Mathematics Asked by Const on January 2, 2021

This is the proof of Theorem 4.19. in Joseph J. Rotman „An Introduction to Algebraic Topology“. We want to show that $H_n(X)=0$ for all $n>0$ and $X$ convex bounded in some euclidean space. I refer to a similar question Homology group of convex sets – boundedness condition for used notation, in short, $gamma=partial cgamma+ cpartialgamma$ for a chain $gammain S_n(X)$. He then says that if $gamma$ is a cycle, we have $gamma =partial cgamma$, a boundary, so homology is trivial. But this I don‘t understand: is the cone over $0$ again $0$? In my eyes, I would see that the cone over $0$ is a boundary, which of course wouldn’t change the conclusion, but the argument confuses me a little.

Thanks in advance for any sort of clarification!

I came to realise my mistake... the $0$-chain is not the zero simplex (which doesn't make any sense anyway, right?) but the neutral element in the group $S_n(X)$. I first deleted the question, but now I think one needs to stand by such wrong thoughts.

Answered by Const on January 2, 2021

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