My question has to do with the chain homotopy that appears in Lee's *Introduction to Topological Manifols* and Rotman's *Introduction to Algebraic Topology* proofs that the inclusion

$$C_\bullet^\mathcal{U}(X)\hookrightarrow C_\bullet(X)$$

induces an isomorphism in singular homology

$$H_p^\mathcal{U}(X)\cong H_p(X)$$

For all $p\geq 0$. In both references, a chain homotopy $h:C_p(X)\longrightarrow C_{p+1}(X)$ between the barycentric subdivision operador and the identity map is given by

If $p=0$, $h$ is the zero homomorphism. If we have defined $h$ up to some $p\in\mathbb{N}$, and $\sigma$ is a $p-$simplex in $X$, then

$$h\sigma=\sigma_\#b_p*(i_p-si_p-h\partial i_p)$$

Where $b_p$ is the barycentre of the standard $p-$simplex $\Delta_p$ and $*$ is the cone operator. We then extend $h$ linearly to singular chains: $h\big(\sum_{i\in I}n_i\sigma_i\big)=\sum_{i\in I}n_ih\sigma_i$

In contrast with the chain homotopy that appears in the proof of the homotopy axiom, this is really less intuitive, and relies heavily on the equation

$$\partial(w*c)=c-w*\partial c$$

So my questions are:

How should we understand geometrically this map $h$? What is the geometric intuition that allows us to choose this a a good chain homotopy for our purposes?

How should we understand the formula $\partial(w*c)=c-w*\partial c$? What is the meaning of this equation geometrically speaking?

How to come up with such a map in the first place? How has this theorem developed historically?

I understand perfectly both demonstrations, since the calculations are easy to follow; I am just concerned with how this map gives no intuition at first glance about the geometry involved.

\mathcal{U}-small chains in $X$, whose elements are chains such that the image of each of its simplices is contained in some element of the cover $\mathcal{U}$. $H_p^\mathcal{U}(X)$ is just the homology of such complex. $\endgroup$2more comments