Novikov Conjecture: Homotopy Invariance of Higher Signatures

Establish that for every compact oriented manifold M with fundamental group Γ, classifying map f: M → BΓ to the classifying space of Γ, and Hirzebruch L-class L(M) ∈ H*(M, Q), the higher signatures ⟨f*(x), L(M) ∩ [M]⟩, defined for all x ∈ H*(Γ, Q) ≅ H*(BΓ, Q), are invariant under oriented homotopy equivalences of manifolds.

Background

The paper opens by recalling the Novikov conjecture, a central problem in geometric topology and operator K-theory concerning the homotopy invariance of higher signatures. For a compact oriented manifold M with fundamental group Γ, one considers the classifying map f: M → BΓ and the Hirzebruch L-class L(M). Using the isomorphism H*(Γ, Q) ≅ H*(BΓ, Q), one defines higher signatures as rational numbers ⟨f*(x), L(M) ∩ [M]⟩ for x ∈ H*(Γ, Q).

This conjecture is historically significant and has driven developments in index theory and operator algebras. While many cases are known (e.g., for groups amenable at infinity, hyperbolic groups, and others via Kasparov theory and coarse geometric methods), the conjecture remains open in general. The current paper advances the program by establishing quantitative assembly-map estimates stable under coarse decompositions and proving the Novikov conjecture for groups with finite decomposition complexity, but it does not resolve the conjecture universally.