Cohomology-Homotopy operator etc.,

Homotopy equivalent manifolds have isomorphic de Rahm cohomology groups.

Suppose $F,G:M\rightarrow N$ are smooth homotopic maps. Suppose $\omega$ is a $k$ form on $N$ and $h$ be an homotopic operator that maps from space of $k$ forms on $N$ to $k-1$ forms on $M$ given by
\begin{equation}
  d(h\omega)+h(d\omega)=G^{*}(\omega)-F^{*}(\omega)
\end{equation}
This means $h:\mathcal{A}^k(N) \rightarrow \mathcal{A}^{k-1}(M)$.

This homotopy is used as a stepping stone for proving homotopy equivalent manifolds have isomorphic homology groups.

I shall write in detail the motivation and how this is used later.

There is deRahm theorem proof of which I shall blog later.