Novikov conjecture

Can numbers built from how pieces of a space intersect change when maps preserve its continuous-deformation structure? Here the spaces are compact without boundary, locally like ordinary coordinates, with a chosen orientation. Higher signatures add information about loops; the comparison preserves orientation and the corresponding loop information.[1]

Can a number combining intersection geometry and loop information change under an orientation-preserving equivalence up to continuous deformation? This is the higher-signature question for closed oriented manifolds.

In dimensions divisible by four, the ordinary signature is positive directions minus negative directions of a symmetric intersection form. For the illustrative form $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, that count is $1-1=0$. Higher signatures add loop-group input: a classifying map $f:M\to B\pi$ carries it to a space representing the fundamental group $\pi$. The numbers are $\langle L(M)\smile f^*c,[M]\rangle$. Here $L(M)$ is the characteristic class used in the signature formula, $c$ is a rational cohomology class on $B\pi$, and $[M]$ supplies the orientation for evaluating the combined class.[1]

Many families are known to work, but the guarantee for all fundamental groups remains open.[1] [2]

The September 25, 2026 v8 preprint reports a rational strong-Novikov theorem for groups admitting a coarse embedding with finite complexity into the specified Banach spaces. That stronger theorem implies the higher-signature conclusion for those groups. An embedding without the finite-complexity condition does not meet its hypothesis.[2]

A dependable number lets us rule out a proposed equivalence when two spaces give different values. Equal values alone would not show that the spaces are equivalent.[1]

Ordinary signature already survives oriented homotopy equivalence, although the individual characteristic classes used in its formula need not. Higher signatures ask how much more reliable information survives after incorporating loops. This connects geometric constructions to algebraic comparison tests while remaining weaker than topological rigidity.[1]