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 …