Timeline for Reference for Ring Structure on Group Cohomology

6 events

when toggle format	what		by	license	comment
Jun 18, 2013 at 16:24	history	edited	Ian Agol	CC BY-SA 3.0	added 94 characters in body; added 95 characters in body
Jun 18, 2013 at 14:19	vote	accept	Peter Crooks
Jun 18, 2013 at 14:09	comment	added	Allen Hatcher		Comment on my previous comment: It is an interesting little exercise to determine which powers of $\alpha$ are nonzero.
Jun 18, 2013 at 14:07	comment	added	Allen Hatcher		Just for the record, here's a minor correction which doesn't affect the overall argument in this answer. In $H^\ast({\mathbb Z}/n,{\mathbb Z}/n)$ the 1-dimensional generater $\alpha$ satisfies $2\alpha^2=0$ by commutativity of cup product, which forces $\alpha^2=0$ when $n$ is odd but not when $n$ is even. In fact $\alpha^2\neq 0$ when $n$ is even. This is well known for $n=2$, and for larger even $n$ this is shown in Example 3.9 in my book. Thus for $n=2k$ we have $H^\ast({\mathbb Z}/n,{\mathbb Z}/n)={\mathbb Z}/n[\alpha,\beta]/(\alpha^2-k\beta)$.
Jun 18, 2013 at 0:38	history	edited	Ian Agol	CC BY-SA 3.0	added 778 characters in body
Jun 17, 2013 at 17:48	history	answered	Ian Agol	CC BY-SA 3.0