Signature operator

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In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four.[1] It is an instance of a Dirac-type operator.

Definition in the even-dimensional case[edit]

Let be a compact Riemannian manifold of even dimension . Let

be the exterior derivative 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 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 .[2]

Definition in the odd-dimensional case[edit]

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

Hirzebruch Signature Theorem[edit]

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

where the right hand side is the topological signature (i.e. the signature of a quadratic form on defined by the cup product).

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

where is the Hirzebruch L-Polynomial,[3] and the the Pontrjagin forms on .[4]

Homotopy invariance of the higher indices[edit]

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.[5]

See also[edit]

Notes[edit]

References[edit]