Signature (topology)
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| Signature (topology) | |
|---|---|
| Name | Signature (topology) |
| Field | Topology, Algebraic topology, Differential topology, Algebraic geometry |
| Introduced | 1930s |
| Notable | Friedrich Hirzebruch, John Milnor, René Thom, Michael Atiyah, Isadore Singer |
- Definition and basic properties
- Signature of quadratic forms and symmetric bilinear forms
- Signature in topology and manifolds (Hirzebruch signature theorem)
- Relation to intersection forms and 4-manifolds
- Signature and cobordism theory
- Computation methods and examples
- Generalizations and applications
Signature (topology) The signature in topology is an integer invariant assigned to a compact oriented smooth manifold that captures the difference between positive and negative definite directions of a nondegenerate symmetric bilinear intersection form. Introduced in the work of Hermann Weyl, refined by Friedrich Hirzebruch and used extensively by John Milnor, René Thom, Michael Atiyah, and Isadore Singer, the signature links algebraic structures like quadratic forms and symmetric bilinear forms to geometric and topological properties of manifolds, cobordism groups, and index theorem phenomena.
Definition and basic properties
For a compact oriented 4k-dimensional manifold M, the middle-dimensional cohomology H^{2k}(M; ℝ) carries a nondegenerate symmetric bilinear pairing given by cup product and evaluation on the fundamental class; the signature is the integer difference b_+ - b_- where b_+, b_- are the dimensions of maximal positive and negative definite subspaces. This invariant is stable under oriented homotopy equivalence and additive under disjoint union and connected sum, and it defines a homomorphism from oriented cobordism groups studied by René Thom to ℤ. Important properties are established in the work of Friedrich Hirzebruch, John Milnor, Raoul Bott, and Lazarus Fuchs.
Signature of quadratic forms and symmetric bilinear forms
Algebraically, the signature of a real symmetric matrix or a nondegenerate quadratic form over ℝ equals the inertia index given by Sylvester's law of inertia, a classical result connected to Carl Gustav Jacob Jacobi and developed further by James Joseph Sylvester. The connection to topology arises by identifying the intersection pairing on H^{2k}(M; ℝ) with such a bilinear form; foundational algebraic treatments appear alongside work by Emmy Noether and Herbrand. For rational and integral coefficients, one considers the Witt group and the classification of quadratic forms over ℚ and ℤ, linking to arithmetic questions addressed by Helmut Hasse, John Tate, and Johan de Jong.
Signature in topology and manifolds (Hirzebruch signature theorem)
The Hirzebruch signature theorem of Friedrich Hirzebruch expresses the signature of a closed oriented 4k-dimensional smooth manifold as the evaluation of a polynomial in the Pontryagin classes—the L-genus—on the fundamental class. This bridges characteristic class theory developed by Shiing-Shen Chern, Atiyah–Bott–Patodi style index results, and the Atiyah–Singer index theorem by Michael Atiyah and Isadore Singer. The theorem plays a central role in connecting global analysis, exemplified in the work of Atiyah, Singer, and Raoul Bott, with algebraic geometry via Grothendieck–Riemann–Roch and the study of characteristic numbers by Alexander Grothendieck and André Weil.
Relation to intersection forms and 4-manifolds
In dimension four, the intersection form on H^2(M; ℤ) is a primary invariant for classification problems of smooth and topological 4-manifolds; results by Michael Freedman and Simon Donaldson show deep links between the signature, unimodular forms, and the existence of smooth structures. The signature detects phenomena such as exotic smooth structures pioneered by C. H. Taubes and used in gauge theory approaches by Edward Witten and Peter Kronheimer. The role of even and odd forms ties into classification theorems of H. S. M. Coxeter style lattice theory and to arithmetic lattices studied by John Conway and N. J. A. Sloane.
Signature and cobordism theory
The signature gives a homomorphism from the oriented cobordism ring studied by René Thom and later algebraic topologists like Ralph Fox and Daniel Quillen to ℤ, and it refines to maps on bordism groups with additional structure (e.g., complex bordism by Quillen). Hirzebruch used this to relate genera and formal group laws, an approach developed further by J. P. Serre and Jean-Pierre Serre in arithmetic contexts and by Douglas Ravenel in stable homotopy theory. Cobordism detection results by William Browder and Frank Adams connect signature considerations to stable homotopy and Adams spectral sequence methods.
Computation methods and examples
Computational techniques for the signature include diagonalization of intersection matrices for explicit manifolds like complex projective spaces ℂP^n and products of spheres, application of the Hirzebruch signature theorem using Pontryagin classes for smooth algebraic varieties studied by Alexander Grothendieck and Kunihiko Kodaira, and surgery and plumbing methods popularized by M. Kervaire and John Milnor. Signature calculations appear in examples involving K3 surfaces (work of Igor Krichever and Shing-Tung Yau), complex surfaces analyzed by Oscar Zariski and Kunihiko Kodaira, and link complements explored by William Thurston and Rolfsen. Computational algebra systems and lattice-theoretic methods by John Conway facilitate explicit matrix-based evaluations.
Generalizations and applications
Generalizations include signatures for singular varieties via intersection homology by Mark Goresky and Robert MacPherson, equivariant signatures in the presence of group actions studied by Raoul Bott and Loring Tu, families index signature theorems by Alain Connes and Atiyah–Singer collaborators, and analytic signatures associated to signature operators on Dirac operator-type complexes by Atiyah, Patodi, and Singer. Applications span classification problems in low-dimensional topology addressed by Simon Donaldson and Michael Freedman, invariants in algebraic geometry influenced by Friedrich Hirzebruch and Alexander Grothendieck, and interactions with mathematical physics via Edward Witten and Seiberg–Witten theory. The signature continues to inform research in surgery theory (work of C. T. C. Wall), L-theory in algebraic K-theory by André Ranicki, and modern studies of manifolds with singularities by Goresky and MacPherson.