Dimensions Of Cohomology Groups Research Articles

(Co)homology of $$\Gamma $$-groups and $$\Gamma $$-homological algebra

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of \(\Gamma \)-simplicial groups, particularly of \(\Gamma \)-equivariant homology and cohomology of \(\Gamma \)-groups. This new direction of homological algebra is called \(\Gamma \)-homological algebra. The abstract kernel of non-abelian extensions of groups, its relationship with the obstruction theory, and with the second group cohomology are extended to the case of non-abelian \(\Gamma \)-extensions of \(\Gamma \)-groups. We compute the rational \(\Gamma \)-equivariant (co)homology groups of finite cyclic \(\Gamma \)-groups. The isomorphism of the group of n-fold \(\Gamma \)-equivariant extensions of a \(\Gamma \)-group G by a \(G\,{\rtimes }\, \Gamma \)-module A with the \((n+1)\)th \(\Gamma \)-equivariant group cohomology of G with coefficients in A is proven. We define the \(\Gamma \)-equivariant Hochschild homology as the homology of the \(\Gamma \)-Hochschild complex when the action of the group \(\Gamma \) on the Hochschild complex is induced by its action on the basic ring. Important properties of the \(\Gamma \)-equivariant Hochschild homology related to Kähler differentials, Morita equivalence, and derived functors are established. Group (co)homology and \(\Gamma \)-equivariant group (co)homology of crossed \(\Gamma \)-modules are introduced and investigated by using relevant derived functors. Relations with extensions of crossed \(\Gamma \)-modules, in particular with relative extensions of group epimorphisms in the sense of Loday are established. Universal and \(\Gamma \)-equivariant universal central \(\Gamma \)-extensions of \(\Gamma \)-perfect crossed \(\Gamma \)-modules are constructed and relevant Hopf formulas are obtained. Finally, applications to the algebraic K-theory, the Galois theory of commutative rings, and the cohomological dimension of groups are given.

On Cohomology Groups of Four-Dimensional Nilpotent Associative Algebras

The study of cohomology groups is one of the most intensive and exciting researches that arises from algebraic topology. Particularly, the dimension of cohomology groups is a highly useful invariant which plays a rigorous role in the geometric classification of associative algebras. This work focuses on the applications of low dimensional cohomology groups. In this regards, the cohomology groups of degree zero and degree one of nilpotent associative algebras in dimension four are described in matrix form.

Abelian extensions and crossed modules of Hom-Lie algebras

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Sectional rank and cohomology

Donovan's conjecture implies a bound on the dimensions of cohomology groups in terms of the size of a Sylow p-subgroup and we give a proof of a stronger bound (in terms of sectional p-rank) for dim⁡H1(G,V). We also prove a reduction theorem for higher cohomology.

On One Property of Bounded Complexes of Discrete $$\mathbb{F}_p[\pi]$$ -modules

The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret $$\mathbb{F}_p[\pi]$$ -modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete $$\mathbb{F}_p[\pi]$$ -modules such that all Li are finite Abelian groups. This is an analog for discrete $$\mathbb{F}_p[\pi]$$ -modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.

Projective Plane Bundles Over an Elliptic Curve

AbstractWe calculate the dimension of cohomology groups for the holomorphic tangent bundles of each isomorphism class of the projective plane bundle over an elliptic curve. As an application, we construct the families of projective plane bundles, and prove that the families are effectively parametrized and complete.

Pro-p completions of groups of cohomological dimension 2

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].

On the Gorenstein and cohomological dimension of groups

Here we relate the Gorenstein dimension of a group G G , G c d R G \mathrm {Gcd}_{R}G , over Z \mathbb Z and Q \mathbb Q to the cohomological dimension of G G , c d R G \mathrm {cd}_{R}G , over Z \mathbb Z and Q \mathbb Q , and show that if G G is in L H F {\scriptstyle \bf {LH}}\mathfrak F , a large class of groups defined by Kropholler, then c d Q G = G c d Q G \mathrm {cd}_{\mathbb Q}G=\mathrm {Gcd}_{\mathbb Q}G and if G G is torsion free, then G c d Z G = c d Z G \mathrm {Gcd}_{\mathbb Z}G= \mathrm {cd}_{\mathbb Z}G . We also show that for any group G G , G c d Q G ≤ G c d Z G \mathrm {Gcd}_{\mathbb Q}G\leq \mathrm {Gcd}_{\mathbb Z}G .

A seven-term exact sequence for the cohomology of a group extension

In this paper we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occurring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).

On the equality between homological and cohomological dimension of groups

On the equality between homological and cohomological dimension of groups

𝔬K0-quasi-abelian varieties with complex multiplication

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

Virtual Moduli Cycles and Gromov-Witten Invariants of Noncompact Symplectic Manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

String equation and quantum cohomology for noncompact symplectic manifolds

We use our Gromov–Witten invariant theory in a previous Note for noncompact geometrically bounded symplectic manifolds to get solutions of the generalized string equation and dilation equation and their variants. More solutions of the WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

On dimensions in Bredon homology

We define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We apply this to describe the Bredon-homological dimension of nilpotent groups in terms of the Hirsch-rank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension.

Infinite Dimensional Cohomology Groups and Periodic Solutions of Asymptotically Linear Hamiltonian Systems

In this paper we study the existence of nontrivial 2π-periodic solutions of asymptotically linear Hamiltonian systems. We consider the case of resonance both at zero and at infinity, and we permit time-dependent asymptotic matrices. Our main tools are an infinite dimensional cohomology theory and a corresponding Morse theory recently constructed by W. Kryszewski and the first author. We develop a method to compute the new critical groups.

Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

The leafwise complex of a reducible non-negative polarization with values in the prequantum bundle on a prequantizable symplectic manifold is studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polarization. The Bohr-Sommerfeld set for a reducible non-negative polarization is defined. A factorization theorem is proved for these reducible non-negative polarizations. For compact symplectic manifolds, it is shown that the above complex has finite dimensional cohomology groups, more-over a Lefschetz fixed point theorem and an index theorem for these non-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any reducible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold.

Unitary representations of Brieskorn spheres

In this article, we commence an investigation of the SU(N) representation space of Seifert fibered homology spheres Σ(a1, . . . , an). Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of Σ(a1, . . . , an) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian group. The latter representation spaces can be studied using the general correspondence between representations of Fuchsian groups and the moduli of parabolic bundles given by Mehta and Seshadri. For example, the inductive procedure of Atiyah-Bott-Nitsure determines the cohomology of this moduli space and it follows that the odd dimensional cohomology groups of any component of irreducible SU(N) representations of Σ(a1, . . . , an) vanish. In particular, any irreducible component of the SU(3) representation space of a Brieskorn spheres Σ(p, q, r) is either a point or a two sphere. By repeated application of the inductive procedure, the precise number of points and two spheres in this representation space is determined. Specific results for the Brieskorn spheres with p = 2 are given, where the representation space is a collection of points. In the last section, the SU(N) spectral flow of irreducible representations of Seifert fibered homology spheres is shown to be even. This gives a calculation of the leading term in a gauge-theoretic definition of the generalized Casson invariants.

On a family of one-relator pro-p-groups

The problem of describing one-relator pro-p-groups of cohomological dimension two (along the lines of Lyndon's description of discrete one-relator groups of cohomological dimension two) is still open. The known method of passing by means of a suitable p-filtration to a graded Lie algebra is not applicable to the family of one-relator pro-p-groups presented in this article, since the relators cannot be separated from the p-th powers in the free pro-p-group. In terms of the p-filtrations, the relators come arbitrarily close to a p-th power, yet the groups they define have cohomological dimension two.

Baumslag-Solitar groups and some other groups of cohomological dimension two

Baumslag-Solitar groups and some other groups of cohomological dimension two

Extensions of pro-pgroups of cohomological dimension two

The purpose of this note is to extend Brumer's characterization of pro-pgroups of cohomological dimension two ([1], corollary 5·3) to presentations more general than free ones. The result is then used to rid a proof in [7] of certain field theoretic ingredients. As a by-product we complete a result of Tsvetkov [6] about group extensions obtained by omitting relations.