Antonio Díaz, Albert Ruiz, Antonio Viruel
In this paper we consider classifying spaces of a family of $p$-groups and we prove that mod $p$ cohomology enriched with Bockstein spectral sequences determines their homotopy type among $p$-completed CW-complexes. We end with some applications to group theory.
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Let $G$ be a finite 2-group of maximal nilpotency class, and let $BG$ be its classifying space. We prove that iterated Massey products in $H^*(BG;\F_2)$ do characterize the homotopy type of $BG$ among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class.
We show that a countable group is locally virtually cyclic if and only if its Bredon cohomological dimension for the family of virtually cyclic subgroups is at most one.
PhD thesis concerning cohomological finiteness conditions of infinite discrete groups. Much of the material in this thesis has also appeared in arXiv:1311.7629, arXiv:1310.6262, arXiv:1311.6156, and arXiv:1207.1597.
Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont.
We compare three definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative group cohomologies, giving examples of strict inequality in all cases where it can occur. We prove and make use of Stallings--Swan type results which characterise the groups of equivariant cohomological dimension one. Some of our examples are relevant to Farber's problem which asks for an algebraic characterisation of the topological complexity of discrete groups. In particular, the topological complexity of a group is not in general given by a rela…
We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the F-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group G is not contained in Kropholler's class HF .
We prove that for any discrete group $G$ with finite $\mathfrak{F}$-cohomological dimension, the Gorenstein cohomological dimension equals the $\mathfrak{F}$-cohomological dimension. This is achieved by constructing a long exact sequence of cohomological functors, analogous to that constructed by Avramov and Martsinkovsky, containing the $\mathfrak{F}$-cohomology and complete $\mathfrak{F}$-cohomology. As a corollary we improve upon a theorem of Degrijse concerning subadditivity of the $\mathfrak{F}$-cohomological dimension under group extensions.
Asymptotic and cohomological dimension of surface braid groups and poly-surface groups
本文确定了所有曲面辫群（包括非定向和无限类型曲面相关群）的渐近维数，以及无 torsion 多有限生成曲面群的渐近维数。证明了这两类群的虚拟上同调维数与渐近维数相同，并展示了这些群在 Bieri-Eckmann 意义下是虚拟对偶群。
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean bui…
Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partialΓ=cd_LΓ-1 $$ connecting the cohomological dimension of a group $Γ$ with the cohomological dimension of its boundary $\partialΓ$. In [Be] Bestvina introduced a notion of $\sZ$-structure on a discrete group and noticed that his formula holds true for all torsion free groups with $\sZ$-structure. Bestvina's notion of $\sZ$-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivi…
Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.
A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension $2$ is closed under taking finitely presented (or more generally $FP_2$) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension $2$ with respect to the family of parabolic subgroups. A class of groups where our result applies consists of $C'(1/6)$ small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness propert…
Small Value of cohomological dimension of group homomorphisms
本文刻画了群同态的上同调维数，并证明了群满同态情形下 Stallings-Swan 定理的类比。
For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors.
We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and gi…
Cohomological and quasi-isometric diversity of groups with property $R_\infty$
本文研究具有性质$R_\infty$的群的上同调与准等距多样性，证明了此类群在强意义上非常丰富，存在不可数多个有限生成群具有$R_\infty$且彼此不准等距。
The (co)homological dimension of homomorphism $ϕ:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $ϕ:G\to H$ of a geometrically finite group $G$ the homological dimension of $ϕ$ equals the cohomological dimension, $hd(ϕ)= cd(ϕ)$. THEOREM 2. For every homomorphism $ϕ:G\to H$ of geometrically finite groups $cd(ϕ\timesϕ)=2cd(ϕ)$.
Let $G$ be a group that admits a cocompact classifying space for proper actions $X$. We derive a formula for the Bredon cohomological dimension for proper actions of $G$ in terms of the relative cohomology with compact support of certain pairs of subcomplexes of $X$. We use this formula to compute the Bredon cohomological dimension for proper actions of fundamental groups of non-positively curved simple complexes of finite groups. As an application we show that if a virtually torsion-free group acts properly and chamber transitively on a building, its virtual cohomological dimension coincides …
A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermak's proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the $p$-completed classifying spaces of finite groups. Our main tool is a 1971 result of th…
Cohomological properties of simple complexes of groups
研究群的简单复形的上同调性质，通过Bridson - Haefliger构造的\(K -\)等变同伦型获取关于群\(K\)分类空间的信息，得到了上同调的有限性性质。
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures…
We study cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$ when $\partial M$ is diffeomorphic to a torus or a sphere. In particular, we show that for a $3$-manifold $M$ with torus boundary which is not diffeomorphic to a solid torus, the torus action on the boundary does not extend to a $C^0$-action on $M$.
Arithmeticity for integral cohomological dimension of configuration spaces of manifolds
本文给出具有非平凡余维一上同调群的流形无序构型空间整上同调维数的精确公式，证明其序列的算术性，并应用于有序构型空间得到尖锐下界。
Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (π\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was late…
In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.
A Cohomological Obstruction to the Existence of $ϕ$-Equivariant Maps
研究紧致李群间同态下\(\phi\)-等变映射存在性的障碍，通过阐述过滤群变换障碍，应用于有限循环群到圆的包含及环面间同态，为相关映射存在性研究提供新方法。
Here we prove some special cases of the following conjecture: that the sum of the Betti numbers of a 1-connected elliptic space is greater than the total rank of its homotopy groups. Our main tool is Sullivan's minimal model.
Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G in a binate (hence acyclic) group. In the other direction, there are no nontrivial homomorphisms from binate groups to groups of finite cohomological dimension. Binate groups are shown to be of significance in relation to a number of important K-theoretic isomorphism conjectures.
A family of cohomological complex projective spaces
本文构造了一族在每个奇数复维数上与复射影空间上同调群相同但不同伦等价的复流形，并讨论了偶数维类比。
We show that the vanishing of certain cohomology groups of polyhedral complexes imply upper bounds on Ramsey numbers. Lovasz bounded the chromatic numbers of graphs using Hom complexes. Babson and Kozlov proved Lovasz conjecture and developed a Hom complex theory. We generalize the Hom complexes to Ramsey complexes. The main theorem states that if certain cohomology groups of the Ramsey complex Ram(dDelta_{p^k}, Sigma) are trivial, then the vertices of the simplicial complex Sigma cannot be n-colored such that every color correspond to a face of Sigma. In a corollary, we give an explicit descr…
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abe…
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