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In [[mathematics]], the '''signature operator''' is an [[elliptic differential operator]] defined on a certain subspace of the space of [[differential form]]s on an even-dimensional [[closed manifold|compact]] [[Riemannian manifold]], whose [[Atiyah-Singer_index_theorem#The_analytical_index|analytic index]] is the same as the [[Signature (topology)|topological signature]] of the manifold if the dimension of the manifold is a multiple of four.<ref>{{Harvnb|Atiyah|Bott|1967}}</ref> It is an instance of a Dirac-type operator.
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==Definition in the even-dimensional case==
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Let <math>M</math> be a compact [[Riemannian manifold]] of even dimension <math>2l</math>. Let
 
:<math> d : \Omega^p(M)\rightarrow \Omega^{p+1}(M) </math>
 
be the [[exterior derivative]] on <math>i</math>-th order [[differential forms]] on <math>M</math>. The Riemannian metric on <math>M</math> allows us to define the [[Hodge star operator]] <math>\star</math> and with it the inner product
 
:<math>\langle\omega,\eta\rangle=\int_M\omega\wedge\star\eta</math>
 
on forms. Denote by
 
:<math> d^*: \Omega^{p+1}(M)\rightarrow \Omega^p(M) </math>
 
the [[Hermitian adjoint|adjoint operator]] of the exterior differential <math>d</math>. This operator can be expressed purely in terms of the Hodge star operator as follows:
 
:<math>d^*=  (-1)^{2l(p+1) + 2l + 1} \star d  \star=  - \star d  \star</math>
 
Now consider <math>d + d^*</math> acting on the space of all forms <math>\Omega(M)=\bigoplus_{p=0}^{2l}\Omega^{p}(M)</math>.
One way to consider this as a graded operator is the following: Let <math>\tau</math> be an [[Involution (mathematics)#Linear algebra|involution]] on the space of ''all'' forms defined by:
 
:<math> \tau(\omega)=i^{p(p-1)+l}\star \omega\quad,\quad\omega \in \Omega^p(M) </math>
 
It is verified that <math>d + d^*</math> anti-commutes with <math>\tau</math> and, consequently, switches the <math>(\pm 1) </math>-[[eigenspace]]s <math>\Omega_{\pm}(M)</math>  of <math>\tau</math>
 
Consequently,
 
:<math> d + d^* = \begin{pmatrix} 0 & D \\ D^* & 0 \end{pmatrix}</math>
 
'''Definition:''' The operator <math> d + d^*</math> with the above grading respectively the above operator <math>D: \Omega_+(M) \rightarrow \Omega_-(M) </math> is called the '''signature operator''' of <math>M</math>.<ref>{{Harvnb|Atiyah|Bott|1967}}</ref>
 
==Definition in the odd-dimensional case==
 
In the odd dimensional case one defines the signature operator to be <math>i(d+d^*)\tau</math> acting
on the even-dimensional forms of <math>M</math>.
 
==Hirzebruch Signature Theorem==
 
If <math> l = 2k </math>, so that the dimension of <math>M</math> is a multiple of four, then [[Hodge theory]] implies that:
 
:<math>\mathrm{index}(D) = \mathrm{sign}(M) </math>
 
where the right hand side is the [[Signature (topology)|topological signature]] (''i.e.'' the [[signature of a quadratic form]] on <math>H^{2k}(M)\ </math> defined by the [[cup product]]).
 
The ''Heat Equation'' approach to the [[Atiyah-Singer index theorem]] can then be used to show that:
 
:<math>\mathrm{sign}(M) = \int_M L(p_1,\ldots,p_l) </math>
 
where <math>L</math> is the [[Hirzebruch_L-polynomial#L_genus_and_the_Hirzebruch_signature_theorem|Hirzebruch L-Polynomial]],<ref>{{Harvnb|Hirzebruch|1995}}</ref> and the <math>p_i\ </math> the [[Pontryagin class|Pontrjagin form]]s on <math>M</math>.<ref>{{Harvnb|Gilkey|1973}}, {{Harvnb|Atiyah|Bott|Patodi|1973}}</ref>
 
==Homotopy invariance of the higher indices==
 
Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.<ref>{{Harvnb|Kaminker|Miller|1985}}</ref>
 
==See also==
*[[Hirzebruch signature theorem]]
*[[Pontryagin class]]
*[[Friedrich Hirzebruch]]
*[[Michael Atiyah]]
*[[Isadore Singer]]
==Notes==
{{reflist|2}}
==References==
*{{Citation | 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&ndash;407}}
*{{Citation | 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&ndash;330}}
*{{Citation |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&ndash;382}}
*{{Citation | 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}}
*{{Citation | last1 = Kaminker | first1 = Jerome | last2 = Miller | first2 = John G. | title = Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras | journal = Journal of Operator Theory | year = 1985 | volume = 14 | pages = 113&ndash;127 | url = http://jot.theta.ro/jot/archive/1985-014-001/1985-014-001-006.pdf}}
 
[[Category:Elliptic partial differential equations]]

Latest revision as of 14:32, 1 January 2015

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