Novikov conjecture

Let $G$ be a discrete group and let $BG$ be a classifying space for $G$, that is, a CW complex of type $K(G,1)$. Let $M$ be a closed, connected, oriented smooth manifold of dimension $n$, with fundamental class $[M]\in H_n(M;\mathbb Q)$ and total Hirzebruch $L$-class

$$L(M)=1+\tfrac13p_1(M)+\tfrac1{45}\bigl(7p_2(M)-p_1(M)^2\bigr)+\cdots\in H^{4*}(M;\mathbb Q).$$

For a continuous map $f\colon M\to BG$ and a class $x\in H^*(BG;\mathbb Q)$, the higher signature is

$$\operatorname{sign}_x(M,f)=\bigl\langle L(M)\cup f^*x,\,[M]\bigr\rangle\in\mathbb Q .$$

For $x=1$ this is the ordinary signature of $M$, by the Hirzebruch signature theorem.

Conjecture. For every discrete group $G$ and every $x\in H^*(BG;\mathbb Q)$ the higher signature is an oriented homotopy invariant. That is, if $M'$ is another closed oriented smooth manifold and $h\colon M'\to M$ is an orientation-preserving homotopy equivalence, then

$$\operatorname{sign}_x(M',f\circ h)=\operatorname{sign}_x(M,f).$$

It suffices to prove this for finitely presented $G$, since one may replace $G$ by $\pi_1(M)$. For a given group $G$ the conjecture is equivalent to rational injectivity of the $L$-theory assembly map

$$A\colon H_n(BG;\mathbf L(\mathbb Z))\otimes\mathbb Q\;\cong\;\bigoplus_{k\in\mathbb Z} H_{n-4k}(BG;\mathbb Q)\longrightarrow L_n(\mathbb ZG)\otimes\mathbb Q .$$

The problem is to prove the conjecture for all groups, or to exhibit a group $G$, a class $x$ and a homotopy equivalence for which it fails. [1, 2, 3, 4]

Background

Novikov proved in 1965 that the rational Pontryagin classes of a manifold are topological invariants. The individual classes are not homotopy invariants, and the signature theorem shows that only the top-degree combination $\langle L(M),[M]\rangle$ is [2]. Novikov stated the conjecture around 1970, after proving it for $G=\mathbb Z^n$ [1, 2]. It identifies which characteristic numbers built from Pontryagin classes and the fundamental group are homotopy invariant.

The conjecture became a central link between surgery theory and noncommutative geometry. In surgery theory it is rational injectivity of the assembly map above. In operator algebras it follows from rational injectivity of the Baum–Connes assembly map $K_*(BG)\to K_*(C^*_rG)$, the strong Novikov conjecture. Methods from coarse geometry, $K$-theory of group $C^*$-algebras and controlled topology have all been brought to bear [2, 3, 4].