Intuition behind prism operators to prove homotopy invariance of homology

Question

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as defined in Hatcher, for example) exactly represent. Should I try to have a precise, geometric intuition of what is happening, or not? I haven't seen much description of the intuition behind this proof in Hatcher (where usually geometric intuition is emphasised, I think), or in any other text.

Thanks in advance.

Answer 1

I have not read Hatcher, but Prism's go back to Eilenberg-Steenrod. The idea is simple if $H(x,t)$ is a homotopy on $\Delta_p$ (just assume for the moment it is constant on the boundary), then we can view $H$ as a function on the prism $I\times \Delta_p$. Now the boundary of the prism can be thought of as the difference of the top and the bottom, (the sides are constant). This can be then extended to sums of simplices. And so it shows that homotopy gives a chain-homotopy which induces and isomorphism.

I think the problem arises because $I\times \Delta_p$ is no longer a simplex. Athough personally I have never really understood why this is such a big issue since one can easlily and explicitly write a prism as a sum of simplices.