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Homology Decompositions for Classifying Spaces of Finite Groups Associated to Modular Representations
Author(s) -
Notbohm D.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002459
Subject(s) - mathematics , functor , homology (biology) , sylow theorems , classifying space , homotopy , combinatorics , isomorphism (crystallography) , centralizer and normalizer , pure mathematics , vector space , finite group , discrete mathematics , group (periodic table) , biochemistry , chemistry , crystal structure , organic chemistry , gene , crystallography
For a prime p , a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod‐ p homology. Associated to a modular representation G → Gl( n ; F p ), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so‐called subgroup, centralizer and normalizer decompositions. For an action of G on an F p ‐vector space V , this collection consists of all subgroups of G with nontrivial p ‐Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG .