On Periodic (Co)Homology of Groups

Abstract

Here we show that a countable group G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H q (G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H q (G, ℤ). In [2] Asadollahi, Hajizamani, and Salarian showed that, if a group G is such that every flat ℤG-module has finite projective dimension, then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H q (G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H q (G, C), where C is the cotorsion envelope of the trivial ℤG-module ℤ.