Periodic flat resolutions and periodicity in group (co)homology

On Algebraic Invariants for Free Actions on Homotopy Spheres

“…In [4], using the notion of flat covers and proper flat resolutions, Asadollahi et al, investigated the notion of periodic homology of period q after k steps, that is, that H i (G, −) and H i+q (G, −) are naturally equivalent for all i > k. They showed that if a group G with the property 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 the 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 the cap product with an element in H q (G, C), where C is the cotorsion envelope of the trivial ZG-module Z. In [32], Talelli showed that a countable group G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by the 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 the cap product with an element in H q (G, Z).…”