wilson, and by Shimomura and his collaborators. This new approach has lead to a better understanding of old results as well as to substantial new results.

The notes are structured as follows. In section 2 and section 3 we give a short survey on certain aspects of chromatic stable homotopy theory. In section 2 we recall Bousfield localization and the chromatic set up. In section 3 we discuss the problem of finding finite resolutions of the trivial G n -module Z p and associated resolutions of the K(n)-local sphere and we describe known resolutions. The form of these resolutions depend on cohomological properties of the groups G n and the remaining sections concentrate on those properties. Section 4 contains an essentially self contained discussion of some basic group theoretical properties of these groups. Section 5 discusses the (co)homology of these groups with trivial coefficients; this is self contained except for the discussion of Poincaré duality and the discussion of the case n = 2 and p = 3 which is only outlined. Section 6 concentrates mostly on the continuous cohomology H * (G 1 , (E 1 ) * ) and gives a fairly detailed account on how the short resolutions of the G 1 -module Z p can be used to understand the homotopy of L K(1) S 0 . This homotopy is closely related to the image of the Jhomomorphism studied in the 1960's by Adams, Mahowald, Quillen, Sullivan, Toda and others. Section 6 also contains some brief comments on how the algebraic resolutions surveyed in section 3 can be used to analyze H * (G 2 , (E 2 ) * ), at least for odd primes.

Bousfield localization and the chromatic set up

This section is a very brief introduction to the chromatic set up. More details with more references can be found in the introduction of [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF].

2.1. Bousfield localization. Let E * be a generalized homology theory. Bousfield localization with respect to E * is a functor L E from spectra to spectra together with a natural transformation λ : X → L E X which is terminal among all E *equivalences. L E exists for all homology theories E * [START_REF] Bousfield | The localization of spectra with respect to homology[END_REF]. Bousfield-localization makes precise the idea to ignore spectra which are trivial to the eyes of E * -homology.

Example Let M G be a Moore spectrum for an abelian group G. Then L MZ (p) resp. L MQ are the homotopy theoretic versions of arithmetic localization with respect to Z (p) resp. Q (e.g. homology groups and homotopy groups of a spectrum get localized by these functors).

2.2. Morava K-theories. Fix a prime p. We are interested in the localization functors L K(n) with respect to Morava K-theory K(n). We recall that K(n) is a multiplicative periodic cohomology theory with coefficient ring K(n

) * = F p [v ±1
n ], where v n is of degree 2(p n -1) if n > 0. In case n = 0 the convention is that K(0) = M Q, independant of p. Furthermore K(n) admits a theory of characteristic classes and the associated formal group law Γ n is the Honda formal group law of height n.

The functors L K(n) are elementary "building blocks" of the stable homotopy category of finite p-local complexes in the following sense.

a) The localization functor L K(n) is "simple" in the sense that the category of K(n)-local spectra contains no nontrivial localizing subcategory, i.e. no non-trivial thick subcategory which is closed under arbitrary coproducts [START_REF] Hovey | Morava K-theories and localisation[END_REF] b) There is a tower of localization functors . . . → L n → L n-1 → . . . (with L n = L K(0)∨...∨K(n) ) together with natural transformations id → L n such that X ≃ holim n L n X for every finite p-local spectrum X. Furthermore, for each n and p there is a homotopy pullback diagram (a "chromatic square")

L n X / / L K(n) X L n-1 X / / L n-1 L K(n) X i.e. L n is determined by L K(n) and L n-1 .
The functors L K(n) do not commute with smash products. Therefore the appropriate smash product of K(n)-local spectra X and Y is given by X

∧ K(n) Y := L K(n) (X ∧ Y ).
2.3. L K(n) S 0 as homotopy fixed point spectrum. The functors L K(n) are controlled by cohomological properties of the Morava stabilizer group S n resp. G n where S n is the group of automorphisms of the formal group law Γ n (extended to the finite field F q with q = p n ). The Galois group Gal(F q : F p ) acts on S n and G n is defined as semidirect product G n = S n ⋊ Gal(F q : F p ). This group acts on the Lubin-Tate ring which classifies deformations of Γ n (in the sense of Lubin-Tate). The Lubin-Tate spectrum E n is a complex oriented 2-periodic cohomology theory whose associated formal group law is a universal deformation of Γ n ; its homotopy groups are given as

(E n ) * = π * (E n ) = π 0 (E n )[u ±1 ] with u ∈ π -2 (E) and π 0 (E n ) ∼ = W Fq [[u 1 , . . . , u n-1 ]
], the ring of power series on n -1 generators over the ring of Witt vectors of F q . The group G n acts on deformations and hence on (E n ) * , and by the Hopkins-Miller-Goerss theorem [START_REF] Goerss | Moduli spaces of commutative ring spectra[END_REF] this action can be lifted to E ∞ -ring spectra, i.e. G n acts on E n through E ∞ -maps. By Devinatz-Hopkins [START_REF] Devinatz | Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups[END_REF] the "homotopy fixed point spectrum" E hGn n can be identified with L K(n) S 0 and its Adams-Novikov spectral sequence can be identified with the associated homotopy fixed point spectral sequence

(2.1) E s,t 2 ∼ = H s cts (G n , (E n ) t ) =⇒ π t-s L K(n) S 0 .
Therefore methods of group theory and group cohomology can be used to study the K(n)-local sphere and more generally the K(n)-local category.

Warning: The "homotopy fixed point spectrum" is taken with respect to the action of a profinite group. We will not try to explain how this is done in detail but we insist that in [START_REF] Devinatz | Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups[END_REF] there is a construction such that there is an associated homotopy fixed point spectral sequence with an E 2 -term which is given in terms of continuous group cohomology as in (2.1).

Resolutions of K(n)-local spheres

The case n = 0 is both exceptional and trivial: K(0) = M Q = HQ (with HQ the Eilenberg-MacLane spectrum for the rationals) and L K(0) is rationalization. From now on we will assume n > 1.

3.1. The example n = 1 and p > 2. The case n = 1 is well understood. In this case we have E 1 = KZ p (p-adic complex K-theory). The formal group law Γ is the multiplicative group law given by 1 + (x

+ Γ y) = (1 + x)(1 + y). The endomorphism ring of Γ over F p is isomorphic to Z p : in fact, the element p ∈ Z p corresponds to the endomorphism [p] Γ (x) = (1 + x) p -1 ≡ x p mod (p) and the canonical homomorphism Z → End(Γ), n → [n](x) extends to an continuous isomorphism Z p → End(Γ). Therefore the group G 1 = S 1 can be identified with Z ×
p , the units in the p-adic integers. The group acts on KZ p by Adams operations, and the action on its homotopy π *

(K) = Z p [u ±1 ] is via graded ring automorphisms determined by (λ, u) → λu. If p is odd then Z × p ∼ = C p-1 × Z p ,
and the homotopy fixed points with respect to Z × p can be formed in two steps, first with respect to the cyclic group C p-1 and then with respect to Z p . Taking homotopy fixed points with respect to C p-1 is quite simple; on homotopy groups it amounts to taking invariants with respect to the action of C p-1 . Hence we get

π * (KZ hCp-1 p ) ∼ = Z p [u ±(p-1) ] .
In fact, KZ hCp-1 p is the Adams summand of KZ p . The Adams operation ψ p+1 still acts on KZ hCp-1 p , taking homotopy fixed points with respect to Z p amounts to taking the fibre of ψ p+1 -id and we get a fibration

(3.1) L K(1) S 0 → KZ hCp-1 p ψ p+1 -id -→ KZ hCp-1 p .
We will get back to this in section 6.1.1.

3.2.

The case that p-1 does not divide n. The fibration (3.1) can be considered as an example of a KZ p -resolution in the sense of Miller [START_REF] Miller | On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space[END_REF].

Following Miller we say that a

K(n)-local spectrum I is E n -injective if the canon- ical map I → L K(n) (E n ∧I) splits, i.e
. it has a left inverse in the homotopy category. A sequence of maps X 1 → X 2 → X 3 is said to be E n -exact if the composition of the two maps is nullhomotopic and if [-, I] * transforms X 1 → X 2 → X 3 into an exact sequence of abelian groups for each E n -injective spectrum I. An E n -resolution of a spectrum X is a sequence

I • : * → X → I 0 → I 1 → . . .
such that the sequence is E n -exact and each I s is E n -injective. If there exists an integer k ≥ 0 such that k is minimal with the property that I s is contractible for all s > k then we say that the E n -resolution is of length k.

The spectrum E n is E n -injective because E n is K(n)-local and a ring spectrum.

The following result is in essence due to Morava. Theorem 3.1. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] If n is neither divisible by p -1 nor by p then L K(n) S 0 admits an E n -resolution of length n 2 in which each I s is a summand in a finite wedge of E n 's.

Remarks a) Suppose G = lim α G α is a profinite group and suppose p is a prime. We write Z p [[G]] = lim α Z p [G α ] for the profinitely completed group algebra over Z p . Likewise, for a profinite set S = lim α S α we write Z p [[S]] for lim α Z p [S α ]. The theorem is derived from the existence of a finite projective resolution of length n 2

P • : 0 → P n 2 → . . . → P 0 → Z p → 0 of the trivial G n -module Z p in the category of profinite Z p [[G n ]]-modules.
A more precise form of the theorem is that the E n -resolution "realizes the projective resolution" in the sense that there is an isomorphism of chain complexes

(3.2) Hom cts (P • , (E n ) * ) ∼ = E * (I • )
where here and elesewhere in these notes we adopt the convention that (

E n ) * X for K(n)-local X means π * (E n ∧ K(n) X).
b) The assumption that n is divisible by p (but not by p -1) is not a very serious restriction. There is still a useful variation of this theorem which holds. However, the assumption that n is not divisible by p -1 is quite crucial.

3.3. The example n = 2 and p > 3. In the case n = 2 and p > 3 (even if p = 3) the group G 2 can be decomposed as a product G 2 ∼ = G 1 2 × Z p (cf. section 5.4). The following two results are analogues of results of Ravenel (cf. chapter 6 of [START_REF] Ravenel | Complex cobordism and stable homotopy groups of spheres[END_REF]).

Theorem 3.2. [12]

There is an exact complex of projective

Z p [[G 1 2 ]]-modules 0 → C 3 → C 2 → C 1 → C 0 → Z p → 0 with C 0 = C 3 = Z 3 [[G 2 /F 2(p 2 -1) ]] and C 1 = C 2 = Z 3 [[G 1 2 ]] ⊗ Zp[F 2(p 2 -1) ] λ 1-p where λ 1-p is a certain projective Z p [F 2(p 2 -1) ]-module of Z p -rank 2 and F 2(p 2 -1) is a maximal finite subgroup of order 2(p 2 -1) of G 1
n . Theorem 3.3. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] There exists a fibration

L K(2) S 0 → E hG 1 2 2 -→ E hG 1 2 2 and an E 2 -resolution * → E hG 1 2 2 → X 0 → X 1 → X 2 → X 3 → * with X 0 = X 3 = E hF 2(p 2 -1) 2 and X 1 = X 2 = Σ 2(p-1) E hF 2(p 2 -1) 2 ∨ Σ 2(1-p) E hF 2(p 2 -1) 2 .
3.4. The example n = 1 and p = 2. This case is again well understood. The isomorphism

G 1 = Z × 2 ∼ = C 2 × Z 2
allows, as before, to form the homotopy fixed points in two stages and we obtain the following fibration

(3.3) I • : L K(1) S 0 → KZ hC2 2 ψ 3 -id -→ KZ hC2 2 .
The homotopy fixed points KZ hC2 2 can be identified with 2-adic real K-theory KOZ 2 . Note that this is not an example of Theorem 3.1, in fact a finite length E n -resolution as in Theorem 3.1 cannot exist in this case because G 1 contains an element of order 2, and hence H * cts (G 1 , F 2 ) is nontrivial in arbitrarliy high cohomological degrees. Nevertheless this is a very useful substitute. We will get back to this in section 6.1.2.

3.5. The general case p -1 divides n. The natural question arises whether there are generalizations of the fibre sequence (3.3) for higher n and p such that p -1 divides n. What could they look like? In other words, can we explain the appearance of KZ hC2 2 in (3.3) so that it fits into a more general framework?

A good point of view is provided by group cohomology as follows:

Applying the functor KZ 2 * to (3.3) gives a short exact sequence

0 → KZ 2 * → KZ 2 * (KZ hC2 2 ) → KZ 2 * (KZ hC2 2 ) → 0 in which KZ 2 * (KZ hC2 2
) can be identified with the group of continuous homomorphisms from the permutation module

Z 2 [[Z × 2 /C 2 ]] to (KZ 2 ) * . The fibre sequence
This suggests that we should look for a resolution of the trivial G n -module Z p in terms of permutation modules Z p [[G n /F ]] with F running through finite subgroups (or summands thereof) and try to realize those in the sense of (3.2). In fact, if F is any finite subgroup of G n there is a canonical isomorphism

(3.5) (E n ) * E hF n ∼ = Hom cts (Z p [[G n /F ]], E n * ) .
This leads to the following questions?

Questions: 1) Are there resolutions of finite length and finite type of the trivial

Z p [[G n ]]-module Z p by (direct summands of) permutation modules of the form Z p [[G n /F ]] for finite subgroups F ⊂ G n ?
2) Can these resolutions be realized by resolutions of spectra where the resolving spectra are the corresponding homotopy fixed point spectra with respect to these finite subgroups?

3) If the answers to (a) and (b) are yes, how unique are these resolutions?

Here we call a sequence of spectra

* → X = X -1 → X 0 → X 1 → . . .
a resolution of X if the composite of any two consecutive maps is nullhomotopic and if each of the maps X i → X i+1 , i ≥ 0, can be factored as

X i → C i → X i+1 such that C i-1 → X i → C i is a cofibration for every i ≥ 0 (with C -1 := X -1 ). We say that the resolution is of length n if C n ≃ X n and X i ≃ * if i > n.
Remark The group S n is of finite virtual mod-p cohomological dimension (vcd p ) equal to n 2 , i.e. there is a finite index subgroup whose continuous mod-p cohomology vanishes in degrees > n 2 . In the case of a discrete group G of finite vcd p there is a geometric source for resolutions of the trivial module Z p by permutation modules of the form Z p [G/F ] with F runing through finite subgroups. In fact, they can be obtained as the cellular chains of a contractible finite dimensional G-CW -complex on which G acts with finite stabilizers. Such spaces always exist (if vcd p (G) < ∞) and hence such resolutions always exist. In our case such spaces are not known to exist and we have to manufactor our resolutions by hand.

3.6. The example n = 2 and p = 3. This is the first new case.

Theorem 3.4. [7]

There is an exact complex of

Z 3 [[G 1 2 ]]-modules 0 → C 3 → C 2 → C 1 → C 1 → Z 3 → 0 with C 0 = C 3 = Z 3 [[G 2 /G 24 ]] and C 1 = C 2 = Z 3 [[G 1 2 ]] ⊗ Z3[SD16]
χ where SD 16 is a maximal finite subgroup of G 2 which is isomorphic to the semidihedral group of order 16, χ is a suitable character χ of SD 16 defined over Z 3 , and G 24 is another maximal finite subgroup of order 24 of G 2 . Theorem 3.5. [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] There exists a fibration

L K(2) S 0 → E hG 1 2 2 -→ E hG 1 2 2 and a resolution of E hG 1 2 2 of length 3 * → E hG 1 2 2 → X 0 → X 1 → X 2 → X 3 → * with X 0 = E hG24 2 , X 1 = Σ 8 E hSD16 2 ≃ X 2 = Σ 40 E hSD16 2 and X 3 = Σ 48 E hG24 2 .
Remarks a) The homotopy fixed point spectrum E hSD16 2 is 16-periodic and the suspensions Σ 8 E hSD16 2 and Σ 40 E hSD16 2 are due to the presence of the character χ in the previous theorem. The (E 2 ) * -homology of E hG24 2 is 24-periodic and this resolution realizes the one of the previous theorem in the same sense as before, i.e. there is an isomorphism of complexes

(E 2 ) * (X • ) ∼ = Hom cts (C • , (E 2 ) * ). However, the spectrum E hG24 2
itself is only 72-periodic and the 48-fold suspension appearing with X 3 is a homotopy theoretic subtlety which is not explained by the algebra.

b) The spectrum E hG24 2 is a version of the Hopkins-Miller higher real K-theory spectrum EO 2 . It is equivalent to L K(2) tmf , the K(2)-localization of the spectrum tmf of topological modular forms at p = 3.

There is a second resolution which can be described as follows: we choose an 8-th primitive roof of unity in W F9 . This defines a one-dimensional faithful representation of C 8 over W F9 which we denote it by λ 1 , and its k-th tensor power by λ k . Then the λ k are naturally Z 3 [SD 16 ]-modules and λ 4 splits as λ 4,+ ⊕ λ 4,-. Furthermore λ 4,-is the representation χ of 3.4.

The following results are implicit in [START_REF] Goerss | The homotopy of L 2 V (1) for the prime 3", Categorical decomposition techniques in algebraic topology (Isle of Skye[END_REF].

Theorem 3.6. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] There is an exact complex of

Z 3 [[G 1 2 ]]-modules 0 → Z 3 [[G 1 2 /SD 16 ]] → Z 3 [[G 1 2 ]] ⊗ Z3[SD16] λ 2 → (Z 3 [[G 1 2 ]]⊗ Z3[G24] χ) ⊕ (Z 3 [[G 1 2 ]] ⊗ Z3[SD16] λ 4,-) → Z 3 [[G 1 2 /G 24 ]] → Z 3 → 0 where χ is a suitable nontrivial one-dimensional character of G 24 defined over Z 3 .
Theorem 3.7. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] There exists a resolution of

E hG 1 2 2 of length 3 * → E hG 1 2 2 → E hG24 2 →Σ 36 E hG24 2 ∨ Σ 8 E hSD16 2 → →Σ 4 E hSD16 2 ∨ Σ 12 E hSD16 2 → E hSD16 2 → * .
Remark As in Theorem 3.5 the suspensions are due to the presence of the characters in the previous theorem. In the case of general profinite groups G work of Symonds [START_REF] Symonds | Permutation complexes for profinite groups[END_REF] suggests that such resolutions exist under suitable finiteness assumptions on G. In the case of the stabilizer group [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] provides a more direct approach to their construction. Theorem 3.9. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] For p odd and n = p -1 there is a resolution of L K(n) S 0 of finite length in which all spectra are summands in finite wedges of spectra of the form E hF n and F is a finite subgroup of G n .

3.8. Applications and work in progress. The pioneering work of Shimomura and collaborators on calculating the homotopy groups π * (L K(2) X) for X = S 0 [START_REF] Shimomura | The homotopy groups π * (L 2 S 0 ) at the prime 3[END_REF] resp. the Moore spectrum V (0) [START_REF] Shimomura | The homotopy groups of the L 2 -localized mod 3 Moore spectrum[END_REF] at the prime 3 and of π * (L K(2) S 0 ) for primes p > 3 [START_REF] Shimomura | The homotopy groups π * (L 2 S 0 )[END_REF] have been poorly understood by the community. Therefore an alternative approach (using group cohomology in a systematic way) is useful. Accomplished respectively ongoing projects include the following: 3.8.1. The exact complex of Theorem 3.4 has been made into an efficient calculational tool in the thesis of Nasko Karamanov [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du Pic 2 de Hopkins[END_REF]. This has lead to calculations at p = 3 of π * (L K(2) X) for X = V (1), the cofibre of the Adams self map of V (0) [START_REF] Goerss | The homotopy of L 2 V (1) for the prime 3", Categorical decomposition techniques in algebraic topology (Isle of Skye[END_REF], as well as for V (0) [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF]. The results in [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] refine Shimomura's results of [START_REF] Shimomura | The homotopy groups of the L 2 -localized mod 3 Moore spectrum[END_REF] and correct some errors. The case of S 0 is a joint project with Goerss, Karamanov and Mahowald. Details should appear in the near future.

The main result of [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] together with partial information from [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] have lead to major structural results on the homotopy category of K(2)-local spectra at the prime 3: the rational homotopy of L K(2) S 0 has been calculated and the chromatic splitting conjecture for n = 2 and p = 3 has been established in [START_REF] Goerss | The rational homotopy of the K(2)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2[END_REF], the Picard group of smash-invertible K(2)-local spectra has been calculated in [START_REF] Karamanov | On Hopkins' Picard group Pic 2 at the prime 3[END_REF] and [START_REF] Goerss | On Hopkins' Picard groups for the prime 3 and chromatic level 2[END_REF] and the Brown-Comenetz dual of the sphere has been determined in [START_REF] Goerss | The Brown-Comenetz dual of the K(2)-local sphere at the prime 3[END_REF].

3.8.2. The exact complex of Theorem 3.2 has been turned into an efficient calculational tool in the thesis of O. Lader [START_REF] Lader | Une résolution projective pour le seconde groupe de Morava pour p ≥ 5 et applications[END_REF]. Among other things he has recovered Shimomura's calculation of π * L K(2) V (0) and Hopkins unpublished calculation of the Picard group P ic 2 , both for primes p > 3.

3.8.3. Resolutions for n = p = 2 which resemble those of section 3.5 were announced in [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] although the precise form of X 3 in the analogue of Theorem 3.5 remained unclear at the time. These resolutions have since been constructed in the recent Northwestern theses of Agnès Beaudry and Irina Bobkova. Beaudry has used this to disprove the chromatic splitting conjecture at n = p = 2 [START_REF] Beaudry | The chromatic splitting conjecture at n = p = 2[END_REF]. The resolutions can be expected to lead to further progress in K(2)-local homotopy at the prime 2 similar to the case of the prime 3 mentioned in subsection 3.8.1 above.

In the remaining sections 4-6 of these notes we will explain some of the algebraic aspects of this story in more detail, in particular group theoretical and cohomological properties of G n . The homotopy theoretic aspects will mostly remain in the background.

The Morava stabilizer groups. First properties

There are different ways to discuss these groups. They arise in stable homotopy theory as automorphism groups of certain p-typical formal group laws Γ n defined over F p . For our purposes it seems best to introduce them as follows.

Definition 4.1. Let p be a prime and let O n be the non-commutative algebra over W(F p n ), the ring of Witt vectors for the field F p n , generated by an element S subject to the relations S n = p and Sw = w σ S for each w ∈ W(F p n ) where w σ is the result of applying the lift of Frobenius on w. In other words

(4.1) O n = W(F p n ) S /(S n = p, Sw = w σ S) .
Remarks (on Witt vectors) a) The ring of Witt vectors W(F p n ) is a Z p -algebra which is a complete local ring with maximal ideal (p). It is an integral domain which is free of rank n as Z p -module. As the notation suggests W is a functor, say from the category of finite field extensions of F p to the category of integral domains which are unramified Z p -algebras.

b) Because of functoriality the Frobenius automorphism of F p n lifts to a Z p -algebra automorphism.

c) By Hensel's lemma each root of unity in F × p n lifts uniquely to a root of unity in W(F p n ). d) Each element of w ∈ W(F p n ) can be uniquely written as i≥0 w i p i where all w i ∈ W(F p n ) satisfy w p n i = w i . (Already for n = 1 this is a non-trivial statement).

e) A concrete construction (which, however, does not immediately reveal the functoriality of the construction) can be given as follows. Over F p [X] the polynomial X p n -X can be factored as product of irreducible polynomials whose degrees divide n. For each divisor d of n there is at least one factor p d of degree d. Then

F p n ∼ = F p [X]/(p n ) and W(F p n ) ∼ = Z p [X]/( p n ) where p n is any lift of p n to a poly- nomial p n ∈ Z p [X]. Remarks (on O n ) a)
The left W(F p n )-submodule of O n generated by S is a two sided ideal with quotient O n /(S) ∼ = F p n and O n is complete with respect to the filtration given by the powers of the ideal (S). In fact, O n is a non-commutative complete discrete valuation ring. The valuation

v is normalized such that v(p) = 1, i.e. v(S) = 1 n . b) O n is a free W(F p n )-module of rank n.
A basis is given by the elements 1, S, . . . S n-1 and every element in x ∈ O n can be uniquely written as

x = n-1 i=0 a i S i
with a i ∈ W(F p n ), and thus as

x = ∞ j=0
x j S j with all x j ∈ W(F p n ) satisfying x p n j = x j . In fact, if a i = ∞ j=0 a i,j p j then x i+jn = a i,j . c) Inverting p makes O n into a division algebra D n which is central over Q p and free of rank n 2 as a vector space over Q p . In fact, O n is a domain and if x = j≥k x j S j with x k = 0, then x = S k x ′ and x ′ is invertible in O n . Inverting p also inverts S and thus every nontrivial element admits an inverse.

d) The Galois group Gal(F n p : F p ) of the extension F p ⊂ F p n acts on O n by algebra automorphisms via (σ, n-1 i=0 x i S i ) → n-1 i=0 x σ j S j where as before x σ j is the result of applying the lift of Frobenius to x j . We note that by the relation in b) It can be shown that S n is the group of automorphisms of a suitable formal group law Γ n (associated to the complex oriented cohomology theory given by Morava K-theory K(n)). The group law Γ n is already defined over F p but S n is its automorphism group considered as a formal group law over the field F p n .

4.1. The Morava stabilizer group as a profinite group. The filtration of O n by powers of (S) leads to a very useful filtration of S n . For i = k n with k ∈ N we let

F i := F i S n := {x ∈ S n |x ≡ 1 mod (S in )}
Then we get a decreasing filtration (4.2)

S n = F 0 ⊃ F 1 n ⊃ F 2
n ⊃ by normal subgroups and S n is complete and seperated with respect to this filtration, i.e. the canonical map S n → lim i S n /F i S n is an isomorphism. In particular S n is a profinite group. Furthermore F 1 n S n is the kernel of the reduction homomorphism

S n = O × n → F × p n .
This group is also denoted by S n and is often called the strict Morava stabilizer group. Furthermore for each i = k n > 0 there are canonical isomorphism which define on grS n the structure of a mixed Lie algebra in the sense of Lazard [START_REF] Lazard | Groupes analytiques p-adiques[END_REF]. If we identify the filtration quotients with F p n as above then the Lie bracket and the map P are explicitly given as follows.

(4.3) F k n /F k+1 n → F p n , x = 1 + aS k → a if a ∈ O
P ā =        ā p pni -1 p ni -1 i < (p -1) -1 ā + ā p pni -1 p ni -1 i = (p -1) -1 ā i > (p -1) -1 . Proof. a) Write i = k n , j = l n and choose representatives x = 1 + aS k ∈ F i S n , y = 1 + bS l ∈ F j S n . Then x -1 = 1 -aS k mod S k+1 , y -1 = 1 -bS l mod S l+1 and the formula xyx -1 y -1 = 1 + ((x -1)(y -1) -(y -1)(x -1))x -1 y -1 shows xyx -1 y -1 = 1 + (aS k bS l -bS l aS k ) mod S k+l+1 .
Because O n /(S) ∼ = W(F p n )/(p) we can choose a and b from W(F p n ). Then Sw = w σ S and w σ ≡ w p mod (p) give the stated formula. b) Again we write i = k n and we choose a representative x = 1+aS k with a ∈ W F p n . Consider the expression x p = r p r (aS k ) r . Because p r is divisible by p for 0 < r < p and because S n = p we get

x p ≡ 1 + aS n+k + . . . + (aS k ) p mod S 2k+n .
Furthermore, modulo S kp+1 we get (aS k ) p = aa σ k . . . a σ (p-1)k S pk ≡ aa p k . . . a p (p-1)k S pk ≡ a 1+p k +...+p (p-1)k S pk . Now we only have to determine whether pk is smaller resp. equal resp. larger than n + k. i.e. whether pi is smaller resp. equal resp. larger than 1 + i. These cases are equivalent to i < (p -1) -1 resp. i = (p -1) -1 resp. i > (p -1) -1 and hence we are done.

O n = Z p , S 1 = {x ∈ Z × p | x ≡ 1 mod (p)}. Furthermore, it is well known that Z × p ∼ = F 1 × F × p n p > 2 F 2 × {±1} p = 2
and F 1 is isomorphic to the additive group Z p if p is odd. For p = 2 it is F 2 which is isomorphic to the additive group Z 2 . c) For n = 2 the group S 2 is nonabelian and its structure is complicated. Non-trivial torsion elements can exist only if p = 2 or p = 3.

For p = 3 a non-trivial torsion element must be nontrivial in F 1 2 /F 1 . An easy calculation shows that if ω is a fixed chosen primitive 8-th root of unity in W F9 then the element

(4.4) a = - 1 2 (1 + ωS) satisfies a 3 = 1. (It is clearly in F 1 2
and its image in F 1 2 /F 1 is ω.) For p = 2 there is always, i.e. for each n, the element -1 = 1 -S n which is in F 1 and is a nontrivial element of order 2. If n = 2 there are elements of order 4 which must be nontrivial in F 1 2 /F 1 . d) If n = 4 and p = 2 there is a chance for the existence of elements of order 8 which are nontrivial in F 1 4 /F 2 4 . In fact, such elements exist and they are in the background of the recent solution of the Kervaire invariant one problem by Hill, Hopkins and Ravenel.

On the cohomology of the stabilizer groups with trivial coefficients

The Here is the crucial lemma on commutators. If k+l n is an integer then k + l is divisible by n and hence bp k+l -b = 0.

Proposition 5.2. Let p be an odd prime and n > 1. Then

H 1 (S n , Z p ) ∼ = Z p ⊕ (Z/p) n .
As topological generator of Z p one can choose 1 + cS n = 1 + 2c where c is in W F p n of valuation 0 with tr(c) = 0 and as generators of the n summands Z/p one can choose the elements 1+ω p j S, j = 0, . . . , n-1 of order p where ω is a fixed primitive root of unity of order p n -1.

Proof. The filtration on S n introduced in (4.2) induces one on S n /[S n , S n ] and Lemma 5.1 shows that gr i (S n /[S n , S n ]) is isomorphic to gr i S n ∼ = F q if i = 1 n , isomorphic to the image of tr : F q → F p ) if i is an integer, and zero otherwise. By Proposition 4.3 the induced p-th power map sends gr i (S n /[S n , S n ]) isomorphically to gr i+1 (S n /[S n , S n ]) if i is an integer, and it is clearly trivial on gr 1 n (S n /[S n , S n ]) except possibly if n = p. Furthermore, if n = p we get tr(P (ā)) = tr(ā 1+p+...p (p-1) ) = 0 because ā1+p+...p (p-1) is fixed by Frobenius and thus the trace is p times this element, hence trivial modulo p. Now Lemma 5.1 implies that the induced p-th power map is always trivial on gr 1 n (S n /[S n , S n ]) and this implies the result. Proposition 5.3. Let p = 2 and n > 1. Then

H 1 (S n , Z 2 ) ∼ = Z 2 ⊕ (Z/2) n+1 .
As topological generator of Z 2 one can choose 1+cS 2n = 1+4c and as generators of the n+1 summands Z/2 one can choose the elements 1+cS n , 1+ω 2 k S, k = 1, . . . , n, where c is in W F 2 n of valuation 0 and tr(c) = 0, and ω is a fixed primitive root of unity of order 2 n -1.

Proof. Again the filtration on S n introduced in (4.2) induces one on S n /[S n , S n ] and the previous lemma shows that gr

i (S n /[S n , S n ]) is isomorphic to gr i S n ∼ = F q if i = 1
n isomorphic the image of tr : F q → F p if i is an integer, and zero otherwise. By Proposition 4.3 the induced p-th power map on gr i (S n /[S n , S n ]) sends gr i (S n /[S n , S n ]) isomorphically to gr i+1 (S n /[S n , S n ]) if i is an integer > 1, and it is clearly trivial on gr i except possibly if i = 1 2 or i = 1. The same argument as in the previous proof shows that the induced p-th power map is trivial on gr 1 2 . For i = 1 Proposition 4.3 gives

P (ā) = ā + ā 2 2n-1 2 n -1 = ā + ā2 n +1 = ā + ā2 .
The trace of this element is again trivial and the result follows once again by Lemma 5.1.

Corollary 5.4. Let p be a prime and n > 1. Then

H 1 (S n , Z/p) ∼ = (Z/p) n+1 p > 2 (Z/2) n+2 p = 2 .
5.2. The chomology of S 1 . This case is fairly easy.

Proposition 5.5.

a) If p is odd then

H * (S 1 , Z p ) ∼ = Z p n = 0, 1 0 else . b) If p = 2 then H * (S 1 , Z p ) ∼ = Z 2 n = 0, 1 Z/2 n ≥ 2 . Proof. We have S 1 = Z p if p > 2 and S 1 = Z × 2 ∼ = Z/2 × Z 2 if p = 2.
The result follows therefore as soon as we know that H n (Z p , Z p ) ∼ = Z p if n = 0, 1 and trivial otherwise. (For p = 2 we use the Kuenneth theorem). Now cohomology is calculated from a resolution of the trivial module Z p by projective modules over the completed group ring Z p [[Z p ]]. There is an obvious algebra homomorphism from the polynomial algebra Z p [T ] to the group algebra Z p [Z p ] which sends T to t -e where t is a topological generator of the group Z p . This map extends to a continuous homomorphism from the power series ring Z p [[T ]] to Z p [[Z p ]] which can be checked to be an isomorphism. In fact, this isomorphism is the starting point for Iwasawa theory in number theory (cf. [START_REF] Neukirch | Cohomology of number fields[END_REF]). Now it is obvious that the trivial

Z p [[T ]]-module Z p admits a projective resolution (5.1) 0 → Z p [[T ]] T -→ Z p [[T ]] → Z p
and the result follows.

5.3.

Structural properties of H * (S n , Z/p). Proposition 5.5 and its proof yield immediately the additive structure of H * (S 1 , Z/p) resp. of H * (Z p , Z p ). In fact, there is a cup product structure which is uniquely determined by the additive result.

Proposition 5.6. Let p be any prime. Then

H * (Z p , Z/p) ∼ = Λ Z/p (H 1 (S 1 , Z/p)) ∼ = Λ(e)
with e ∈ H 1 given by the reduction homomorphism Z p → Z/p considered as an element in H 1 (Z p , Z/p) = Hom(Z p , Z/p) and Λ Z/p denotes the exterior algebra over Z/p.

Via the Kuenneth theorem we get the following corollary.

Corollary 5.7. Let p be any prime. Then

H * (Z n p , Z/p) ∼ = Λ Z/p (H 1 (Z n p , Z/p)) ∼ = Λ Z/p (e 1 , . . . , e n )
. with e i ∈ H 1 for i = 1, . . . , i = n, a dual basis of Z n p /(p) and Λ Z/p denoting the exterior algebra over Z/p.

An interesting feature of the stabilizer groups is that although they do not contain abelian subgroups of rank > n (i.e. free Z p submodules of rank > n) they do contain finite index subgroups which look abelian of rank n 2 from the point of view of mod-p cohomology. The following result follows from [START_REF] Lazard | Groupes analytiques p-adiques[END_REF].

Proposition 5.8. a) Let p > 2 and let i = k n ≥ 1. Then H * (F i , Z/p) ∼ = Λ Z/p (H 1 (F i , Z/p)) ∼ = Λ Z/p (e i,j )
where 0 ≤ i, j ≤ n -1 and e i,j is dual to to 1 + ω i S k+j . b) For p = 2 the same result holds if i = k n > 1.

Corollary 5.9. The mod-p cohomology ring of S n is a noetherian algebra over Z/p.

Proof. This follows from Proposition 5.8 by analyzing the spectral sequence of the group extension 1

→ F i → S n → S n /F i → 1.
Definition 5.10. Let p be any prime. A profinite p-group is called a Poincaré duality group of dimension d if

• H s (G, Z/p) is finite dimensional for each s ≥ 0 • H d (G, Z/p) ∼ = Z/p • The cup product H s (G, Z/p)×H d-s (G, Z/p) → H d (G, Z/p) is a nondegenerate bilinear form for each s > 0. Examples a) Z d p is a Poincaré duality group of dimension d. b) F i S n is a Poincaré duality group of dimension n 2 whenever i = k n ≥ 1 if p > 2, and whenever i > 1 if p = 2.
Theorem 5.11. [START_REF] Lazard | Groupes analytiques p-adiques[END_REF] Suppose that G is a profinite p-group without torsion which contains a finite index subgroup which is a Poincaré duality group of dimension n. Then G is itself a Poincaré duality group of dimension n.

5.4.

The reduced norm and a decomposition of S n . If n = 2 and p > 3 then S 2 is torsionfree and hence it is a Poincaré duality group of dimension 2 2 = 4. In fact, we can even reduce to the case of a Poincaré duality group 3 as follows.

In the case of general n and p we consider O n as a left W(F p n )-module of rank n. Multiplying on the right gives a multiplicative homomorphism

O n → M n (W(F p n ))
and hence

S n → GL n (W(F p n ))
. Following this by the determinant gives a homomorphism S n → (W(F p n )) × which is invariant with respect to the natural actions of Gal(F p n : F p ). On the other hand we have noted in the remark preceeding Definition 4.2 that the Galois action on S n is induced by conjugation by the element S in D × n . It follows that the determinant restricted to S n takes its values in the Galois invariant part

Z × p of W(F p 2 ) × .
If n = 2 this can also be seen by an easy calculation as follows. If we choose 1 and S as W(F p 2 ) basis for O 2 then right multiplication defines

O 2 → M 2 (W(F p 2 )), a + bS → a pb σ b a σ
with obviously Galois-invariant determinant.

The resulting homomorphism S n → Z × p is often called the reduced norm. Restricted to the central Z × p in S n the reduced norm is given by the n-th power map. By restricting to the p-Sylow subgroup and assuming that p does not divide n we get a splitting of the sequence

1 → S 1 n → S n → P (Z × p ) → 1 where P (Z × p ) is the p-Sylow subgroup of Z × p .
Proposition 5.12. Suppose p does not divide n. Then the group S n is isomorphic to the direct product of its subgroups S 1 n and P (Z × p ), i.e.

S n ∼ = S 1 n × {g ∈ Z × p | g ≡ 1 mod (p)} p > 2 S 1 n × Z × 2 p = 2 .
5.5. Cohomology in case n = 2 and p > 2.

5.5.1. The case p > 3. In this case we have

S 2 ∼ = S 1 2 × {g ∈ Z × p | g ≡ 1 mod (p)} ∼ = S 1 2 × Z p
The group S 2 is a Poincaré duality group of dimension 4, hence S 1 2 is a Poincaré duality group of dimension 3. Calculating its mod p-cohomology is therefore easy.

By Poincaré duality it is enough to calculate H 1 (S 1 2 , F p ). From Corollary 5.4 we obtain the following result. Theorem 5.13. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] Let p > 3. Then

H * (S 1 2 , Z/p) ∼ =      Z/p * = 0, 3 (Z/p) 2 * = 1, 2 0 * > 3 .
5.5.2. The case p = 3. The cases n = 2 and p = 2, 3 are considerably more complicated. In this case the groups S n do contain p-torsion and they are no longer Poincaré duality groups. In fact, their vcd p is infinite. We will be content to discuss the case p = 3. For p = 3 we still have the decomposition

S 2 ∼ = S 1 2 × {g ∈ Z × p | g ≡ 1 mod (p)} ∼ = S 1
2 × Z p and the problem is again reduced to the case of S 1 2 . Even though the group S 1 2 is not a Poincaré duality group it contains one of index 9, namely the group

F 1 S 1 2 = F 1 S 2 ∩ S 1 2 .
In fact, it even contains one of index 3. In order to see this we consider the formula for the 3-rd power map

P : F 1 2 → F 1 , ā → ā + ā1+3+9
. This shows that if if there is an element g ∈ S 1 2 of order 3 then it has the form g = 1 + aS mod F 1 with ā4 = -1. Thus if we define K to be the kernel of the homomorphism S 1 2 → S 1 2 /F 1 S 1 2 ∼ = F 9 → F 9 /F 3 then K is torsion-free and by Theorem 5.11 it is a Poincaré duality group of dimension 3. Proof. Because K is without torsion Theorem 5.11 implies that it is a Poincaré duality group (of dimension 3). So it is enough to calculate H 1 (K, Z/3) ∼ = H 1 (K, Z/3). For this we consider the filtration on K given by is the element of order 3 of (4.4) and ω is a primitive 8th root of unity in W(F 9 ). This implies the desired result.

F i K := K ∩ F 1 S 2 . It is not hard to check that H 1 (K, Z 3 ) ∼ = Z/9 ⊕ Z/
The cohomology of S 1 2 can now be calculated by using the (non-central) split exact sequence 1 → K → S 1 2 → Z/3 → 1 . The quotient map S 1 2 → Z/3 makes H * (S 1 2 , Z/3) into a module over the polynomial algebra generated by y ∈ H 2 (Z/3, Z/3). It is true (but far from obvious) that this spectral sequence degenerates at E 2 . In fact, it is equivalent to knowing that H * (S 1 2 , Z/3) is a free module over the polynomial algebra Z/3 [y]. Using that we obtain the following result.

Theorem 5.15. [START_REF] Henn | Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups[END_REF] Let p = 3. Then H * (S 1 2 , Z/3) is a free module over Z/3[y] on 8 generators in degrees 0, 1, 1, 2, 2, 3, 3, 4.

The cup product structure is also known. It can be approached as follows. Up to conjugacy there are two subgroups of order 3 in S 1 2 , namely the subgroup a generated by a and the subgroup ωaω -1 . The centralizers of these elements are isomorphic and C S 1 2 a ∼ = a × Z 3 . The cup product structure is detemined by the following result.

Theorem 5.16. [START_REF] Henn | Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups[END_REF] a) The restriction homomorphisms induce a monomorphism

H * (S 1 2 , Z/3) → H * (C S 1 2 a , Z/3) × H * (C S 1 2 ωaω -1 , Z/3) whose target is isomorphic to 2 i=1 Z/3[y i ] ⊗ Λ Z/3 (x i , a i )
where the elements y i are of degree 2 and x i and a i are of degree 1.

b) This map is an isomorphism in degrees > 2. Its image in degree 0 is the diagonal, in degree 1 it is the subspace generated by x 1 and x 2 and in degree 2 the subspace generated by y 1 , y 2 and x 1 a 1 -x 2 a 2 .

c) The image is a free module over Z/3[y 1 + y 2 ] on the following 8 generators: 1,

x 1 , x 2 , y 1 -y 2 , x 1 a 1 -x 2 a 2 , y 1 a 1 , y 2 a 2 , y 1 x 1 a 1 + y 2 x 2 a 2 .
5.5.3. The case p = 2. The case of the prime 2 is even more complicated but it is also understood (cf. [START_REF] Henn | On finite resolutions of K(n)-local spheres[END_REF] and the recent Northwestern theses of Beaudry and Bobkova).

Cohomology with non-trivial coefficients and resolutions

For homotopy theoretic applications we will be interested in calculating cohomology with certain non-trivial coefficients, in particular H * (G n , (E n ) * ). For this we use explicit resolutions of the trivial module. In this section we will discuss the classical case n = 1 in fair detail and briefly comment on the case n = 2.

6.1. The case n = 1. In the case n = 1 we have already seen such resolutions for the group S 1 , at least if p > 2. More precisely we have seen in (5.1) seen that there is a free resolution 0

→ P 1 t-e -→ P 0 → Z p → 0 of the trivial Z p [[Z p ]]-module with P 0 = P 1 = Z p [[Z p ]
] and t a topological generator of Z p . In the case of S 1 = G 1 we can use the same resolutions but enriched as resolutions by Z p [[G 1 ]]-modules. In fact because of the product decomposition

G 1 = Z × p ∼ = Z p ×F where F = {±1} if p = 2 resp. F = Z/p × if p > 2, every Z p [[Z p ]]- module resp. every Z p [[Z p homomorphism can be considered as a Z p [[G 1 ]]- module resp. Z p [[G 1 homomorphism via the projection map G 1 → Z p .
Of course, in this case the modules are no longer free modules. However, if p is odd they are still projective and in case p = 2 they are at least permutation modules. Indeed as Z p [[G 1 ]]-modules we always have P 0

= P 1 = Z p [[G 1 /F ]] = Z p [[G 1 ]]⊗ Zp[F ] Z p and the trivial Z p [F ]-module Z p is projective if p > 2 because
the order of F is prime to p in this case. But even in the case p = 2 this permutation resolution is useful for calculating group cohomology. In fact, it gives rise to a long exact sequence (with

R = Z p [[G 1 ]] and ψ denoting a topological generator of Z × p /F ∼ = Z p ) . . . → Ext s R (Z 2 , M ) → Ext s R (P 0 , M ) ψ-id -→ Ext s R (P 1 , M ) → Ext s+1 R (Z 2 , M ) → . . .
which can be identified by definition of H * and by using Shapiro's Lemma with

(6.1) . . . → H s (G 1 , M ) → H s (F, M ) ψ-id -→ H s (F, M ) → H s+1 (G 1 , M ) → . . . . 6
= (E 1 ) * = Z p [u ±1 ] with |u| = -2.
The action of G 1 = Z × p on this graded algebra is by algebra homomorphisms and is thus specified by the action on the polynomial generator u. It is the tautological action (g, u) → g.u. Then we get

H * (F, Z p [u ±1 ]) = Z p [u ±(p-1) ] s = 0 0 s = 0 .
For ψ we can take the element p + 1 ∈ Z × p . Then (ψ -id) * (u t(p-1) ) = ((p + 1) t(p-1) -1)u t(p-1) = cp νp(t)+1 u t(p-1)

where ν p (t) is the p-adic valuation of the integer t and c is a unit modulo p. This proves the following result.

Theorem 6.1. Let p be an odd prime. Then

H s (S 1 , (E 1 ) t ) =      Z p t = 0, s = 0, 1 Z/p νp(t ′ )+1 t = 2(p -1)t ′ , s = 1 0 else .
Then the homotopy fixed point spectral sequence (2.1)

E s,t 2 = H s (S 1 , (E 1 ) t ) =⇒ π t-s (L K(1) S 0 )
collapses by sparseness, and we get the following result which is essentially equivalent to Ravenel's calculation of π * L 1 S 0 (cf. [START_REF] Ravenel | Localization with respect to certain periodic homology theories[END_REF]).

Theorem 6.2. Let p be an odd prime. Then

π n (L K(1) S 0 ) ∼ = Z p n = 0, -1 Z/p νp(t ′ )+1 n = 2(p -1)t ′ -1 .
6.1.2. The case p = 2. For p = 2 we get (6.2)

H s (C 2 , Z 2 [u ±1 ]) =      Z 2 [u ±2 ] s = 0 Z/2[u ±2 ]{y s ′ } s = 2s ′ Z/2[u ±2 ]{y s ′ x} s = 2s ′ + 1 with y ∈ H 2 (C 2 , Z 2 ), x ∈ H 1 (C 2 , Z 2
{u}). We note that this group cohomology is the E 2 -term for the homotopy fixed point spectral sequence converging to π * KOZ 2 = π * KU hC2 . The bidegree of u, y and x are |u| = (0, -2), |y| = (2, 0) and |x] = (1, -2). The full multiplicative stucture is determined by the relation x 2 = yu 2 . One can thus rewrite this E 2 -term as

(6.3) E * , * 2 = Z/2[u ±2 , η]/(2η) with η = xu -2 .
The notation is chosen so as to agree with usual notation in homotopy theory, i.e. η ∈ E 1,2 2 is a permanent cycle which represents the image of η ∈ π S 1 in π 1 (KOZ 2 ). In the homotopy fixed point spectral sequence there is a single differential. In fact, in the sphere we have η 4 = 0 so η 4 has to be hit by a differential. There is only one way how this can happen, namely via (6.4)

d 3 (u -2 ) = η 3 .
The spectral sequence is multiplicative. Therefore we get

E s,t 4 = Z 2 [u ±4 ]{2u 2 , η, η 2 }/( 2η
) , the spectral sequence degenerates at E 4 and we find the well known homotopy groups of π * (KOZ 2 ) given as

π s (KOZ 2 ) ∼ =      Z 2
s ≡ 0, 4 mod (8) Z/2 s ≡ 1, 2 mod (8) 0 s ≡ 3, 5, 6, 7 mod [START_REF] Goerss | The rational homotopy of the K(2)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2[END_REF] .

Independently of this homotopy theoretic calculation we can use (6.2) and the long exact sequence (6.1) in order to calculate H s (G 1 , (E 1 ) * ). The induced homomorphism in H s is trivial if s > 0 because ψ necessarily acts trivially on H s (C 2 , (E 1 ) t ) if s > 0 (because the group is either trivial or Z/2). For s = 0 we have a similar phenomenon as before, namely we take for ψ the element 3 = 1 + 2 ∈ Z × 2 . Because of (ψ -id) * (u 2t ) = ((1 + 2) 2t -1)u 2t = ((1 + 8) t -1)u 2t = c2 ν2(t)+3 u 2t with c a unit modulo 2 we obtain the following result. Theorem 6.3. Let p = 2. Then ). In cohomological degree s = 0 the E 2 -term is isomorphic to Z 2 concentrated in internal degree t = 0. In bidegrees (1, 4t ′ + 2) it is isomorphic to Z/2 generated by ηu -2t ′ and in bidegrees (1, 4t ′ ) it is isomorphic to Z/(2 ν2(t ′ )+3 ) resp. Z 2 generated by ζu -2t ′ if t ′ = 0 resp. by ζ if t ′ = 0.

H s (S 1 , (E 1 ) t ) ∼ =                Z 2 t = 0, s = 0, 1 Z/(2 ν2(t ′ )+3 ) s = 1, t = 4t ′ = 0 Z/2 s = 1, t = 4t ′ + 2 Z/2 s ≥
To get at the homotopy of L K(1) S 0 we still need to understand the differentials in this spectral sequence. They are determined via naturality and via the geometric boundary theorem [START_REF] Bruner | Algebraic and geometric connecting homomorphisms in the Adams spectral sequence[END_REF] by those for the homotopy fixed point spectral sequence for KOZ 2 , i.e. by (6.4). The d 3 -differential is linear with respect to the permanent cycles η and ζ, and it is determined by

d 3 (ηu -2t ) = η 4 u -2t+2 t ≡ 1 mod (2) 0 t ≡ 0 mod (2)
and d 3 (ζu -2t ) = ζη 3 u -2t+2 t ≡ 1 mod (2) 0 t ≡ 0 mod [START_REF] Bousfield | The localization of spectra with respect to homology[END_REF] .

By sparseness all higher differentials are trivial. In dimension ≡ 1, 3 mod (8) there are still extension problems to be solved. This can be done, for example, by comparing with the calculation for the mod-2 Moore space. In dimension ≡ 1 mod (8) the extensions turn out to be trivial while in dimensions ≡ 3 mod (8) they are non-trivial. The final result reads as follows and is essentially once again equivalent to Ravenel's calculation of π * L 1 S 0 in [START_REF] Ravenel | Localization with respect to certain periodic homology theories[END_REF]. i ≡ 3 mod (8) Z 2 i = -1 Z/(2 ν2(s)+4 ) i = 8s -1, i = -1 0 otherwise .

6.2. Some comments on the case n = 2. As a first step one needs to establish the resolutions of the trivial module described in sections 3.3 and 3.6 and make them into effective calculational tools.

For p > 3 these resolutions are projective minimal resolutions which are constructed from the calculations of H * (S 1 2 , Z/p) discussed in section 5.5. For example, one can construct a resolution of the trivial S 1 2 -module Z p of the form 0 → P 3 → P 2 → P 1 → P 0 → Z p → 0 such that P 0 = P 3 = Z p [[S 1 2 ]] and P 1 = P 2 = Z p [[S 1 2 ]] ⊕2 . As in the case n = 1 this projective resolution can be promoted to one of G 1 2 . However, it is not true that S 1 2 is a quotient of G 1 2 so the details of promoting this resolution to one of G 1 2 are not as straightforward.

Nevertheless the existence of the resolution is fairly formal and knowledge of the existence of such resolutions is already quite useful. However, mere existence is not good enough to carry out actual calculations like that of H * (G 1 2 , (E 2 ) * ). A great deal of work is necessary in order to describe the homomorphism in the resolution explicitly, or at least closely enough such that actual calculations can be carried out. Another problem is that of getting sufficient control of the action of G 2 on (E 2 ) * . At least modulo p all these problems have been resolved in the thesis of O. Lader [START_REF] Lader | Une résolution projective pour le seconde groupe de Morava pour p ≥ 5 et applications[END_REF] (cf. section 3.8.2).

In the case of the prime 3 the resolution has a similar form except that the modules are no longer projective (just as in the case p = 2 and n = 1 before). Nevertheless in the thesis of Karamanov [15] and in [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] this resolution has been made into a very effective computational tool (cf. section 3.8.2).

As noted in section 3.8.3 the case of the prime 2 is currently actively developped.

3. 7 .

 7 Permutation resolutions and realizations. Proposition 3.8. [12] Let p be an odd prime and n = k(p -1) with k ≡ 0 mod p. Then the trivial Z p [[G n ]]-module Z p admits a resolution of finite length in which all modules are finite direct sums of modules which are of the form Z p [[G n /F ]] with F a finite subgroup of G n .

Definition 4 . 2 .

 42 (4.1) this action of Frobenius can be realized by conjugation by S inside D × n . The n-th Morava stabilizer group at p is defined as the group of units in O n . It is denoted S n , i.e. S n = O × n . The extended n-th Morava stabilizer group at p is the semidirect product G n := S n ⋊ Gal(F n p : F p ). Remarks a) Because O n is a complete (non-commutative) discrete valuation ring, an element x ∈ O n is invertible in O n if and only if v(x) = 0.

Proposition 4 . 3 . [ 11 ]

 4311 Let ā ∈ gr i S n , b ∈ gr j S n . With respect to the isomorphism (4.3) the mixed Lie algebra structure maps are given by a) [ā, b] = āb p nibā p nj ∈ gr i+j S n b)

4. 3 .

 3 Torsion in the Morava stabilizer groups. As an immediate consequence of Proposition 4.3 we obtain the following result.

  Corollary 4.4. a) If g ∈ F i has finite order and i > (p -1) -1 then g = 1. b) S n is torsionfree if n is not divisible by p -1. Examples a) In particular, if n = 1 and p > 2 and n = 2 and p > 3 then the groups S n are torsionfree. b) For n = 1 we have

Lemma 5 . 1 .

 51 Let p be any prime and let k and l be integers > 0. a) If k+1 n is not an integer then the commutator map gr k n S n ⊗ gr 1 n S n → gr k+1 n S n is onto. b) If k+1 n is an integer then the image of the commutator map gr k n S n ⊗ gr 1 n S n → gr k+1 n S n is equal to the kernel of the trace tr : F p n → F p c) If k+l n is an integer then the image of the commutator map gr k n S n ⊗ gr l n S n → gr k+l n S n is contained in the kernel of the trace tr : F p n → F p . Proof. a) By Proposition 4.3 the commutator map is given by the formula [ā, b] = āb p kbā pBy taking b = 1 one sees that all elements of the form ā -āp belong to the image. This is an F p -linear subspace of F p n of F p -codimension 1 which is contained in and therefore equal to the kernel of the trace. Furthermore, if k+l n is not an integer, it is enough to exhibit a couple (ā, b) such that tr(ā bp kbā p ) = tr(ā p ( bp k+1 -b) = 0 . Now, if k + 1 is not divisible by n there exists b such that c := bp k+1 -b = 0. Because the trace is a nontrivial linear form and becauseF p n → F p n , ā → āp c is bijective we are done. b) If k+1n is an integer, i.e. k + 1 divisible by n, then bp k+1 -b = 0 for all b and therefore tr(ā bp kbā p ) = tr(ā p ( bp k+1 -b) = 0 .On the other hand we have already seen in the proof of (a) that the kernel of the trace is in the image of the commutator map. c) In general the commutator map gr k n S n ⊗ gr l n S n → gr k+l n S n is given by [ā, b] = āb p kbā p l and hence tr(ā bp kbā p l ) = tr(ā p l ( bp k+l -b)) .

3

 3 generated by b := [a, ω] and c = [a, b] where as before

Theorem 6 . 4 .Z 2 ⊕

 642 Let p = 2. Then π n (L K(1) S 0 ) ∼ =                             

  n and if a denotes the residue class of a in O n /(S) ∼ = F p n . In particular S n /F i is a finite p-group for each i > 0 and S n is a profinite p-group. As S n is also normal in S n , S n is the p-Sylow subgroup of the profinite group S n . Furthermore the exact sequence1 → S n → S n → F × p n → 1 splits, i.e. S n ∼ = S n ⋊ F ×p n is a semidirect product. In fact, the splitting is given by Remark c on Witt vectors above. 4.2. The associated mixed Lie algebra of S n . The associated graded object grS n with respect to the above filtration with gr i S n := F k = k n becomes a graded Lie algebra with Lie bracket [ā, b] induced by the commutator [x, y] := xyx -1 y -1 in S n . Furthermore, if we define a function ϕ from

	n	S n /F k+1 n	S n
	for i { k

n |k = 1, 2, . . .} to itself by ϕ(i) := min{i + 1, pi} then the p-th power map on S n induces maps P : gr i S n -→ gr ϕ(i) S n

  stabilizer groups are examples of p-adic Lie groups. For such groups the category of profinite modules over Z p[[G]] has enough projectives and one can define continous cohomology with coefficients in a profinite Z p [[G]]-module M sim-, N ). In the sequel cohomology resp homology will always be continuous cohomology resp continuous homology and we will simply write is asH * (G, M ) resp. H * (G, M ).5.1. H 1 . The stabilizer group made abelian. The commutator formula in Proposition 4.3 can be used to calculate the abelianization of the groups S n . In this profinite setting it is the quotient S n /[S n , S n ] which identifies with the homology H 1 (S n , Z p ). (Here E denotes the closure of a given subset E ⊂ S n ). Likewise H 1 (S n , Z/p) identifies with the quotient S n / [S n , S n ], S p n .

	ply as H s cts (G, M ) = Ext s Zp[[G]] (Z p , M ). Likewise one can define continuous ho-
	mology with coefficients in a profinite Z p [[G]]-module N simply as H cts s (G, M ) = Tor Zp[[G]] s (Z p

  .1.1. The case p > 2. If p > 2 the groups in the middle of (6.1) are trivial unless s = 0. Now we consider the graded module M

  2, t even 0 else . In cohomological degrees s ≥ 2 the E 2 -term of the homotopy fixed point spectral sequence (2.1) E s,t 2 = H s (S 1 , (E 1 ) t ) =⇒ π t-s (L K(1) S 0 ) agrees with the algebra Z 2 [u ±2 , η, ζ]/(2η, ζ 2 ) with ζ ∈ H 1 (S 1 , Z 2 [u ±1 ] 0


Kazuyoshi Kitazawa
email: kkitazawa@hotmail.co.jp

On a Singular Solution in Higgs Field (III) -Condensates and Representation of Certain f 0 Mesons' Masses *

Keywords: Higgs boson mass, glueball, gluon, f 0 meson, Bethe-Salpeter equation Nomenclature isospinor scalar Higgs field φ : scalar Higgs field φ Br (p, P B ): BS amplitude modified Feynman propagators ν : vacuum expectation value α, ξ : gauge parameter ε λ : infinitesimal Grassmsnn number ε photon ; energy of emitted photon

We have recently discussed the mass and the basic structure of SM Higgs boson (H 0 ) by obtaining asymptotic solution for the equation of motion of nonlinear Klein-Gordon type partial differential equation. In this paper, we will treat with above in mind, -masses of glueball (GB) of ground state and of certain f 0 mesons, ur-Higgs boson (ur-H 0 ) which will consist of a number of GBs and/or f 0 above for respective fullerene structure, a representation of these f 0 mesons' masses by masses of π octet and GB, and transformation of ur-H 0 into H 0 .

Introduction

In preceding paper [START_REF] Kitazawa | On a Singular Solution in Higgs Field[END_REF] the mass and the basic structure of H 0 were discussed by obtaining asymptotic solution for the Euler-Lagrange equation of nonlinear Klein-Gordon type, in Higgs field with newly developed mass triangle method, and also the basic structure of H 0 by referring to a tightly bound virtual top quark-pair (tt) * . Though we saw an intimate relation between H 0 and (tt) * , in our calculation H 0 had a smaller mass than the predicted one by the dynamical strong coupling theory of top quark condensation [START_REF] Nambu | Proceedings of XI Warsaw Symposium on Elementary Particle Physics[END_REF], and also seemed to have a truncated-Octahedron (tr-O) mass structure composed of heavy and light pseudo-scalar mesons' masses of all spin 0. There we considered that the mass deference would come from different phase state of each H 0 . Therefore, to understand this in detail we shall hereafter start with investigating the phase transition of H 0 via a relativistic energy equation with describing a phase transition diagram of H 0 , after review of former result. Then we at first see that the ground state mass of GB is calculated at 502.55 MeV/c 2 which is expected as f 0 (500) meson's mass. The GBs will attract mutually with neighbors among original their components of gluons in different colors, so that they could gradually form cluster. And we show that our computed masses of f 0 (1370), f 0 (1500) and f 0 (1710) are within each f 0 meson's mass from experiment while they will construct respective fullerene structure for ur-H 0 as well as f 0 (500), provided that the mass of ur-H 0 (120.611 GeV/c 2 ) will consist of a number of masses of GB or f 0 in which all (pure) GB-fullerene may have an icosahedral (I h ) rotational symmetry. Finally we propose a representation by which f 0 mesons masses above are reproduced respectively with masses of several light pseudo-scalar mesons such as η, K 0 , K 0 _bar, K ± , π 0 , π ± and GB, under the consideration of those junction networks. Where the mass of f 0 (1500) is described only by the mass of GB. And also ur-H 0 will transform into H 0 under mass invariance through, for instance, γf 0 reaction to η c as its component via radiative decay of J/Ψ. Along with these discussions, a massive gluon propagator for virtual top quark-pair decay is calculated by Bethe-Salpeter equation.

Review -formula of SM Higgs boson mass

SM Higgs boson mass formula

EOM of Higgs field [START_REF] Kitazawa | On a Singular Solution in Higgs Field[END_REF] should have a solution at the point of vacuum expectation value (φ= ν), or φ = 0. When we choose an asymptotic form for s → 0 as,

    3 0 0 ( ) 1 exp , s s avs s s    (1)
where

22 i i s c t x x   .
Then asymptotically. φ(0) ~ 0, φ'(0) ~ 0. And expanding near s → 0, we can take an asymptotic form near singular solution

(φ=0) as, 2 , ( 0) v     (2)
So we have a Higgs mass formula from EOM as

        2 2 2 2 2 (3) 2 2 (Z ) 2 W W g G m                  
Finally we get the value of rest mass for H 0

0 0.023 2 0.022 H 2 2 2 (4) 2 2 M 120.611 GeV 1 cos W W Z W W Z M M M c M M            

Top quark mass formula

Extended EOM of Higgs scalar field [START_REF] Kitazawa | On a Singular Solution in Higgs Field[END_REF] from Euler-Lagrange equation is

        2 2 2 3 2 2 2 2 2 2 1 1 2 4 1 2 3 1 1 0 i i i W Z d u s i i i c t b g W W G Z t M W W GM Z bb v m g m m cc m tt v m dd m uu m ss v                                                          (5)
Then extended Higgs boson mass formula is

        2 2 2 2 ( ) 2 2 2 2 2 2 bsd bsd F i bsd q W W g q q m G m                            2 2 2 2 2 2 ( ) + (Z ) 2 2 2 2 2 2 2 2 2 (6) i cu cu F F t i cu q G q q G G m tt m               
It will be understood that first and second terms in right-hand side of Eq.( 6) are both related to decay of top quark to weak boson and to other quark. And we assume that the probability of m t -decay process obeys binominal distribution of being k-times in n-trials (-particles) with r 0 (0< r 0 <1) as decay-mode parameter,

  0 0 1 1 0 0 ( ) 0 0 ( ) 1 1 0 0 0 ( ) 0 ( )
( ) 0 ( ) (1 )
(1 )

(1 ) (1 ) , / / (1

k n k n k t k n k k n k n k W bsd n k Z cu k n k n k W bsd Z cu t W bsd Z cu C r r m C r r m C r r m C r r r m r m m m r m r                      0 (7)
)

Thus the stationary mass value of top quark is

        2 1 1 4 2 2 2
W b s d Z c t u b M M M M M M M M M                   (8)
which is consistent with CDF/D0's experimental result [4]. Since H 0 is expected to be a composite scalar meson,

t tt H H tt M M M M M M                  (9)
which is little smaller than masses of K ± ,0 mesons, and is smaller than mass of η meson. Therefore it is expected that H 0 is to be a scalar meson after emitting one photon from the virtual top quark pair: (tt)*, * 0

(tt) H .

 

Basic structure of SM Higgs boson mass

Let us consider that H 0 mass is constructed basically by heavy mesons' masses of all spin 0, such as

    0 0 S S C C S S B B B B D D     (11)
It is expected that they will form a polyhedron composed of planes of hexagon in space. Because effective number of the planes turns out to be 4 (four), basic structure of H 0 mass is represented as

       
  0 2 2 . H 2 t bound photon t V M c M c     ( 14
)
because we will treat the situation in next subsection that total mass of the system is zero, we here set its tightly bound (total) energy to equalize to the mass of H 0 which will be produced at second stage from massless vector particle (gluon). Then by substituting Eq.( 14) into Eq.( 13), we have a naïve (approximate) relativistic energy equation for the system as

  2 2 . / Total photon t M c r r       e (15)
So if we take r= 0.4 fm, then e Total =-48.0 GeVfm where we adopt σ = 1.5 GeV / fm [START_REF] Eichten | The Spectrum of Charmonium[END_REF]. Here we consider that there would be the latent heat between the molecular-like state: M =2M t of the dynamical strong coupling theory of top quark condensation and pure liquid state: M =M t / √2 of our calculation. It is expected that the former would not be in tightly bound state. Then we could describe the diagram of phase transition into H 0 as shown in Fig. 3. We will later return to this diagram. It is interesting that the rate of outgoing energy from the system to the space (= +Q) by deficit of mass is fairly large during the condensation:

    2 2 2 1 2 2 0.646 Q t t M c M c      (16)

Bethe-Salpeter equation with Goldstein approximation

Tightly bound fermion-antifermion coupling which exchanges with vector particle by Bethe-Salpeter equation (BS) [START_REF] Goldstein | Properties of the Salpeter-Bethe Two Nucleon Equation[END_REF] has long been investigated. [START_REF] Kummer | Exact Solution of the Bethe-Salpeter Equation for Fermions[END_REF][START_REF] Kugo | The chiral Ward-Takahashi identity in the ladder approximation[END_REF][START_REF] Nakanishi | General Survey of the Theory of the Bethe-Salpeter Equation[END_REF] Firstly, Salpeter and Bethe constructed the relativistic equation for two interacted nucleons. Goldstein studied its solution by ladder approximation and discovered the continuous spectral solution with relevant discrete ones. [START_REF] Goldstein | Properties of the Salpeter-Bethe Two Nucleon Equation[END_REF] There Goldstein argued the lack of physical interpretation for the continuous solution of highly singular behavior at the origin of coordinate space. Later, Kummer; Higashijima and Nishimura; Fukui and Seto; and others discussed the continuous spectral solution in the fermionantifermion or in the spinor-spinor interactions. They excluded it from the reason of each difficulty of interpreting physical meaning, except that Higashijima and Nishimura considered it as a renormalized vertex function of the solution for the homogeneous BS. [START_REF] Kummer | Exact Solution of the Bethe-Salpeter Equation for Fermions[END_REF] Thus we shall hereafter apply BS for tightly bound fermion (top quark)-antifermion (anti-top quark) coupling which exchanges with vector particle, and reconsider the physical meaning of continuous solution. The general form of BS is [START_REF] Nakanishi | General Survey of the Theory of the Bethe-Salpeter Equation[END_REF] Then BS for fermion-antifermion bound state with total four momentum is given explicitly [START_REF] Goldstein | Properties of the Salpeter-Bethe Two Nucleon Equation[END_REF] in the Bjorken-Drell metric, with

x  as 4×4 matrix BS-amplitude of spinor having two legs;

        1 1 4 4 ; , 1 1 , 2 2
,; 2 , qP q P q P q P q q P S S

dq K x x                                    (20)   4 where 1 1 , 2 2 0 . qP P d x T x x x                            (21)
Hence we will have the Goldstein equation for abelian vector gluon model with the ladder approximation, putting 0

P   and       5 ,0 q F q x     , [7]         2 2 2 2 4 . 4 dq q q i qq mq i F F            (22)
After the Wick rotation and then the Fourier transform regarding Eq.( 22), we will see that it has the continuous spectrum solution λ > 0, putting K as modified Bessel function of ν -th order, f(r) = (mr) -1 K ν (mr).

(

) 23
On the other hand, recently Iritani et al calculated gluon propagator's functional forms [START_REF] Iritani | Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function[END_REF] in the Landau gauge in SU(3), which fit their result of lattice QCD (LQCD), one of whose candidates has similar form of Eq.( 23). However they abandoned this form because of the deviation from LQCD for small r . We consider in this case, it should rather be adopted that

    2 2
where

3 4 3 , 1 , 0< 1, g                 (24)
according to the continuous solution of Goldstein for BS and also the behaviors of modified Bessel function of Kν(x). Since the value of Kν(x) is more gradually decreasing along with decreasing of ν, the relevant value of ν will be certainly obtained by comparing to LQCD. We prefer that ν = 0, λ= 1. So this is just the case that Higashijima and Nishimura have interpreted it as a renormalized vertex function [START_REF] Kummer | Exact Solution of the Bethe-Salpeter Equation for Fermions[END_REF]. Then we expect that the glueball would be produced successively by two gluons each of which has a length of and is made by the gluon from the vertex respectively, as shown in Fig. 4.

Therefore the final stage of massive gluon propagator into glueball (  r of Fig. 4) in the Landau gauge in SU(3) which is in accord with LQCD is determined from Eq.( 23) as ()

         
K mr mr r m Dr                     (25)
Here the Compton wavelength of glueball in ground state is

c c             (26)
Thus c r   , and the value of m will be actually given in next subsection.

Clustering force between glueballs

Since the number of the kind for colored gluon is 8 (eight), the color valence of glueball should be 4 (four) according to Fig. 5, which is same as carbon and is expected to be self-assembling force between GBs.

In LQCD it is now believed that there might be several scalar mesons of f 0 (1370), f 0 (1500), f 0 (1710) all of which are supposed to have some contents of glueball of ground state. Then we can expect similar structure of the carbon fullerenes for these scalar mesons. After setting N 1 , N 2, N 3 as the fullerene number whose fullerene consists of f 0 mesons above respectively, under the consideration of similar structure to the carbon fullerenes of C 90 , C 80 and C 70 , we put , It should be noted that f 0 (1500) may be also a glueball for each element of GB 80 since 0.333×3 =1. And f 0 (1370), f 0 (1500) and f 0 (1710) will have respective fullerene structure according to their N i -number: 90, 80, 70. The calculated f 0 mesons' masses by Eq.( 27) and Eq.( 28) are well consistent with (or at least within) experimental values as shown in Table 1.

G GB M       

Fullerene structure of ur-SM Higgs Boson

The fullerene structure of GB 240 (≡ ur-H 0 ) of I h symmetry [START_REF] Chancey | The Jahn-Teller effect in C 60 and other icosahedral complexes[END_REF] which consists of 110-hexagons and 12-pentagons, obeying Euler's theorem [START_REF] Goldberg | The isoperimetric problem for polyhedra[END_REF], as shown in Fig. 6 with whose each hexagon of GB from two gluons on it.

As far as carbon fullerenes, C 20 , C 60 , C 80 , C 180 , C 240 have a common point group: I h which is of the icosahedral symmetry. Thus we expect that GB 80 (f 0 (1500)) and GB 240 (ur-H 0 ) also have I h . Therefore, inversely, we could expect that the f 0 meson which has a fullerene structure of I h symmetry, it may consist of pure GB.

A representation of certain f 0 masses

Over three decades ago, Rossi and Veneziano, also Igarashi et al [START_REF] Rossi | A Description of Baryon Dynamics in Dual and Gauge Theories[END_REF] have described gauge invariant junction type baryonium , where j, i and m denote respective number of junction, junction-pair (equivalent to GB) and quarkpair (meson), as shown in Fig. 7. Recently, Csörgó et al [START_REF] Csörgó | Buckyballs and gluon junction networks on the femtometre scale[END_REF] showed gluon junction networks of truncated-polyhedrons. Therefore we shall apply this junction type to be able to describe H 0 with f 0 mesons of the fullerene structure. For H 0 with f 0 (1500) of pure GB's we may construct it as an aggregation of 12 units of 10 0 S and 30 units of S . For H 0 with f 0 (1370) or with f 0 (1710) we must construct it as mixed one from a GB and several certain light pseudo-scalar mesons, because f 0 (1370) and f 0 (1710) have both been interpreted that they are not consisted only of GB's. We propose their mass structure formulae: in Eq.(29-3) is expected from that 3× [fullerene number of H 0 with f 0 (1500) of pure GB] = 240. Because, H 0 with f 0 (1500) might have resonant mass of 3×(mass of GB for ground state) at each 80 vertexes.

             

Transformation (decay) of ur-H 0 into H 0

Reminding Eq.( 10) and Eq.( 12

Concluding remarks

So far, we have shown the condensation (molecular selfassemblage) of H 0 into certain fullerene structure which is to be constructed from a number of glueballs (f 0 (500)) of ground state as well as of heavier f 0 mesons, with phase transition diagram and then Bethe-Salpeter equation. And we proposed a representation by which certain f 0 mesons masses are reproduced respectively with masses of several light pseudoscalar mesons. The relation between recent experiments of LHC and our result for H 0 mass will be discussed in next paper. [15]

3 .e

 3 Structure of ur-Higgs boson -Condensates and a representation of certain f 0 mesons' masses3.1 A naïve relativistic energy equation for the systemBefore we later apply Bethe-Salpeter equation on (tt)*, we shall at first make a naïve relativistic energy equation for (tt)* with considering Cornell potential V of quarkonium in lattice QCD of Wilson loop)[START_REF] Eichten | The Spectrum of Charmonium[END_REF] in Figs.1 and 2.distance from the centre of the string Hence we shall write for a tightly bound energy as[START_REF] Kitazawa | On a Singular Solution in Higgs Field[END_REF]

Fig. 2

 2 Fig. 2 Quenched Wilson action SU(3) potential [5], normalized to V (r0) = 0.

.

Fig. 1 Fig. 3

 13 Fig.1 String Tension σ

Fig. 4

 4 Fig.4 Glueball producing process

Fig. 5 Fig. 6

 56 Fig.5 Color valence of glueball (= 4)

  values of 1340.1 and 1723.0 MeV/c 2 respectively, reproducing the calculated values in Table1which have been obtained from Eq.(27) and Eq.(28), as already we have seen. Here we should remind that H 0 would be constructed by 70×f 0 (1710)-or 90× f 0 (1370)-fullerene. So the factor (70/90) for K 0 or K  in Eq. (29-1) is considered. While the factor (1/3)

Table 1 . Comparison of f0 mesons' mass values

 1

			 	MeV/c	2	 
	f 0 meson	Our calculation	Experiment [4]
	f 0 (500)	502.55	400-1200
	f 0 (1370)	1340.1	1200-1500
	f 0 (1500)	1507.6	1505±6
	f 0 (1710)	1723.0	1720±6