Periodicity and Jumps in Cohomology of R-torsion-free Groups
نویسنده
چکیده
A discrete group G has periodic cohomology over R if there is an element α ∈ Ext RG (R,R) of positive degree and an integer n ≥ 0 such that the cup product map by α induces an isomorphism of Ext RG (R,−) and Ext i+|α| RG (R,−) for i ≥ n. Adem and Smith [1] showed if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It was conjectured by Olympia Talelli in [14], that if G is also torsion-free then it must have finite cohomological dimension. In this paper we investigate the implied condition of jump cohomology over R: G has jump cohomology of height k over R if any subgroup H with finite cohomological dimension over R, has cdR(H) ≤ k. We prove that all solvable and HF groups with jump cohomology over R must have finite cohomological dimension over R if they are R-torsion-free.
منابع مشابه
Jumps in Cohomology and Free Group Actions
A discrete group G has periodic cohomology over R if there is an element in a cohomology group, cup product with which induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CWcomplex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is ...
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