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At the bottom of page 122 of Hatcher, he defines a map $S:C_{n}(X)\rightarrow C_{n}(X)$ by $S\sigma=\sigma_{\#}S\Delta^{n}$. What is the $S$ on the right hand side and how does it act on the simplex $\Delta$? I'm having trouble deconstructing the notation here.

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It's the $S$ he defines in (2) of his proof: barycentric subdivision of linear chains.

I think you will benefit from reading the following supplementary notes.

update: here's the idea: first he deals with linear chains so he knows how to define the map $S: LC_n(\Delta^n) \to LC_n(\Delta^n)$, i.e. he knows how to find the barycentric subdivision of the standard $n$-simplex (thought of as the identity map onto itself). Then he uses the map $\sigma_\#$ composed with the subdivided standard $n$-simplex to define a barycentric subdivision of singular $n$-chains.

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