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 caseedit
In the odd-dimensional case one defines the signature operator to be acting on the even-dimensional forms of .
Hirzebruch Signature Theoremedit
If , so that the dimension of is a multiple of four, then Hodge theory implies that:
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, ISBN978-3-540-58663-0
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
April 13, 2024
signature, operator, mathematics, signature, operator, elliptic, differential, operator, defined, certain, subspace, space, differential, forms, even, dimensional, compact, riemannian, manifold, whose, analytic, index, same, topological, signature, manifold, d. 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 Contents 1 Definition in the even dimensional case 2 Definition in the odd dimensional case 3 Hirzebruch Signature Theorem 4 Homotopy invariance of the higher indices 5 See also 6 Notes 7 ReferencesDefinition in the even dimensional case editLet M displaystyle M nbsp be a compact Riemannian manifold of even dimension 2l displaystyle 2l nbsp Let d Wp M Wp 1 M displaystyle d Omega p M rightarrow Omega p 1 M nbsp be the exterior derivative on i displaystyle i nbsp th order differential forms on M displaystyle M nbsp The Riemannian metric on M displaystyle M nbsp allows us to define the Hodge star operator displaystyle star nbsp and with it the inner product w h Mw h displaystyle langle omega eta rangle int M omega wedge star eta nbsp on forms Denote by d Wp 1 M Wp M displaystyle d Omega p 1 M rightarrow Omega p M nbsp the adjoint operator of the exterior differential d displaystyle d nbsp This operator can be expressed purely in terms of the Hodge star operator as follows d 1 2l p 1 2l 1 d d displaystyle d 1 2l p 1 2l 1 star d star star d star nbsp Now consider d d displaystyle d d nbsp acting on the space of all forms W M p 02lWp M displaystyle Omega M bigoplus p 0 2l Omega p M nbsp One way to consider this as a graded operator is the following Let t displaystyle tau nbsp be an involution on the space of all forms defined by t w ip p 1 l w w Wp M displaystyle tau omega i p p 1 l star omega quad quad omega in Omega p M nbsp It is verified that d d displaystyle d d nbsp anti commutes with t displaystyle tau nbsp and consequently switches the 1 displaystyle pm 1 nbsp eigenspaces W M displaystyle Omega pm M nbsp of t displaystyle tau nbsp Consequently d d 0DD 0 displaystyle d d begin pmatrix 0 amp D D amp 0 end pmatrix nbsp Definition The operator d d displaystyle d d nbsp with the above grading respectively the above operator D W M W M displaystyle D Omega M rightarrow Omega M nbsp is called the signature operator of M displaystyle M nbsp 2 Definition in the odd dimensional case editIn the odd dimensional case one defines the signature operator to be i d d t displaystyle i d d tau nbsp acting on the even dimensional forms of M displaystyle M nbsp Hirzebruch Signature Theorem editIf l 2k displaystyle l 2k nbsp so that the dimension of M displaystyle M nbsp is a multiple of four then Hodge theory implies that index D sign M displaystyle mathrm index D mathrm sign M nbsp where the right hand side is the topological signature i e the signature of a quadratic form on H2k M displaystyle H 2k M nbsp defined by the cup product The Heat Equation approach to the Atiyah Singer index theorem can then be used to show that sign M ML p1 pl displaystyle mathrm sign M int M L p 1 ldots p l nbsp where L displaystyle L nbsp is the Hirzebruch L Polynomial 3 and the pi displaystyle p i nbsp the Pontrjagin forms on M displaystyle M nbsp 4 Homotopy invariance of the higher indices editKaminker and Miller proved that the higher indices of the signature operator are homotopy invariant 5 See also editHirzebruch signature theorem Pontryagin class Friedrich Hirzebruch Michael Atiyah Isadore SingerNotes edit Atiyah amp Bott 1967 Atiyah amp Bott 1967 Hirzebruch 1995 Gilkey 1973 Atiyah Bott amp Patodi 1973 Kaminker amp Miller 1985References editAtiyah 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 Retrieved from https en wikipedia org w index php title Signature operator amp oldid 1117767679, wikipedia, wiki, book, books, library,