My question (see below for the definitions used):
Barycentric subdivision $B$ should not change the homology. In other words, we want to show that the chain map $B$ is a chain homotopy equivalence (and thus particularly a quasi-isomorphism).To show this it suffices to prove that $B$ is chain homotopic to the identity. We construct this chain homotopy $\psi_n\colon S^{\text{aff}}_n(\Delta^p) \rightarrow S^{\text{aff}}_{n+1}(\Delta^p)$ inductively on generators as follows: $$\psi_0([v])=-[v,v],$$ $$\psi_n(\alpha)=(-1)^{n+1}K_{b(\alpha)}(\alpha -\psi_{n-1}(\partial \alpha)) \text{ for } n>0.$$
- Where does this chain homotopy come from? It is mysterious to me. While I have geometric intuition for barycentric subdivision, I do not for the chain homotopy. How should I visualize the chain homotopy? I can verify algebraically that it gives a chain homotopy, but don't think I would have come up with it on my own.
Definitions used:
- A singular $n$-simplex $\alpha: \Delta^n\rightarrow \Delta^p$ in $\Delta^p$ is called affine if $\alpha(\sum_{i=0}^n t_i e_i)= \sum_{i=0}^n t_i\alpha(e_i)$ holds for all $t_i$ with $\sum_{i=0}^n t_i=1$ and $t_i\geq 0$. Following Hatcher, given an affixe $n$-simplex $\alpha$, we set $v_i=\alpha(e_i)$ and write $[v_0,\ldots, v_n]$ for $\alpha$. The free abelian group generated by the affine $n$-simplices in $\Delta^p$ is denoted by $S^{\text{aff}}_n(\Delta^p)$. Restricting the differential of the singular chain complex of $\Delta^p$ makes this a subcomplex of the singular chain complex of $\Delta^p$.
- For $v\in\Delta^p$ and $\alpha=[v_0,\ldots,v_n]$ an affine $n$-simplex in $\Delta^p$ we let the cone $K_v(\alpha)$ of $\alpha$ with respect to $v$ be given by the affine $(n+1)$-simplex $[v_0,\ldots,v_n,v]$. Extending $K_v$ linearly, i.e. $K_v(\sum_i \lambda_i \alpha_i)=\sum_i \lambda_i K_v(\alpha_i)$, turns the cone with respect to $v$ into a group homomorphism $S^{\text{aff}}_n(\Delta^p)\rightarrow S^{\text{aff}}_{n+1}(\Delta^p)$.
- The barycentric subdivision of an affine $n$-simplex in $\Delta^p$ is a map $B\colon S^{\text{aff}}_n(\Delta^p)\rightarrow S^{\text{aff}}_n(\Delta^p)$ defined inductively by $B(\alpha)=\alpha$ for $\alpha\in S^{\text{aff}}_0(\Delta^p)$ and $B(\alpha)=(-1)^n K_{b(\alpha)}(B(\partial \alpha))$ for $\alpha\in S^{\text{aff}}_n(\Delta^p)$ with $n>0$. Here $b(\alpha)$ denotes barycenter of $\alpha$. One shows that $B$ is a chain map.