Homology decompositions for classifying spaces of finite groups

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 sporadi