In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential form

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s on an even-dimensional compact

Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

 Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

, whose analytic index is the same as the topological signature

Signature (topology)

In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....

 of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator.

Definition in the even-dimensional case

Let be a compact Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 of even dimension . Let


be the exterior derivative

Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

 on -th order differential forms on . The Riemannian metric on allows us to define the Hodge star operator  and with it the inner product


on forms. Denote by


the adjoint operator

Hermitian adjoint

In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...

 of the exterior differential . This operator can be expressed purely in terms of the Hodge star operator as follows:


Now consider acting on the space of all forms .
One way to consider this as a graded operator is the following: Let be an involution on the space of all forms defined by:


It is verified that anti-commutes with and, consequently, switches the -eigenspaces of

Consequently,


Definition: The operator with the above grading respectively the above operator is called the signature operator of .

Definition in the odd-dimensional case

In the odd dimensional case one defines the signature operator to be acting
on the even-dimensional forms of .

Hirzebruch Signature Theorem

If , so that the dimension of is a multiple of four, then Hodge theory

Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...

 implies that:


where the right hand side is the topological signature

Signature (topology)

In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....

 (i.e. the signature of the quadratic form on defined by the cup product

Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...

).

The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:


where is the Hirzebruch L-Polynomial, and the the Pontrjagin form

Pontryagin class

In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...

s on .

Homotopy invariance of the higher indices

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.

See also

• Hirzebruch signature theorem
• Pontryagin class
  Pontryagin class
  In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
  
  
• Friedrich Hirzebruch
  Friedrich Hirzebruch
  Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
  
  
• Michael Atiyah
  Michael Atiyah
  Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
  
  
• Isadore Singer
  Isadore Singer
  Isadore Manuel Singer is an Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology...