Two manifolds $M$ and $N$ with the same homotopy type have the same de Rham cohomology

A homotopy relation ($M\sim N$) is a pair of maps $f:M\longrightarrow N$, $g:N\longrightarrow M$ with $g\circ f\sim \text{id}_M$, $f\circ g\sim \text{id}_N$. In this setting

$$H^\ast(M)\xrightarrow{g^\ast}H^\ast(N)$$ $$H^\ast(M)\xleftarrow{f^\ast}H^\ast(N)$$

are inverse isomorphisms

$$H^\ast(M)\xrightarrow{g^\ast}H^\ast(N)$$ $$H^\ast(M)\xleftarrow{f^\ast}H^\ast(N)$$

are inverse isomorphisms