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Let $\mathcal{K}$ be a simplical complex. Define (simplical) $n$-chains as maps from $n$-simplices in $\mathcal{K}$ to $\mathbb{Z}$ such that they vanish cofinitely many times. We then get a basis for a free abelian group "for free" since every chain is a sum of characteristic functions over the support for some coefficients and this sum is guaranteed to be finite by the cofinite vanishing condition.

What if we remove this requirement? Could we fix it by using $\mathbb{Z}_2$? I can't find anything about it and all the references I looked at (Munkres, Hatcher, Rotman, Hocking & Young) do it either with this requirement or even define them as finite formal sums straight away. I verified that $\partial$ behaves as expected. Where does this approach exactly fail, apart from that it doesn't benefit from "niceness" of having a free abelian group as the output.

locally_convex
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1 Answers1

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I think there's not any interesting mathematical content here... more that things simply won't work right for not-cofinitely-vanishing.

For example, consider a $0$-simplex $z_o$ with (countably) infinitely-many $1$-oriented simplices $t_1, t_2, t_3, ...$ having boundaries $\partial t_i=z_o-z_i$ with $0$-simplices $z_i$. (Can do this over $\mathbb Z/2$, as well.) Then the boundary of $\sum_i t_i$ is $\sum_{i=1}^\infty z_o-\sum_{j\not=0} z_j$. The latter sum might be ok, but what could the former mean? Doing things mod $2$ does not seem to help.

So, unless we enlarge the possible meanings of these sums (which is not out of the question), it's simply unclear what the meaning would be, in the context of boundaries and so on.

paul garrett
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  • Sorry for a late reply. It's a good point, thanks! I didn't really think about the geometric meaning, as this is still an abelian group under pointwise addition (just not free abelian) and image on n+1 is still a subgroup of kernel on n so it's still a chain complex. By the way, I found an exercise in Munkres that asks to do exactly this. It calls these "simplical homology groups based on infinite chains" and asks to compute them for some spaces where they end not coinciding with usual simplical homology. So I guess E-S axioms fail somewhere down the line. – locally_convex Feb 23 '21 at 13:32