Novikov's conjecture on homotopy invariance of higher signatures

Let $\Gamma$ be a finitely generated group and let $M$ be a compact oriented manifold with fundamental group $\Gamma$. Let $B_{\Gamma}$ be the classifying space for $\Gamma$, and let $f_{\Gamma}:M\to B_{\Gamma}$ be the classifying map. For $x\in H^*(\Gamma,\mathbb Q)$, regard $f^*_{\Gamma}(x)$ as an element of $H^*(M,\mathbb Q)$ using the natural isomorphism $H^*(\Gamma,\mathbb Q)\cong H^*(B_{\Gamma},\mathbb Q)$. If $\mathbb{L}(M)\in H^*(M,\mathbb Q)$ denotes the Pontryagin-Hirzebruch class and $[M]$ the fundamental class, the higher signatures are the rational numbers

$$\langle f^*_{\Gamma}(x),\mathbb{L}(M)\cap [M]\rangle.$$

Novikov's conjecture. The higher signatures are oriented homotopy invariant. The conjecture is a central problem in index theory and topology. In the paper's setting, it is related to quantitative assembly map estimates, and the abstract states that the authors prove it for groups with finite decomposition complexity; the general conjecture remains unresolved.