Graham Ellis, Roman Mikhailov
We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere.
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The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces.…
In this paper we study the colimit N_2(G) of abelian subgroups of a discrete group G. This group is the fundamental group of a subspace B(2,G) of the classifying space BG. We describe N_2(G) for certain groups, and apply our results to study the homotopy type of the space B(2,G). We give a list of classes of groups for which B(2,G) is not an Eilenberg--Maclane space of type K(π,1).
We construct an analogue of the normaliser decomposition for p-local finite groups (S,F,L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.
Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf t\|$ for corresponding ideals $\bf r,\bf s, \bf t$ in the integral group ring $\mathbb Z[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\|\bf r,\bf s,\bf t\|)$ has exponent 2 modulo $\|R,S,T\|.$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subg…
We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.
By examining commensurators of virtually cyclic groups, we show that for each natural number n, any locally finite-by-virtually cyclic group of cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.
We prove that, for $n\geq 3$, the minimal dimension of a model of the classifying space of the full braid group $B_n$, and of the pure braid group $P_n$, with respect to the family of virtually cyclic groups is $n$.
We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.
Among the generalizations of Serre's theorem on the homotopy groups of a finite complex we isolate the one proposed by Dwyer and Wilkerson. Even though the spaces they consider must be 2-connected, we show that it can be used to both recover known results and obtain new theorems about p-completed classifying spaces.
We describe a cocompact model for the classifying space for proper actions of the mapping class group of a surface with punctures and boundary components. Our construction relies on a known model for the case of a closed surface and uses an induction on the number of punctures and boundary components.
One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.
In this paper we complete the description of the $B\mathbb{Z} /p$-cellularization of the classifying spaces of all finite groups, for all primes $p$. The techniques are based in a careful analysis of the $p$-fusion structure of the groups involved -with special attention to their strongly closed subgroups- and Chachólski's description of the $A$-cellular approximation.
In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements formula for generalized cohomological invariants of p-local finite groups, which is used to show the existence of unitary embeddings of p-local finite groups. Finally, we show that the augmentation map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the co…
We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space of G as an object under the stable p-completed classifying space of S via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino-Priddy conjecture, we obtain the surprising result that the unstable p-completed homotopy type of BG is determined by the stable p-completed inclus…
Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.
The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces
本文提出了一种灵活的范畴框架，用于研究度量Rips过滤和分层超几何空间的通用准测地锥体，通过显式Rips-元组配方提供规范模型。
Realizing wedges of Moore spaces as classifying spaces of finite semigroups
针对Fiedorowicz提出的猜想，证明有限生成阿贝尔群的单连通Moore空间的有限楔和同伦等价于有限半群的分类空间，表明仅同调群无法排除有限单连通CW复形等价于有限半群分类空间的可能。
We show that every discrete subgroup of $\mathrm{GL}(n,\mathbb{R})$ admits a finite dimensional classifying space with virtually cyclic stabilizers. Applying our methods to $\mathrm{SL}(3,\mathbb{Z})$, we obtain a four dimensional classifying space with virtually cyclic stabilizers and a decomposition of the algebraic $K$-theory of its group ring.
A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the framework of knot theory the notion of classifying spaces for families pops out naturally and we provide explicit models for them. The study of their topological properties hints at the existence of arcane connections between knot theory and algebraic number theory. Trying to track these connections, we discovered a behavioural analogy between the fundamental group…
Classifying spaces of finite groups of tame representation type
本文研究有限群中具有tame表示类型的块的p-完成分类空间及其循环空间，重点在于$A_\infty$代数结构和奇点余奇点范畴。
Given a group G, we consider its classifying space for the family of virtually cyclic subgroups. We show for many groups, including for example, one-relator groups, acylindrically hyperbolic groups, 3-manifold groups and CAT(0) cube groups, that they do not admit a finite model for this classifying space unless they are virtually cyclic. This settles a conjecture due to Juan-Pineda and Leary for these classes of groups.
We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for this classifying space. We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G-CW-complex.
We investigate the structure of the minimal displacement set in $8$-located complexes with the SD'-property. We show that such set embeds isometrically into the complex. Since $8$-location and simple connectivity imply Gromov hyperbolicity, the minimal displacement set in such complex is systolic. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on an $8$-located complex with the SD'-property.
Given a group action on a simplicial complex such that each simplex stabiliser admits a cocompact model of classifying space for proper actions, we give conditions implying the existence of a cocompact model of classifying space for proper actions for the whole group. This is used to generalise previous combination results for boundaries of groups and hyperbolic groups.
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Szép product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our m…
The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height.
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