Math Conjectures

Novikov higher-signature conjecture

OPENLandmarkConjectureProposed 1970 · Full conjecture

Canonical statement

Let MM be a closed connected oriented smooth manifold, let Γ=π1(M)\Gamma=\pi_{1}(M), let u:M→BΓu:M\to B\Gamma classify its universal cover, and let α∈H∗(BΓ;Q)\alpha\in H^{*}(B\Gamma;\mathbb Q). The higher signature
⟨L(M)⌣u∗α,[M]⟩∈Q \left\langle L(M)\smile u^{*}\alpha,[M]\right\rangle\in\mathbb Q
is invariant under orientation-preserving homotopy equivalence. Here L(M)L(M) is the total Hirzebruch LL-class of TMTM, and only the top-degree component is evaluated.
View source LaTeX
Let \(M\) be a closed connected oriented smooth manifold, let \(\Gamma=\pi_{1}(M)\), let \(u:M\to B\Gamma\) classify its universal cover, and let \(\alpha\in H^{*}(B\Gamma;\mathbb Q)\). The higher signature \[ \left\langle L(M)\smile u^{*}\alpha,[M]\right\rangle\in\mathbb Q \] is invariant under orientation-preserving homotopy equivalence. Here \(L(M)\) is the total Hirzebruch \(L\)-class of \(TM\), and only the top-degree component is evaluated.
GEO-TOP-007:[Novikov1970PontryaginClasses][FerryRanickiRosenberg1995Novikov][Weinberger2024HigherSignatures]
Geometry and topologyAnalysis and functional analysis

Notes

The Novikov conjecture concerns the homotopy invariance of higher signatures. For a closed connected oriented smooth manifold MM with fundamental group Γ\Gamma, classifying map u:M→BΓu:M\to B\Gamma, and a rational cohomology class α\alpha of BΓB\Gamma, the higher signature is the rational number obtained by evaluating L(M)⌣u∗αL(M)\smile u^{*}\alpha against the fundamental class, where L(M)L(M) is the total Hirzebruch LL-class. The conjecture, formulated by Novikov around 1970 in his study of Hermitian analogues of KK-theory [Novikov1970PontryaginClasses], predicts that all these numbers are invariant under orientation-preserving homotopy equivalence. For α=1\alpha=1 this recovers the classical homotopy invariance of the signature.

The standard strategy is to prove that a rational assembly map is injective, and the conjecture is by now known for many classes of groups through two complementary routes: controlled topology and surgery theory on one side, and operator KK-theory and index theory on the other. The state of the art in the mid-1990s is documented in [FerryRanickiRosenberg1995Novikov], with later developments tied to higher invariants and the surgery structure set [Weinberger2024HigherSignatures].

For general finitely presented groups the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.