A connection between cellularization for groups and spaces via two-complexes (original) (raw)
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Group-theoretic algebraic models for homotopy types
Journal of Pure and Applied Algebra, 1991
In this paper a nonabelian version of the Dold-Kan-Puppe theorem is provided. showing hou the Moore-complex functor defines a full equivalence between the category of simplicial groups and the category of what is called 'hypcrcrcssed complexes of groups'. i.e. chain complexes of nonabelian groups (G,,. S,,) with an additional structure in the form of binary operations G, x G,* G,. We associate to a pointed topological space X a hypercrossed complex A(X): and the functor 2 induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationshjp between s(X) and Whitehead's crossed complex II(X) is established by a cano:!ical surjection p : ii(X)-* II(X). which is a quasi-isomorphism if and only if X is a J-complex. Algebraic rnodels consisting of truncated chain-complexes with binary operations I;re deduced for rl-types. and as an application we deduce a group-theoretic interpretation of the cohomology groups H"(G. A).
On catn-groups and homotopy types
Journal of Pure and Applied Algebra, 1993
Bullejos, M., A.M. Cegarra and J. Duskin, On cat"-groups and homotopy types, Journal of Pure and Applied Algebra 86 (1993) 135-154. We give an algebraic proof of Loday's 'Classification theorem' for truncated homotopy types. In particular we give a precise construction of the homotopy cat"-group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusie's 'total Dee'. modules.
Eprint Arxiv 0802 4357, 2008
The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
Homotopy colimits of classifying spaces of abelian subgroups of a finite group
The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q, G), q ≥ 2, 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 ⊂ B(q, G) defined for a fixed prime p. Then 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. Extraspecial 2-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups of order 2 2n+1 , n ≥ 2, B(2, G) does not have the homotopy type of a K(π, 1) space. For a finite group G, we compute the complex K-theory of B(2, G) modulo torsion.
K-Theory, 2000
The results of a previous paper [3] on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from [19], and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.
Cellularization of classifying spaces and fusion properties of finite groups
Journal of the London Mathematical Society, 2007
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.
On H-groups and their applications to topological homotopy groups
This paper develops a basic theory of H-groups. We introduce a special quotient of H-groups and extend some algebraic constructions of topological groups to the category of H-groups and H-maps. We use these constructions to prove some advantages in topological homotopy groups. Also, we present a family of spaces that their topological fundamental groups are indiscrete topological group and find out a family of spaces whose topological fundamental group is a topological group.
Application of group cohomology to space constructions
Transactions of the American Mathematical Society, 1987
From a short exact sequence of crossed modules 1 → K → H → H ¯ → 1 1 \to K \to H \to \bar H \to 1 and a 2 2 -cocycle ( ϕ , h ) ∈ Z 2 ( G ; H ) (\phi ,\,h) \in {Z^2}(G;\,H) , a 4 4 -term cohomology exact sequence H a b 1 ( G ; Z ) → H ( ϕ ¯ , h ¯ ) 1 ( G ; H ¯ , Z ¯ ) → δ ⋃ { H ψ 2 ( G ; K ) : ψ o u t = ϕ o u t } → H a b 2 ( G ; Z ) H_{ab}^1(G;Z) \to H_{(\bar {\phi }, \bar {h})}^1 (G; \bar {H}, \bar {Z}) \stackrel {\delta }{\to } \bigcup \{ H_\psi ^2 (G;K) : \psi _{\mathrm {out}} = \phi _{\mathrm {out}} \} \to H_{ab}^2(G;\,Z) is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and Z Z is the center of the crossed module H H . The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.