Is there a general simple theorem for the third cohomology of cyclic groups $H_3(\mathbb{Z}_n, U(1))= ?$. In particular, I am interested in finding $H_3(\mathbb{Z}_8, U(1))$. I know the answer can be found using GAP, but I wanted a formal theorem, and also I don't have access to the HAP package of GAP, which is used for such computations.
1 Answer
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For cyclic groups, the modified cohomology groups are periodic with period 2. In particular, there are isomorphisms $$H_3(\mathbb Z/n\mathbb Z,U(1))=\hat H^{-4}(\mathbb Z/n\mathbb Z,U(1))\cong \hat H^0(\mathbb Z/n\mathbb Z,U(1))=U(1)^{\mathbb Z/n\mathbb Z}/N(U(1))=U(1)/n=0$$
Here $N$ is the norm map $x\mapsto \sum_{a\in \mathbb Z/n\mathbb Z} a\cdot x$.
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$\begingroup$ Thanks a lot for the answer. Please could you provide some background or give me a reference for how you got each of those isomorphisms. I don't have much knowledge of group cohomology. $\endgroup$– user35360Commented Apr 12, 2016 at 9:41
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$\begingroup$ This should be contained in any reasonably complete treatment of group cohomology. Google led me to these notes, for example: math.arizona.edu/~sharifi/groupcoh.pdf. Here the Tate cohomology groups (what I have called modified cohomology in the answer) are defined in 1.6, and from the definition all of the equality signs in the answer (but not the isomorphism!) follow immediately. The isomorphism follows from the discussion in 1.10. $\endgroup$ Commented Apr 12, 2016 at 14:28