Math 2.0

Global analysis, analysis on manifolds Index theory Theorem

G-signature theorem

Builds on Hirzebruch signature theoremIntersection formSignature of a quadratic formEquivariant index theoremSignature operator

Statement

Let MM be a closed oriented smooth manifold of dimension 4k4k. Let a compact Lie group GG act smoothly on MM, and let g∈Gg \in G act by an orientation-preserving diffeomorphism. The intersection form on H2k(M;ℝ)H^{2k}(M; \R) is gg-invariant, so H2k(M;ℝ)=H+⊕H−H^{2k}(M; \R) = H^+ \oplus H^- with gg-invariant subspaces on which the form is positive and negative definite. The gg-signature is sign⁡(g,M)=tr⁡(g∣H+)−tr⁡(g∣H−)\operatorname{sign}(g, M) = \operatorname{tr}(g \mid H^+) - \operatorname{tr}(g \mid H^-). Let MgM^g be the fixed point set of gg, a disjoint union of closed submanifolds FF. Then sign⁡(g,M)=∑F⟨L(g,F),[F]⟩.\operatorname{sign}(g, M) = \sum_{F} \langle L(g, F), [F] \rangle. Here each L(g,F)∈H∗(F;ℂ)L(g, F) \in H^*(F; \C) is a characteristic class. It is determined by the tangent bundle of FF, the normal bundle of FF in MM and the action of gg on that normal bundle. In particular:

  1. If gg has no fixed points, then sign⁡(g,M)=0\operatorname{sign}(g, M) = 0.
  2. If gg is the identity, the formula is the Hirzebruch signature theorem.

Remarks

The theorem is the equivariant index theorem for the signature operator, whose equivariant index evaluated at gg is sign⁡(g,M)\operatorname{sign}(g, M); the topological side localizes to the fixed point set. For an isolated fixed point the contribution is an explicit product of cotangents of half the rotation angles of gg on the tangent space, up to a power of ii. Example: let a finite group Γ\Gamma act freely and preserving orientation on MM. Then every g≠1g \ne 1 has sign⁡(g,M)=0\operatorname{sign}(g, M) = 0, and averaging the characters gives σ(M)=∣Γ∣ σ(M/Γ)\sigma(M) = \abs{\Gamma} \, \sigma(M / \Gamma), where σ\sigma is the signature of the intersection form.

Going deeper

For a student

The GG-signature theorem of Atiyah and Singer is a basic tool in the study of group actions on manifolds. It gives obstructions to actions with prescribed fixed point data, through the cotangent expressions that appear at isolated fixed points. It is a model for later fixed point formulas from the equivariant index theorem.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

Show the path as a graph
Analytic index, concept, no evidence levelAnalytic indexconceptCharacter of a representation, concept, no evidence levelCharacter of a repres…conceptCobordism, concept, no evidence levelCobordismconceptCompact space, concept, no evidence levelCompact spaceconceptContinuous map, concept, no evidence levelContinuous mapconceptElliptic complex, concept, no evidence levelElliptic complexconceptElliptic operator on a compact manifold, concept, no evidence levelElliptic operator on …conceptEuler characteristic, concept, no evidence levelEuler characteristicconceptIntersection form, concept, no evidence levelIntersection formconceptLefschetz number, concept, no evidence levelLefschetz numberconceptLie group, concept, no evidence levelLie groupconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptPontryagin class, concept, no evidence levelPontryagin classconceptRepresentation of a Lie group, concept, no evidence levelRepresentation of a L…conceptRetract, concept, no evidence levelRetractconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSignature operator, concept, no evidence levelSignature operatorconceptSimplicial complex, concept, no evidence levelSimplicial complexconceptSingular homology, concept, no evidence levelSingular homologyconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTopological index, concept, no evidence levelTopological indexconceptVector bundle, concept, no evidence levelVector bundleconceptLefschetz fixed-point theorem, statement, no evidence levelLefschetz fixed-point…statementHirzebruch signature theorem, statement, no evidence levelHirzebruch signature …statementAtiyah–Bott fixed-point theorem, statement, no evidence levelAtiyah–Bott fixed-poi…statementEquivariant index theorem, statement, no evidence levelEquivariant index the…statementG-signature theorem, statement, no evidence levelG-signature theoremstatement
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.
  • no evidence level
  1. Analytic index concept
  2. Character of a representation concept
  3. Cobordism concept
  4. Compact space concept
  5. Continuous map concept
  6. Elliptic complex concept
  7. Elliptic operator on a compact manifold concept
  8. Euler characteristic concept
  9. Intersection form concept
  10. Lefschetz number concept
  11. Lie group concept
  12. Orientation of a manifold concept
  13. Pontryagin class concept
  14. Representation of a Lie group concept
  15. Retract concept
  16. Signature of a quadratic form concept
  17. Signature operator concept
  18. Simplicial complex concept
  19. Singular homology concept
  20. Smooth manifold concept
  21. Topological index concept
  22. Vector bundle concept
  23. Lefschetz fixed-point theorem statement
  24. Hirzebruch signature theorem statement
  25. Atiyah–Bott fixed-point theorem statement
  26. Equivariant index theorem statement
  27. G-signature theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence
About this page
TagW6DAWQ6H48
URIm2://W6DAWQ6H48
Kindstatement, theorem
WritingW0 stub
MSC58J20, 57S15, 57R20
StatusThis copy of the site has no build data. Run m2 status W6DAWQ6H48.
AboutHirzebruch signature theorem
UsesHirzebruch signature theorem, Intersection form, Signature of a quadratic form, Equivariant index theorem, Signature operator

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