Homology decompositions for classifying spaces of finite groups - PDF document

Homology decompositions for classifying spaces of finite groups Silvia Onofrei Department of Mathematics, University of California at Riverside Thursday, October 20 , 2005 1. Outline of the talk and motivation Let G be a finite group and p a prime dividing its order. A homology decomposition of a classifying space BG is a process of glueing together classifying spaces of suitable chosen subgroups of G in order to obtain a space X and a map X → BG which induces an isomorphism in mod- p homology. The glueing information is encoded in the form of a homotopy colimit. The mod- p homology of BG can then be determined by an exact sequence of the homology groups of the subgroups if certain higher derived functors associated to the homotopy colimit are zero. Homology decompositions provide a systematic way for approaching the cohomology of simple groups. The finite simple groups can be divided into three classes: the alternating groups (for n ≥ 5), the groups of Lie type and the 26 sporadic groups. Results obtained in the past two decades show that the mod p (especially for p = 2) cohomology of the sporadic groups exibit interesting features and finding the cohomology of these groups remains an area of active research. In an attempt to generalize the theory of buildings for Lie groups, several p -local geometries (geometric structures which have the property that the stabilizers of the objects are normalizers of p -subgroups of the group G ) have been constructed for the sporadic finite simple groups but there is no uniform treatment of the p -local geometries of these groups. However, many sporadic groups, have a p -local geometry whose barycentric subdivision corresponds to a certain family of p -subgroups called the Bouc collection (this contains all the p -subgroups such that N G ( P ) /P does not have non-trivial normal p -subgroups). A very recent work of Benson and Smith [ BS04 ] focuses on the p -local approach to the structure of the mod- p cohomology of the sporadic groups. For p = 2, they exhibit for each sporadic group an ample collection of 2-subgroups related to a 2-local geometry. By a case by case analysis, they 1

2 describe homotopy equivalences between most of these collections and one of the standard collections, such as the Quillen or the Bouc collection. Our work (joint with J. Maginnis) began with an analysis of the homotopy relation between a 2-local geometry and the aforementioned Bouc collection for the third sporadic group of Conway. We were able to define a new collection which is in the same homotopy class as the 2-local geometry. Motivated by this result we generalized the definition of this collection. The novelty of our approach resides in the fact that we impose conditions on the groups which are studied, instead of imposing conditions on their collections of subgroups. In this talk I shall present these new collections of p -subgroups which can be used to obtain homology decompositions for finite groups. For the 26 sporadic simple groups these collections are related to p -local geometries. I shall start with few definitions, followed by a brief outline of the development of the area from historic point of view; then continue with the simple example of the group GL (3 , 2). Next, I shall adumbrate a general description for the construction of the homotopy decomposition adapted for the case of the subgroup decomposition. Finally the statement of the main theorem and an outline of its proof will be given. 2. Some terminology A collection C is a family of subgroups of G which is closed under conjugation and ordered by inclusion; thus a collection is a G -poset, under the conjugation action of the group. To this G -poset one can associate a G -simplicial complex (and thus a topological space) |C| whose vertices are the elements of C and whose simplices are the nonempty finite chains in C . This is called the subgroup complex associated to C . In finite group theory, the subgroup complexes play an important role, they are regarded as geometries and generalizations of buildings. If C is viewed as a small category, then |C| is the classifying space of C ; the functor C − → |C| assigns topological concepts to a posets. Probably the best known group complexes are the buildings . If we let G to be a Lie group in natural characteristic p , then the building of G is the simplicial complex of the poset of parabolic subgroups (the overgroups of the Borel subgroup). The theory of buildings was initiated by Tits [ T74 ] in order to give a systematic approach for the geometric interpretation of the semisimple Lie groups and in particular those of exceptional type. The understanding of groups is simplified through the use of geometries on which these groups act and the theory of buildings provides a tool in approaching these geometries directly. The topology of buildings is relevant to the representation theory of the underlying Lie group: the top dimensional homology module is the Steinberg module, the only irreducible projective module of G .

3 The following collections are standard in the literature: A p ( G ) = { E | E nontrivial elementary abelian p-subgroup of G } ; S p ( G ) = { P | P nontrivial p-subgroup of G } ; B p ( G ) = { R | R nontrivial p-radical subgroup of G } . The simplicial complex |A p ( G ) | is known as the Quillen complex, |S p ( G ) | is known as the Brown complex and |B p ( G ) | is called the Bouc complex. The three complexes are G -homotopy equivalent [ ThWb91 ]. 3. Historical context Simplicial complexes from posets of subgroups of a group Brown [ Br75, Br76 ] studied the influence of the elements of finite p -power order on group cohomology 1 , in groups which were not necessarily finite. 2 Restricted to the case of a finite group G , his work focused on the poset S p ( G ) of all nontrivial p -subgroups of G ordered by inclusion and its associated simplicial complex |S p ( G ) | , obtained from the inclusion chains of p -subgroups. One of the most interesting results he obtained refered to the Euler characteristic of this complex: χ ( |S p ( G ) | ) ≡ 1 (mod | G | p ) where | G | p is the highest power of p which divides the order of G , thus the order of a Sylow p - subgroup if G is finite. This is also known as the homological Sylow theorem , due to the resemblance to the Sylow theorem from group theory which affirms that the number of Sylow p -subgroups of a finite group G is congruent to 1 mod p. Quillen [ Qu78 ] studied in his seminal paper the homotopy properties of S p ( G ). He developed the fundaments for the study of the homotopy properties of posets of subgroups. In fact, Quillen introduced the poset of non-trivial elementary abelian p -subgroups, which is usually denoted A p ( G ) and it is known today as the Quillen poset. He proved that the inclusion of posets A p ( G ) ⊆ S p ( G ) induces a homotopy equivalence. Another common collection was introduced by Bouc [ Bou84 ]; this collection consists of non- trivial radical p -subgroups (also known as p -stubborn subgroups). These are p -subgroups R with the property that N G ( R ) /R does not have nontrivial normal p -subgroups. Bouc proved that this new collection has the same homotopy type as the previously defined collections of Brown and Quillen. For a finite Chevalley group in characteristic p , the Bouc collection is the collection of unipotent 1 In fact, he was interested in the cohomology of the group and the relation to the p -part of the Euler characteristic. 2 Brown considered infinite groups, in particular he was interested in discrete groups of virtually finite cohomolog- ical dimension, but we will focus on finite groups only.