Novikov's higher-signature conjecture

Let $M^n$ be a closed oriented manifold, with fundamental class $[M]\in H_n(M,\mathbb Z)$ and total $L$-class $\mathcal L(M)\in H^*(M,\mathbb Q)$. Let $\pi$ be a discrete group, let $x\in H^*(B\pi,\mathbb Q)$, and let $u\colon M\to B\pi$ be a map to the classifying space of $\pi$. The expression $\langle \mathcal L(M)\cup u^*(x),[M]\rangle$ is the higher signature. Novikov's conjecture. If $\widehat M$ is another closed oriented manifold and $h\colon\widehat M\to M$ is an orientation-preserving homotopy equivalence, then

$$\langle \mathcal L(M)\cup u^*(x),[M]\rangle=\langle \mathcal L(\widehat M)\cup h^*\circ u^*(x),[\widehat M]\rangle.$$

Thus higher signatures should be oriented homotopy invariants. The conjecture generalizes homotopy invariance of the Hirzebruch signature and would imply strong rigidity for Pontryagin classes of manifolds with large fundamental group; it is not asserted as solved in the supplied text.