Schur homotopy of the simplest group

Arxiv preprint arXiv:0901.0137, 2008

Let Γ q denote the q-th stage of the descending central series of the free group on n generators F n . For each q ≥ 2 and every topological group G, a simplicial space B * (q, G) is constructed where B n (q, G) = Hom(F n /Γ q , G) and the realizations B(q, G) = |B * (q, G)| 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, including their description as homotopy colimits. Cohomology calculations are provided for compact Lie groups. The spaces B(2, G) are described in detail for transitively commutative groups. Stable homotopy decompositions of the B(q, G) are also provided with a formula giving the cardinality of Hom(F n /Γ q , G) for finite discrete groups G (and thus the cardinality of Hom(Z n , G) in case q = 2) in terms of the ranks of the homology groups for the associated filtration quotients of B(q, G). Specific calculations for H 1 (B(q, G); Z) are shown to be delicate in case G is finite of odd order in the sense that resulting topological properties (which are not yet fully understood) are equivalent to the Feit-Thompson theorem.

arXiv: Group Theory, 2015

We prove that for a finitely generated group $G$, the second stable homotopy group $\pi_2^S(K(G,1))$ of the Eilenberg-Maclane space $K(G,1)$ is completely determined by the Schur multiplier $H_2(G)$. We also prove that the second stable homotopy group $\pi_2^S(K(G,1))$ is equal to the Schur multiplier $H_2(G)$ for a torsion group $G$ with no elements of order $2$ and show that for such groups, $\pi_2^S(K(G,1))$ is a direct factor of $\pi_{3}(SK(G,1))$, where $S$ denotes suspension and $\pi_2^S$ the second stable homotopy group. We compute $\pi_{3}(SK(G,1))$ and $\pi_2^S(K(G,1))$ for symmetric, alternating, general linear groups over finite fields and some infinite general linear groups $G$. We also obtain a bound for the Schur multiplier of all finite groups $G$ analogous to Green's bound for $p$-groups.

Journal of Mathematics of Kyoto University, 1992

On homotopy associative mod 2 H-spaces' By Jo h n McCleary I n [1], [3 ], a n d [6 ] th e following question is considered: If Y is a m od 2 H-space, when does Y x S 7 adm it the structure of a homotopy associative m od 2 H-space ? Among the sim ple L ie groups, th e results i n [ 1 ] reveal that th e only possible examples a re th e following: Spin(7) (2) .' _-_' (G 2 x S 7)(2) Spin(8) (2) ._^2 (Spin(7) x S 7) (2) a n d SO(8)(2) (S O (7) x S 7) (2). The focus o f [6 ] is o n generalizing th e results o f [1 ] to finite H-spaces. Here th e Hopf algebra over th e m od 2 Steenrod algebra, 4 '2 , given by A = F2 [X3]/X1 A(Sg 2 x 3) L , ' H*(G 2 ; F2) plays a crucial role. T he m ain results o f [6 ] a re summarized in th e following Theorem (Lin-W illiam s). L et Y be a finite simply-connected CW-complex and suppose H*(Y; F 2) contains no subalgebras isomorphic to A. T hen Y x S 7 cannot b e a homotopy associative H-space. Suppose H*(Y; F 2) contains at m ost one subalgebra isom orphic to A. T hen Y x (S 7)" cannot be a homotopy associative H-space f o r k 3. T h e m ethod o f proof o f this theorem suggests a n e x te n sio n th a t is the principal result o f this paper. Main Theorem. L et Y be a finite simply-connected CW-complex and suppose that H*(Y; F 2) contains g " f or some e > O. Then Y x (S 7)" cannot be a homotopy associative H-space f o r k > 2(+ 1. This show s th a t, a t the prim e 2, from e copies of G 2 a n d k copies of 5 7 , th e examples o f Spin(7) a n d Spin(8) above a r e th e only homotopy associative

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.

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.

Manuscripta Mathematica, 1987

K-theory, 1998

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic Kand L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced C * -algebras of groups. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.

Archiv der Mathematik, 1983

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.

Mathematische Zeitschrift, 2017

We prove that for any group G, π S 2 (K (G, 1)), the second stable homotopy group of the Eilenberg-Maclane space K (G, 1), is completely determined by the second homology group H 2 (G, Z). We also prove that the second stable homotopy group π S 2 (K (G, 1)) is isomorphic to H 2 (G, Z) for a torsion group G with no elements of order 2 and show that for such groups, π S 2 (K (G, 1)) is a direct factor of π 3 (SK (G, 1)), where S denotes suspension and π S 2 the second stable homotopy group. For radicable (divisible if G is abelian) groups G, we prove that π S 2 (K (G, 1)) is isomorphic to H 2 (G, Z). We compute π 3 (SK (G, 1)) and π S 2 (K (G, 1)) for symmetric, alternating, dihedral, general linear groups over finite fields and some infinite general linear groups G. For all finite groups G, we obtain a sharp bound for the cardinality of π S 2 (K (G, 1)).

Proceedings of the American Mathematical Society, 1978

A certain Samelson (commutator) product is computed in the homotopy groups of a finite H-space. This result is applied to the study of i/-maps in rr3 of that space, and to the study of multiplications on Lie groups. 0. Introduction. Let m: S3 xS3^53bea multiplication on the 3-sphere S3 and let a E ^(S3) aZbea generator. Recall that Arkowitz and Curjel [2] have shown that the Samelson (commutator) product (a, a)m E ir6(S3) ss Z/12 is a generator precisely when m is homotopy-associative. In this paper we shall consider a generalization (Theorem 0.1) of that computation and also present some applications. We now give specific results. Unless otherwise stated X will denote a finite CW //-space with tr3(X) = Z. Further X must possess at least one homotopy-associative multiplication and satisfy a technical condition 0.4 (which holds for all known finite //-spaces). Included in our study are all compact, connected, simple, nonabelian Lie groups. The following is proved in §1. 0.1 Theorem. Let a E tr3(X) be a generator. If p is any homotopy-associative multiplication on X then (a, a) generates ttA^X). As an application we consider the question raised at Neuchâtel [9, Problem 24, p. 127] as to whether or not a is an //-map, and prove in §2: 0.2 Theorem, (a) Given any multiplication m on S3 there exists a multiplication pon X such that a: (S3, m)-» (X, p) is an H-map. (b) Given any multiplication p on X there exists a multiplication m on S3 such that a: (S3, m)-* (X, p) is an H-map. In a different direction Theorem 0.1 has an application to the study of properties of multiplications on Lie groups. Recall that Mimura [6] has shown that there are 1215 • 39 • 5 • 7 distinct multiplications on SU(3). We show in §3 0.3 Theorem. At least one-third of the multiplications on SU(3) are not homotopy-associative. Let us now state the condition we assume (except in §3) all finite //-spaces satisfy in this paper. Let X denote the universal cover of X.

Topology, 1984

$1. INTRODUCTION THE RELEVANCE of crossed modules to problems on second homotopy groups, and to some difficult problems in combinatorial group theory, is well known (see [5]). The difficulties are essentially those of understanding free crossed modules, and, more generally, colimits of crossed modules. The algebraic purpose of this paper is to give a simple description of the coproduct of two crossed P-modules. The application of this algebra to homotopy theory comes from the generalisation of the van Kampen theorem to dimension two given by Brown and Higgins[3]. This theorem shows that certain unions of pairs of spaces give rise to pushouts of crossed modules. A simple special case of our main result (Corollary 3.2) concerns the union of

H n (G, M) where n = 0, 1, 2, 3,. . ., called the nth homology and cohomology of G with coefficients in M. To understand this we need to know what a representation of G is. It is the same thing as ZG-module, but for this we need to know what the group ring ZG is, so some preparation is required. The homology and cohomology groups may be defined topologically and also algebraically. We will not do much with the topological definition, but to say something about it consider the following result: THEOREM (Hurewicz 1936). Let X be a path-connected space with π n X = 0 for all n ≥ 2 (such X is called 'aspherical'). Then X is determined up to homotopy by π 1 (x). If G = π 1 (X) for some aspherical space X we call X an Eilenberg-MacLane space K(G, 1), or (if the group is discrete) the classifying space BG. (It classifies principal G-bundles, whatever they are.) If an aspherical space X is locally path connected the universal cover˜X is contractible and X = ˜ X/G. Also H n (X) and H n (X) depend only on π 1 (X). If G = π 1 (X) we may thus define H n (G, Z) = H n (X) and H n (G, Z) = H n (X) and because X is determined up to homotopy equivalence the definition does not depend on X. As an example we could take X to be d loops joined together at a point. Then π 1 (X) = F d is free on d generators and π n (X) = 0 for n ≥ 2. Thus according to the above definition H n (F d , Z) = Z if n = 0 Z d if n = 1 0 otherwise. Also, the universal cover of X is the tree on which F d acts freely, and it is contractible. The theorem of Hurewicz tells us what the group cohomology is if there happens to be an aspherical space with the right fundamental group, but it does not say that there always is such a space.

Forum Mathematicum, 2009

We establish sufficient conditions of the nerve of the linking system of a p-local finite group (S, F , L) to have the homotopy type of an Eilenberg-MacLane space K(Γ, 1) for a group Γ which contains S. We prove that in this situation the entire p-local finite group can be reconstructed from Γ. Examples of p-local finite groups that our theorem applies to are the exotic ones which were constructed by the second author and Ruiz.

Forum Mathematicum, 1999