Signature set by Hirzebruch

The signature operator is a statement from the mathematical branch of the global analysis . It is named after the mathematician Friedrich Hirzebruch and can be understood as a special case of the Atiyah-Singer index theorem applied to the signature operator . The signature set gives a connection between the signature and the L-gender of a manifold. It was proven in 1953 by Friedrich Hirzebruch using the cobordism theory .

Statement of the signature sentence

Let be an oriented compact smooth manifold of dimension . With is the signature of , which is defined as the signature of the cut shape . Then applies ${\displaystyle X}$ ${\displaystyle n:=4k}$ ${\displaystyle \sigma (M)}$ ${\displaystyle M}$

${\displaystyle \sigma (M)=(\pi i)^{-{\frac {n}{2}}}\int _{M}L(M)\,,}$

where is the L gender of . It is defined as ${\displaystyle L(M)}$ ${\displaystyle M}$

${\displaystyle L(M):=\det \nolimits ^{\frac {1}{2}}\left({\frac {R/2}{\tanh(R/2)}}\right)\in {\mathcal {A}}^{4\bullet }(M)\,,}$

wherein the space of the - differential forms and the Riemannian curvature is. ${\displaystyle {\mathcal {A}}^{4\bullet }(M)}$ ${\displaystyle 4k}$ ${\displaystyle R}$

Signature operator and signature complex

This section defines a specific Dirac operator called the signature operator. It is closely related to the signature set. His Fredholm index is precisely the signature that appears on the left side of Hirzebruch's sentence. To define the signature operator, is - graduation on the space of differential forms required. The signature operator is now a Dirac operator that takes this graduation into account. With it a complex with two terms can be induced, which is an elliptical complex . This complex is called the signature complex. ${\displaystyle \mathbb {Z} _{2}}$

In this section, a compact, orientable Riemannian manifold of dimension is designated. The complexified cotangent bundle of is therefore noted with and the algebra of the differential forms over the complexified cotangential bundle is noted with. ${\displaystyle X}$ ${\displaystyle 2k}$ ${\displaystyle X}$ ${\displaystyle T^{*}X\otimes \mathbb {C} }$ ${\displaystyle {\mathcal {A}}(X):=\Gamma (\Lambda (T^{*}X\otimes \mathbb {C} ))}$

Graduation of the algebra of differential forms

A graduation of is induced by the involutive operator for , where is the Hodge star operator . So graduation is through with ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle {\mathcal {A}}(X)}$ ${\displaystyle \tau \phi :=i^{p(p-1)+k}\star \phi }$ ${\displaystyle \phi \in {\mathcal {A}}^{p}(T^{*}X\otimes \mathbb {C} )}$ ${\displaystyle \star }$ ${\displaystyle {\mathcal {A}}(X)={\mathcal {A}}^{+}(X)\oplus {\mathcal {A}}^{-}(X)}$

${\displaystyle {\mathcal {A}}^{+}(X):=\left(v\in {\mathcal {A}}(X)|\;\tau v=v\right)}$

and

${\displaystyle {\mathcal {A}}^{-}(X):=\left(v\in {\mathcal {A}}(X)|\;\tau v=-v\right)}$

given.

Signature operator

The signature operator is now the Dirac operator associated with the outer bundle . With the help of the external derivation , this can be specified in concrete terms. The operator adjoint to the outer derivative is denoted by. This is also called the Hodge derivative . The relation holds for compact orientable Riemann manifolds . The operator respects the previously defined graduation and is therefore the signature operator sought ${\displaystyle \mathrm {d} }$ ${\displaystyle \mathrm {d} ^{t}}$ ${\displaystyle \mathrm {d} ^{t}=-\tau \mathrm {d} \tau }$ ${\displaystyle \mathrm {d} +\mathrm {d} ^{t}\colon {\mathcal {A}}(X)\to {\mathcal {A}}(X)}$

${\displaystyle (\mathrm {d} +\mathrm {d} ^{t})\colon {\mathcal {A}}^{\pm }(X)\to {\mathcal {A}}^{\mp }(X)\,.}$

Because it is a Dirac operator, it is also elliptical and has an analytic index . This is given by the signature of the manifold . ${\displaystyle \sigma (X)}$ ${\displaystyle X}$

Signature complex

The signature complex is the complex

${\displaystyle 0\longrightarrow {\mathcal {A}}^{+}(X){\stackrel {\mathrm {d} +\mathrm {d} ^{*}}{\longrightarrow }}{\mathcal {A}}^{-}(X)\longrightarrow 0}$.

This is an elliptical complex , that is, in addition to the complex presented above, there is also the complex of its symbols

${\displaystyle 0\longrightarrow \pi ^{*}({\mathcal {A}}^{+}(X)){\stackrel {\sigma _{\mathrm {d} +\mathrm {d} ^{*}}}{\longrightarrow }}\pi ^{*}({\mathcal {A}}^{-}(X))\longrightarrow 0}$

exactly .

Idea of ​​proof and reference to the Atiyah-Singer index theorem

Using the Hodge theory, it can be shown that the signature of the compact Riemannian manifold agrees with the Fredholm index of the signature operator. From the Atiyah-Singer index theorem or its special case for Dirac operators it follows that the index of a Dirac operator can be represented as a polynomial in the Pontryagin classes . In the case of the signature operator, this polynomial is precisely the L gender. Hirzebruch himself proved the signature theorem in 1953 using methods from the cobordism theory before Atiyah and Singer published their index theorem. ${\displaystyle \sigma (M)}$ ${\displaystyle M}$

generalization

Hirzebruch's signature theorem was generalized in 1975 by Michael Francis Atiyah , Vijay Kumar Patodi, and Isadore M. Singer for a special class of Riemannian manifolds with a margin . In contrast to the Hirzebruch signature sentence, a further invariant , the -invariant , appears in this generalized version . This invariant is neither topological nor differential geometric. The invariant is calculated from the eigenvalues ​​of a differential operator associated with the signature operator, the so-called tangential signature operator. ${\displaystyle \eta }$ ${\displaystyle \eta }$

Individual evidence

1. ^ Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , p. 161.
2. ^ Hirzebruch The Signature Theorem. Reminiscences and recreation . Prospects in Mathematics, Annals of Mathematical Studies, Volume 70, 1971, pp. 3-31.
3. ^ ^{A b} Nicole Berline , Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , p. 146.
4. ↑ ^{a b c d} Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of the mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , pp. 127-129.
5. ^ Charles Nash: Differential Topology and Quantum Field Theory . 1992, p. 110 .
6. ↑ Atiyah, Patodi, Singer: Spectral asymmetry and Riemannian Geometry. I . Math. Proc. Camp. Phil. Soc. (1975), 77, 43-69.
7. ^ Peter B. Gilkey : Invariance theory, the heat equation, and the Atiyah-Singer index theorem (= Mathematics Lecture Series 11). Publish or Perish Inc., Wilmington DE 1984, ISBN 0-914098-20-9 , p. 269 ( online )