Novikov Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\pi$ be a discrete group and $B\pi$ a classifying space with fundamental group $\pi$ and contractible universal cover. For a closed connected oriented smooth manifold $M$, write $[M]$ for its fundamental class and $L(TM)\in H^{4*}(M;\mathbb Q)$ for the total Hirzebruch $L$-class of its tangent bundle, the characteristic class determined by the power series $x/\tanh x$. For a continuous map $f:M\to B\pi$ and a rational cohomology class $a\in H^*(B\pi;\mathbb Q)$, the higher signature is $\langle L(TM)\smile f^*a,[M]\rangle$, taking the component in degree $\dim M$.

Statement recorded in the supplied source.

For every $\pi$, every orientation-preserving homotopy equivalence $h:M'\to M$ of closed connected oriented smooth manifolds, every $f:M\to B\pi$, and every $a\in H^*(B\pi;\mathbb Q)$, is \[ \langle L(TM')\smile(f\circ h)^*a,[M']\rangle =\langle L(TM)\smile f^*a,[M]\rangle? \] Equivalently, are the higher-signature classes $f_*(L(TM)\cap[M])\in H_*(B\pi;\mathbb Q)$ invariant under oriented homotopy equivalence? [S2]

Short English statement

Are all higher signatures, obtained by pairing the manifold’s L-class with classes from a group classifying space, preserved by orientation-preserving homotopy equivalences?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1334 (TOP-004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jonathan Rosenberg, Novikov's Conjecture (2015), arXiv:1506.05408v1, Section 1, Definition 1.1 and Conjecture 1.2, pp. 3--4. https://arxiv.org/abs/1506.05408v1.