Cohomology Of Finite Groups Research Articles

Curves are algebraic K(π,1)$K(\pi,1)$: Theoretical and practical aspects

AbstractWe prove that any geometrically connected curve over a field is an algebraic , as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of ‐modules, with invertible in , is canonically isomorphic to the cohomology of its corresponding ‐module. To this end, we explicitly construct some Galois coverings of corresponding to Galois coverings of the normalisation of its irreducible components. When is finite or algebraically closed, we precisely describe finite quotients of that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products and in étale cohomology in terms of finite group cohomology.

Generalized periodicity in group cohomology

Given a finite group G, we introduce “encoding pairs,” which are a pair of G-modules M and M ′ equipped with a shifted natural isomorphism between the cohomological functors H • ( G , Hom Z ( M , − ) ) and H • ( G , Hom Z ( M ′ , − ) ) . Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere.

Negligible degree two cohomology of finite groups

For a finite group G, a G-module M and a field F, an element u∈Hd(G,M) is negligible over F if for each field extension L/F and every group homomorphism Gal(Lsep/L)→G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). We determine the group of negligible elements in H2(G,M) for every abelian group M with trivial G-action.

Cohomology of Finite Groups: Interactions and Applications

The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the fifth in a series that has emphasized the interactions of group cohomology with other areas. In spite of the Covid-19 epidemic, this hybrid meeting ran smoothly with about half the participants physically present and the other half participating via Zoom.

Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology

Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B$-infinity algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate-Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in non-negative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate-Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin-Vilkovisky algebra.

Quillen’s stratification for fusion systems

ABSTRACTThe purpose of this note is to provide a reference for the fact that the proof of Quillen’s stratification for finite group cohomology carries over to fusion system. As in the case of Quillen’s stratification for block varieties, the proof is similar to the usual proof for group cohomology except for the use of fusion stable bisets, whose existence is due to Broto et al.

Cohomology of Finite Groups: Interactions and Applications

The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the fourth in a series that has emphasized the interactions of group cohomology with other areas.

Bounding Cohomology for Finite Groups and Frobenius Kernels

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible kG(σ)-modules \(L,L^{\prime }\), with a similar result holding for Gσ. In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.

Extensions for finite Chevalley groups III: Rational and generic cohomology

Let G be a connected reductive algebraic group and B be a Borel subgroup defined over an algebraically closed field of characteristic p>0. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(Fq) are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.

Shifted generic cohomology

AbstractThe idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [E. Cline, B. Parshall, and L. Scott, Cohomology of finite groups of Lie type, I, Publ. Math. Inst. Hautes Études Sci. 45 (1975), 169–191] and [E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143–163]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large $r$, depending only on the root system and $m$, and not on the prime $p$ or the irreducible module $L$, there are isomorphisms ${\mathrm{H} }^{m} (G({p}^{r} ), L)\cong {\mathrm{H} }^{m} (G({p}^{r} ), {L}^{\prime } )\cong { \mathrm{H} }_{\mathrm{gen} }^{m} (G, {L}^{\prime } )\cong {\mathrm{H} }^{m} (G, {L}^{\prime } )$, where the subscript ‘gen’ refers to generic cohomology and ${L}^{\prime } $ is a constructibly determined irreducible ‘shift’ of the (arbitrary) irreducible module $L$ for the finite Chevalley group $G({p}^{r} )$. By a famous theorem of Steinberg, both $L$ and ${L}^{\prime } $ extend to irreducible modules for the ambient algebraic group $G$ with ${p}^{r} $-restricted highest weights. This leads to the notion of a module or weight being ‘shifted $m$-generic’, and thus to the title of this paper. Our approach is based on questions raised by the third author in [D. I. Stewart, The second cohomology of simple ${\mathrm{SL} }_{3} $-modules, Comm. Algebra 40 (2012), 4702–4716], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of ${ \mathrm{Ext} }_{G({p}^{r} )}^{m} $ with irreducible coefficients.

On automorphisms groups of cyclic p-gonal Riemann surfaces

In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p⩾3 is a prime integer and the genus of the surfaces is at least (p−1)2+1. We use Fuchsian and NEC groups, and cohomology of finite groups.

Symmetric continuous cohomology of topological groups

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic (J. Algebra 322 (2009), 1360-1378), where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results.

First cohomology for finite groups of Lie type: Simple modules with small dominant weights

Letkkbe an algebraically closed field of characteristicp>0p > 0, and letGGbe a simple, simply connected algebraic group defined overFp\mathbb {F}_p. Givenr≥1r \geq 1, setq=prq=p^r, and letG(Fq)G(\mathbb {F}_q)be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology groupH1⁡(G(Fq),L(λ))\operatorname {H}^1(G(\mathbb {F}_q),L(\lambda )), whereL(λ)L(\lambda )is the simpleGG-module of highest weightλ\lambda. Under certain very mild conditions onppandqq, we are able to completely describe the first cohomology group whenλ\lambdais less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case whenλ\lambdais a minimal non-zero dominant weight.

Dyer–Lashof operations on Tate cohomology of finite groups

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in negative degree are non-trivial. As an application, we prove that at the prime 2 these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.

Second cohomology for finite groups of Lie type

Let G be a simple, simply-connected algebraic group defined over Fp. Given a power q=pr of p, let G(Fq)⊂G be the subgroup of Fq-rational points. Let L(λ) be the simple rational G-module of highest weight λ. In this paper we establish sufficient criteria for the restriction map in second cohomology H2(G,L(λ))→H2(G(Fq),L(λ)) to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on p and q provided λ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe H2(G(Fq),L(λ)) in terms of rational cohomology for G. We apply our techniques to compute H2(G(Fq),L(λ)) in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

On the product in negative Tate cohomology for finite groups

Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of $BG$: {\[H_{n}(BG,\mathbb{Z})\otimes H_{m}(BG,\mathbb{Z})\rightarrow H_{n+m+1}(BG,\mathbb{Z})\]} for $n,m>0$. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where $G$ is a compact Lie group (with an additional dimension shift).

On the Vanishing Ranges for the Cohomology of Finite Groups of Lie Type

Abstract. The computation of the cohomology for finite groups of Lie type in the describing characteristicis a challenging and difficult problem. In [BNP], the authors constructed an induction functor which takesmodules over the finite group of Lie type, G(F q ), to modules for the ambient algebraic group G. In particularthis functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in[BNP] to determine the first non-trivial cohomology class when the underlying root system is of type A n orC n . In this paper the authors extend these results toward locating the first non-trivial cohomology classesfor the remaining finite groups of Lie type (i.e., the underlying root system is of type B n , C n , D n , E 6 , E 7 ,E 8 , F 4 , and G 2 ) when the prime is larger than the Coxeter number. 1. Introduction1.1. Let Gbe a simple algebraic group scheme over a field kof prime characteristic pwhich is definedand split over the prime field F p , and F : G→ Gdenote the Frobenius map. The fixed points of therth iterate of the Frobenius map, denoted G(F

Cohomology of Finite Groups: Interactions and Applications

The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the third in a series that has emphasized the interactions of group cohomology with other areas.

$p$-local finite group cohomology

We study cohomology for $p$-local finite groups with non-constant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences.

WZW Orientifolds and Finite Group Cohomology

The simplest orientifolds of the WZW models are obtained by gauging a \({\mathbb{Z}_2}\) symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion \({g\mapsto(\zeta g)^{-1}}\), where ζ is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in [31]. More generally, one may gauge orientifold symmetry groups \({\Gamma=\mathbb{Z}_2\times Z}\) that combine the \({\mathbb{Z}_2}\)-action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-Γ cohomology that we solve for all simple simply connected compact Lie groups G and all orientifold groups \({\Gamma=\mathbb{Z} _2\times Z}\).