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It is well-known that $H^{3}(\mathbb{Z}/n\mathbb{Z};U(1))$ (the 3rd cohomology group of the cyclic group of $n$ in coefficient $U(1)$) is isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Does there exist an explicit formula for a 3-cocycle representing a generator of $H^{3}(\mathbb{Z}/n\mathbb{Z};U(1))$? What I mean precisely is: does there exist such a formula that is uniformly expressed in terms of $n$?

In fact, I need an expression to compute the Dijkgraaf-Witten invariant invariant of a 3-manifold.

Thanks!

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    $\begingroup$ If $G$ is cyclic, then $H^1(G,M)$ is isomorphic to $H^3(G,M)$ for any $M$, and an isomorphism is given by cup-product with a generator of $H^2(G,\mathbb{Z})$. So one approach to your question would be to write down a generator for $H^1(\mathbb{Z}/n\mathbb{Z},U(1))$ and a generator for $H^2(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})$, which are both easy, and then try to compute an explicit cocycle representing their cup product. This may or may not satisfy your "uniformity" condition, depending on exactly what you mean. $\endgroup$
    – Martin Bright
    Commented Feb 7, 2013 at 13:37
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    $\begingroup$ An explicit formula is given in Proposition 2.3 in this paper: arxiv.org/pdf/1206.5402.pdf $\endgroup$
    – Ralph
    Commented Feb 7, 2013 at 23:01
  • $\begingroup$ These answers are both very useful! $\endgroup$
    – user31234
    Commented Feb 8, 2013 at 2:09

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The corresponding 3-cocycles of $H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n$ are very simple: $$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{n^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$ with $p_I \in \mathbb{Z}_n$ labels the element in $H^3(\mathbb{Z}_n,U(1))$. Also $a,b,c \in \mathbb{Z}_n$. $[b^{}+c^{}] \equiv (b^{}+c^{})$mod $n$. You can check explicitly it satisfies 3-cocycles conditions.

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  • $\begingroup$ Idear: The formula is indeed not too complicated. But I'd like to not just see it, but to also understand it. How did you find it? What makes you believe that it represents non-trivial elements of $H^3$? $\endgroup$
    – André Henriques
    Commented Feb 8, 2014 at 5:21
  • $\begingroup$ To understand the cocycles, my method is by using the relation between Chern-Simons theory (partition function or path integral $Z$)and cohomology group('s cocycles): this 3-cocycle term actually shows us the so-called 2+1 dimensional Abelian Chern-Simons theory at the level $p_I$, which has a partition function: $Z=\int [DA]\exp[i\; p_{I} \int A \wedge d A]$, where A is 1-connection of a gauge group U(1) (or $\mathbb{Z}_n$ subgroup). I hope this physics field theory intuition helps. (p.s. I am a physicist.) $\endgroup$
    – wonderich
    Commented Feb 8, 2014 at 6:01
  • $\begingroup$ Well... your reply still doesn't give me a way to independently check that your formula is correct. $\endgroup$
    – André Henriques
    Commented Feb 8, 2014 at 8:47
  • $\begingroup$ Plug it into n-cocycle conditions, you will know it is right. Each of $p_I$ are independent, not equivalent via n-coboundary conditions, so each of $p_I$ are distinct. It will be nice Andre may post an answer how mathematician derives this. (I have some TQFT intuition to derive it.) many thanks. $\endgroup$
    – wonderich
    Commented Feb 8, 2014 at 18:38

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