# Two forms related by an automorphism are in the same cohomology class?

Let $$f: M \to M$$ define an automorphism on the smooth manifold M.

Given a differential form $$\omega \in \Omega^k$$ is it true that the de Rham cohomology class of $$\omega$$ and $$f^*\omega$$ are the same? That is, does $$[\omega]=[f^*\omega]$$.

• Here's another type of example: Consider the antipodal map ($f(x)=-x$) on $S^n$ with $n$ even. – Ted Shifrin Mar 14 at 16:29

No. One example: take the torus $$X = \mathbb{R}^2/\mathbb{Z}^2$$. The flip-flop on the factors interchanges the closed forms $$dx$$ and $$dy$$ which are linearly independent in $$H^1(X)$$.