- Signature operator
Let X be a 4k dimensional compact
Riemannian manifold . The signature operator is aelliptic differential operator defined on a subspace of the space ofdifferential form s 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 ith orderdifferential 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::egin{alignat}{2} d^{star} & = (-1)^{2l(i+1) + 2l + 1} star d star\ & = - star d star end{alignat}
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
eigenspace s Omega_{pm} of auConsequently,
:d + d^{star} = egin{pmatrix} 0 & A \ A^{star} & 0 end{pmatrix}
"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
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