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A differentiable map $f:M\longrightarrow N$ induces pullback functions $f^*:\Omega^k(N)\longrightarrow \Omega^k(M)$. But since the identity $\mathrm{d}\circ f^*=f^*\circ\mathrm{d}$, we indeed have a complex homomorphism

$$f^*:\Omega^*(N)\longrightarrow \Omega^*(M)$$

that is,

$$ \require{AMScd} \begin{CD} \cdots @>>> \Omega^{k-1}(N) @>{\mathrm{d}^{k-1}}>> \Omega^{k}(N) @>{\mathrm{d}^{k}}>> \Omega^{k+1}(N) @>>> \cdots\\ @. @V{f^{k-1}}VV @V{f^{k}}VV @V{f^{k+1}}VV @. \\ \cdots @>>> \Omega^{k-1}(M) @>{\mathrm{d}^{k-1}}>> \Omega^{k}(M) @>{\mathrm{d}^{k}}>> \Omega^{k+1}(M) @>>> \cdots\\ \end{CD} $$

which induces homomorphisms in cohomology

$$f^*:H^k(N)\longrightarrow H^k(M)$$

for each $k$. The usual properties $(f\circ g)^*=g^*\circ f^*$, $(\mathrm{id}_M)^*=\mathrm{id}_{H(M)}$ hold. Therefore a diffeomorphism $f:M\longrightarrow N$ induces isomorphisms in the cohomology groups