Singular cohomology is homotopy invariant

Set $f^{\ast }\phi =\phi f_{\mathrm{\#}}$. The chain equation [F1] gives $\delta f^{\ast }=f^{\ast }\delta$, hence $f^{\ast }$ preserves cocycles and coboundaries and induces a real-linear map on the quotient [F2]. The generator identity and composition laws in [F1] become ${\mathrm{id}}^{\ast }=\mathrm{id}$ and $(gf{)}^{\ast }=f^{\ast }{g}^{\ast }$.

Put $K^k\phi =\phi P_{k-1}$ for $k\ge 1$ and $K^k=0$ for $k\le 0$. For a degree-$k$ cochain and $c\in C_k(X;\mathbb{R})$, [F3] gives $(g^{\ast }-f^{\ast })\phi (c)=\phi (\mathrm{\partial }{P}_{k}c+P_{k-1}\mathrm{\partial }c)=(K^{k+1}\delta \phi +\delta K^k\phi )(c)$. In degree zero the second summand is zero, exactly as in [F3]; negative degrees are zero.

If $\phi$ is a cocycle, step 1.2 says $g^{\ast }\phi -f^{\ast }\phi =\delta K^k\phi$, so their classes coincide. If $f:X\to Y$ has a supplied homotopy inverse $h:Y\to X$, apply this equality to $hf\simeq {\mathrm{id}}_X$ and $fh\simeq {\mathrm{id}}_Y$; step 1.1 yields $f^{\ast }{h}^{\ast }=\mathrm{id}$ and $h^{\ast }{f}^{\ast }=\mathrm{id}$. Thus $f^{\ast }$ is an isomorphism.

Empty spaces and negative groups have only zero maps, and on a point the identity acts identically on $H^0=\mathbb{R}$. Constant homotopies and degenerate simplices need no normalization: [F3] applies to their full signed prisms as well. Only the given homotopy and its explicit finite prism are used, so no AC is needed.