Math 2.0

Manifolds and cell complexes Cobordism and surgery Conjecture

Novikov conjecture

Builds on Classifying space of a groupHomotopy equivalenceSignaturePontryagin classHirzebruch signature theoremSingular cohomologyFundamental classHyperbolic group

Statement

Let GG be a discrete group and let BGBG be its classifying space. Let MM be a closed oriented smooth manifold with fundamental class [M][M], and let f ⁣:M→BGf \colon M \to BG be a continuous map. Let L(M)∈H∗(M;ℚ)L(M) \in H^*(M; \Q) be its Hirzebruch LL-class, a polynomial in the rational Pontryagin classes. For x∈H∗(BG;ℚ)x \in H^*(BG; \Q), the higher signature is sign⁡x(M,f)=⟨L(M)⌣f∗(x),[M]⟩∈ℚ,\operatorname{sign}_x(M, f) = \ip{L(M) \smile f^*(x)}{[M]} \in \Q, computed in rational singular cohomology. The conjecture states that higher signatures are oriented homotopy invariants. That is, for every orientation-preserving homotopy equivalence h ⁣:N→Mh \colon N \to M of closed oriented smooth manifolds and every x∈H∗(BG;ℚ)x \in H^*(BG; \Q), sign⁡x(N,f∘h)=sign⁡x(M,f).\operatorname{sign}_x(N, f \circ h) = \operatorname{sign}_x(M, f).

Remarks

For x=1x = 1, the higher signature is the ordinary signature of MM by the Hirzebruch signature theorem. It is a homotopy invariant because it is defined from the cup product pairing on H∗(M;ℚ)H^*(M; \Q). The individual Pontryagin classes are not homotopy invariants, so the conjecture singles out special combinations of them.

The conjecture is open in general. It is known for many classes of groups. Among them are free abelian groups, discrete subgroups of connected Lie groups (Kasparov), and hyperbolic groups (Connes and Moscovici). Yu proved it for groups that coarsely embed into a Hilbert space and have a classifying space with a finite model; the finiteness hypothesis was later removed. The main methods go through operator K-theory: injectivity of the rational assembly map in the Baum–Connes conjecture, or of the coarse assembly map, implies the conjecture.

Reading path

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Classifying space of a group, concept, no evidence levelClassifying space of …conceptCobordism, concept, no evidence levelCobordismconceptFundamental class, concept, no evidence levelFundamental classconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptHyperbolic group, concept, no evidence levelHyperbolic groupconceptIntersection form, concept, no evidence levelIntersection formconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptPontryagin class, concept, no evidence levelPontryagin classconceptSignature, concept, no evidence levelSignatureconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSingular cohomology, concept, no evidence levelSingular cohomologyconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptHirzebruch signature theorem, statement, no evidence levelHirzebruch signature …statementNovikov conjecture, conjecture, no evidence levelNovikov conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Classifying space of a group concept
  2. Cobordism concept
  3. Fundamental class concept
  4. Homotopy equivalence concept
  5. Hyperbolic group concept
  6. Intersection form concept
  7. Orientation of a manifold concept
  8. Pontryagin class concept
  9. Signature concept
  10. Signature of a quadratic form concept
  11. Singular cohomology concept
  12. Smooth manifold concept
  13. Hirzebruch signature theorem statement
  14. Novikov conjecture conjecture

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Novikov_conjecture
About this page
TagKEF3ZDF0QH
URIm2://KEF3ZDF0QH
Kindconjecture
WritingW0 stub
MSC57R67, 19J25, 20F65
StatusThis copy of the site has no build data. Run m2 status KEF3ZDF0QH.
AboutClassifying space of a group
UsesClassifying space of a group, Homotopy equivalence, Signature, Pontryagin class, Hirzebruch signature theorem, Singular cohomology, Fundamental class, Hyperbolic group
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Changed3 November 2026

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