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

$d:{\mathrm{\Omega }}^p(M)\to {\mathrm{\Omega }}^{p+1}(M)$

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

$⟨\omega ,\eta ⟩=\underset{M}{\int }\omega \wedge \star \eta$

on forms. Denote by

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

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 $\mathrm{\Omega }(M)=\underset{p=0}{\overset{2l}{⨁}}{\mathrm{\Omega }}^p(M)$. 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:

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

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

$d+d^{*}=\left(\begin{array}{cc}0& D\\ D^{*}& 0\end{array}\right)$

Definition: The operator $d+d^{*}$ with the above grading respectively the above operator $D:{\mathrm{\Omega }}_{+}(M)\to {\mathrm{\Omega }}_{-}(M)$ 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:

$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(D)=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(M)$

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:

$\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(M)=\underset{M}{\int }L(p_1,\dots ,p_l)$

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.

See also

• Hirzebruch signature theorem
• Pontryagin class
• Friedrich Hirzebruch
• Michael Atiyah
• Isadore Singer

References

• Atiyah, M.F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics, 86 (2): 374–407, doi:10.2307/1970694, JSTOR 1970694
• Atiyah, M.F.; Bott, R.; Patodi, V.K. (1973), "On the heat equation and the index theorem", Inventiones Mathematicae, 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/bf01425417, S2CID 115700319
• Gilkey, P.B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes", Advances in Mathematics, 10 (3): 344–382, doi:10.1016/0001-8708(73)90119-9
• Hirzebruch, Friedrich (1995), Topological Methods in Algebraic Geometry, 4th edition, Berlin and Heidelberg: Springer-Verlag. Pp. 234, ISBN 978-3-540-58663-0
• Kaminker, Jerome; Miller, John G. (1985), "Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras" (PDF), Journal of Operator Theory, 14: 113–127