# Rigidity implication of the Novikov conjecture

Part of my own research is related to the Novikov conjecture, and hence I am always interested to see applications of it. When reading the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892) I saw another one of these applications (Theorem D therein).

I do not want to rephrase this particular result here, but instead I will focus on the concrete topological implication of the Novikov conjecture which is used by them in the proof. It is a rigidity result that can be found at the end of the paper A fixed point theorem for periodic maps on locally symmetric manifolds by Shmuel Weinberger and reads as follows:

Assume that $$M$$ is an oriented and rationally essential manifold and such that the Novikov conjecture holds for its fundamental group. Then any continuous map $$M’ \to M$$, where $$M’$$ is also oriented, which has positive sign on top homology, which induces an isomorphism on $$\pi_1$$ and which induces an isomorphism on all higher rational homotopy group, has actually degree $$1$$.

Of course, the ‘proof’ that Weinberger provides is extremely bare …