Signature operator

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. It is an instance of a Dirac-type operator.

Definition in the even-dimensional case
Let $$M$$ be a compact Riemannian manifold of even dimension $$2l$$. Let


 * $$ d : \Omega^p(M)\rightarrow \Omega^{p+1}(M) $$

be the exterior derivative on $$i$$-th order differential forms on $$M$$. The Riemannian metric on $$M$$ allows us to define the Hodge star operator $$\star$$ and with it the inner product


 * $$\langle\omega,\eta\rangle=\int_M\omega\wedge\star\eta$$

on forms. Denote by


 * $$ d^*: \Omega^{p+1}(M)\rightarrow \Omega^p(M) $$

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


 * $$d^*= (-1)^{2l(p+1) + 2l + 1} \star d  \star=  - \star d  \star$$

Now consider $$d + d^*$$ acting on the space of all forms $$\Omega(M)=\bigoplus_{p=0}^{2l}\Omega^{p}(M)$$. One way to consider this as a graded operator is the following: Let $$\tau$$ be an involution on the space of all forms defined by:


 * $$ \tau(\omega)=i^{p(p-1)+l}\star \omega\quad,\quad\omega \in \Omega^p(M) $$

It is verified that $$d + d^*$$ anti-commutes with $$\tau$$ and, consequently, switches the $$(\pm 1) $$-eigenspaces $$\Omega_{\pm}(M)$$   of $$\tau$$

Consequently,


 * $$ d + d^* = \begin{pmatrix} 0 & D \\ D^* & 0 \end{pmatrix}$$

Definition: The operator $$ d + d^*$$ with the above grading respectively the above operator $$D: \Omega_+(M) \rightarrow \Omega_-(M) $$ is called the signature operator of $$M$$.

Definition in the odd-dimensional case
In the odd-dimensional case one defines the signature operator to be $$i(d+d^*)\tau$$ acting on the even-dimensional forms of $$M$$.

Hirzebruch Signature Theorem
If $$ l = 2k $$, so that the dimension of $$M$$ is a multiple of four, then Hodge theory implies that:


 * $$\mathrm{index}(D) = \mathrm{sign}(M) $$

where the right hand side is the topological signature (i.e. the signature of a quadratic form on $$H^{2k}(M)\ $$ defined by the cup product).

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


 * $$\mathrm{sign}(M) = \int_M L(p_1,\ldots,p_l) $$

where $$L$$ is the Hirzebruch L-Polynomial, and the $$p_i\ $$ the Pontrjagin forms on $$M$$.

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