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If I have a homology group $H_{1}(G,A)$ where $G$ is a group and $A$ is a $G$-module, the what does a 1-cycle on $G$ with values in $A$ look like? I know what the 1-cocycles look like but 1-cycles seem a bit of a mystery to me. Thank you

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Maybe you want to read this and this. –  P.. Nov 25 '12 at 13:38
    
You can use the bar resolution to get a description analogous to cocycles, though it won't be as nice because of the nature of the tensor product. I wrote a brief description in my MathOverflow answer: mathoverflow.net/questions/94383 –  Jason Polak Nov 29 '12 at 18:04

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