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Periodicity in Group Cohomology and Complete Resolutions
Author(s) -
Talelli Olympia
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004273
Subject(s) - mathematics , cohomology , group cohomology , conjecture , group (periodic table) , resolution (logic) , cup product , product (mathematics) , element (criminal law) , étale cohomology , pure mathematics , injective function , homotopy , infimum and supremum , combinatorics , equivariant cohomology , de rham cohomology , geometry , physics , quantum mechanics , artificial intelligence , computer science , political science , law
A group G is said to have periodic cohomology with period q after k steps, if the functors H i ( G , −) and H i+q ( G , −) are naturally equivalent for all i > k . Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups G that admit a finite‐dimensional, free G ‐CW‐complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in H q ( G , Z ). It is expected that the periodicity isomorphisms will always be given by the cup product with an element in H q ( G , Z ); this paper shows that this is the case if and only if the group G admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in H q ( G , Z ) is equivalent to silp G being finite, where silp G is the supremum of the injective lengths of the projective ZG ‐modules. 2000 Mathematics Subject Classification 20J05, 57S25.