Signature operator

Let $X$ be a $4k$ dimensional compact Riemannian manifold. The signature operator is a elliptic differential operator defined on a subspace of the space of differential forms on $X$, whose analytic index is the same as the topological signature of the $X$. [Harvnb|Atiyah|Bott|1967]

Preliminaries

Let $X$ be a compact Riemannian manifold of dimension $2l$. Let $Omega^i(X)$ denote the space of $i$th order differential forms on $X$. Let

:$d\; :\; Omega^i(X)\; ightarrow\; Omega^\{i+1\}(X)$ be the exterior derivative on forms, and:

:$d^\{star\}\; :\; Omega^\{i+1\}(X)\; ightarrow\; Omega^i(X)$ its adjoint.

$d^\{star\}$ can be expressed in terms of the Hodge star operator $star$:

:

Consider $d\; +\; d^\{star\}$ acting on the space of all forms $Omega(X)\; =\; oplus\; sum\_\{i=0\}^\{2l\}Omega^\{i\}(X)$

Let $au$ be an involution on the space of "all" forms defined by:

:$au(omega)\; =\; iota^\{p(p-1)+l\}\; star\; omega,\; omega\; in\; Omega^\{p\}$

Definition

It is verified that $d\; +\; d^\{star\}$ anti-commutes with $au$ and, consequently, switches the $pm\; 1$eigenspaces $Omega\_\{pm\}$ of $au$

Consequently,

:

"Definition:" The operator: $A\; :\; Omega^+\; ightarrow\; Omega^-$ is called the signature operator. [Harvnb|Atiyah|Bott|1967]

Hirzebruch Signature Theorem

If $l\; =\; 2k$ then Hodge theory implies that:

:$mathrm\{index\}(A)\; =\; mathrm\{sign\}(X)$

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

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

$mathrm\{sign\}(X)\; =\; int\_X\; L(p\_1,ldots,p\_l)$

where $L$ is the Hirzebruch L-Polynomial, [Harvnb|Hirzebruch|1995] and the $p\_i$ the Pontrjagin forms on $X$. [Harvnb|Gilkey|1973, Harvnb|Atiyah|Bott|Patodi|1973]

ee also

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

Notes

References

*Harvard reference | last1 = Atiyah | first1 = M.F. | last2 = Bott | first2 = R. | title = A Lefschetz fixed-point formula for elliptic complexes I | journal = Annals of Mathematics | volume = 86 | year = 1967 | pages = 374-407
*Harvard reference | last1 = Atiyah | first1 = M.F. | last2 = Bott |first2= R. | last3 = Patodi |first3 = V.K.| title = On the heat equation and the index theorem |journal = Inventiones Math. | volume = 19 | year = 1973 | pages = 279-330
*Harvard reference |last1 = Gilkey | first1 = P.B. | title = Curvature and the eigenvalues of the Laplacian for elliptic complexes | journal = Advances in Mathematics | year = 1973 | volume = 10 | pages = 344-382
*Harvard reference | last = Hirzebruch | first = Friedrich | title = Topological Methods in Algebraic Geometry, 4th edition|year = 1995 | publisher = Berlin and Heidelberg: Springer-Verlag. Pp. 234|isbn= 3-540-58663-6