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In usual cases, the chain group and cochain group in homology/cohomology are free, where the basis are n-simplices and elementary cochains respectively. (Is that correct? I am not very sure about cohomology.)

Are there cases that the chain/cochain groups are not free?

For instance, bounded cohomology, I think the bounded cochain group is not free?

Thanks.

yoyostein
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1 Answers1

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Cochain groups are actually very often not free. Indeed, if $X$ is a space, then the group of singular $n$-cochains on $X$ is a product $\prod\mathbb{Z}$, with one factor for each singular $n$-simplex in $X$. For most spaces $X$, there are infinitely many singular simplices in $X$, so this product is an infinite product of copies of $\mathbb{Z}$, which is not a free abelian group.

Eric Wofsey
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