Atiyah-Bott Fixed Point Theorem

The Atiyah – Bott fixed point theorem was proved in 1966 by Michael Atiyah and Raoul Bott and generalizes Lefschetz's fixed point theorem for smooth manifolds .

Preliminary remarks

Let be a smooth, closed manifold, then the Lefschetz number ${\displaystyle M}$

${\displaystyle L(f):=\sum _{i\geq 0}(-1)^{i}\mathrm {Tr} (f_{i}|H_{i}(M,\mathbb {Q} ))}$

defined by a continuous self-mapping . The image induced by is denoted by. The Lefschetz number is well-defined, because the singular homologies of a smooth, compact manifold are finite-dimensional as vector spaces. The Atiyah-Bott Fixed Point Theorem now generalizes this statement to a class of cohomologies and gives a formula for calculating the Lefschetz number. ${\displaystyle f\colon M\to M}$ ${\displaystyle f_{i}}$ ${\displaystyle f}$ ${\displaystyle f_{i}\colon H_{i}(M,\mathbb {Q} )\to H_{i}(M,\mathbb {Q} )}$ ${\displaystyle H_{i}(M,\mathbb {Q} )}$

Be an elliptical complex . That is, is a sequence of smooth vector bundles and a sequence of (geometric) differential operators such that ${\displaystyle (E,\mathrm {d} )}$ ${\displaystyle E=(\pi _{i}\colon E_{i}\to M)_{i\in \mathbb {Z} }}$ ${\displaystyle \mathrm {d} =(\mathrm {d} _{i}\colon \Gamma (E_{i})\to \Gamma (E_{i+1}))_{i\in \mathbb {Z} }}$

1. ${\displaystyle \mathrm {d} _{i+1}\circ \mathrm {d} _{i}=0}$ applies and
2. the sequence is exact. Here referred to the vector bundle over the cotangent bundle formed by induced, and the main symbol of ${\displaystyle \ldots \to \pi ^{*}(E_{i}){\stackrel {\sigma (\mathrm {d} _{i})}{\longrightarrow }}\pi ^{*}(E_{i+1})\to \ldots }$ ${\displaystyle \pi ^{*}(E_{i})}$ ${\displaystyle T^{*}M,}$ ${\displaystyle \pi \colon T^{*}M\to M}$ ${\displaystyle \sigma (\mathrm {d} _{i})}$ ${\displaystyle \mathrm {d} _{i}.}$

Due to the first property, a cohomology can be obtained from every elliptical complex , and due to the second property, the cohomologies are finite-dimensional. Be a chain endomorphism . This induces an endomorphism of cohomologies In analogy to the Lefschetz number one defines ${\displaystyle K}$ ${\displaystyle T=(T_{i})_{i\in \mathbb {Z} }\colon (E,\mathrm {d} )\to (E,\mathrm {d} )}$ ${\displaystyle K(T)\colon K(\Gamma (E))\to K(\Gamma (E)).}$

${\displaystyle L(T)\colon {=}\sum _{i\geq 0}(-1)^{i}\mathrm {Tr} (K^{i}(T)).}$

Let be a differentiable function whose graph is transversal to the diagonal . The fixed points of are the points of intersection of the graph with the diagonal. From the transversality it follows for all fixed points that applies, where the derivative of is at the point . A lift from above an elliptical complex is a sequence of bundle homomorphisms , so that for with ${\displaystyle f\colon M\rightarrow M}$ ${\displaystyle M\times M}$ ${\displaystyle f}$ ${\displaystyle x,}$ ${\displaystyle \det(I-Df_{x})\neq 0}$ ${\displaystyle Df_{x}}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle \phi =(\phi _{i})_{i}}$ ${\displaystyle f}$ ${\displaystyle (\phi _{i}\colon f^{*}E_{i}\to E_{i})_{i}}$ ${\displaystyle T_{i}:=\Gamma \phi _{i}\circ f^{*}\colon \Gamma (E_{i})\to \Gamma (E_{i})}$

${\displaystyle \Gamma (E_{i}){\stackrel {f^{*}}{\longrightarrow }}\Gamma (f^{*}E_{i}){\stackrel {\Gamma \phi _{i}}{\longrightarrow }}\Gamma (E_{i})}$

the identity applies. In particular, there is then an endomorphism of sections in the elliptical complex . ${\displaystyle T_{i+1}\mathrm {d} _{i}=\mathrm {d} _{i}T_{i}}$ ${\displaystyle T\colon {=}(T_{i})_{i}:\Gamma (E)\to \Gamma (E)}$ ${\displaystyle (E,\mathrm {d} )}$

Atiyah-Bott fixed point formula

Let be a smooth, closed manifold and a differentiable map such that its graph is transverse to the diagonal of . Also be an elliptical complex, a lift from and the endomorphism defined by . Then the Lefschetz number is through ${\displaystyle M}$ ${\displaystyle f\colon M\to M}$ ${\displaystyle M\times M}$ ${\displaystyle (E,\mathrm {d} )}$ ${\displaystyle \phi }$ ${\displaystyle f}$ ${\displaystyle T:\Gamma (E)\to \Gamma (E)}$ ${\displaystyle T:=\Gamma \phi \circ f^{*}}$ ${\displaystyle L(T)}$

${\displaystyle L(T)=\sum _{\{x\in M|f(x)=x\}}{\frac {\sum _{i}(-1)^{i}\,\operatorname {Spur} (\phi _{i}|_{x})}{|\det(I-Df_{x})|}}}$

determined, wherein the track of at a fixed point of said and the derivation of in is. ${\displaystyle \operatorname {Spur} (\phi _{j}|_{x})}$ ${\displaystyle \phi _{j}}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle Df_{x}}$ ${\displaystyle f}$ ${\displaystyle x}$

An application of the Atiyah-Bott fixed point theorem is a simple proof of Weyl's character formula for the representation of Lie groups .

Special case

Let be the De Rham complex , here is the algebra of differential forms and the Cartan derivative . This is an elliptical complex, so the fixed point formula can be applied to this complex. Let be a differentiable map again, so that its graph is transverse to the diagonal of and the corresponding lift. Then applies to the index ${\displaystyle ({\mathcal {A}}(M),\mathrm {d} )}$ ${\displaystyle {\mathcal {A}}(M)=\Gamma (\Lambda (T^{*}M))}$ ${\displaystyle \mathrm {d} }$ ${\displaystyle f\colon M\to M}$ ${\displaystyle M\times M}$ ${\displaystyle \phi \colon \Lambda (T^{*}M)\to \Lambda (T^{*}M)}$

${\displaystyle L(T)=\sum _{\{x\in M|f(x)=x\}}{\frac {\det(I-Df_{x})}{|\det(I-Df_{x})|}}.}$

Since it is differentiable and has only isolated fixed points, this corresponds to Lefschetz's fixed point formula. ${\displaystyle f}$

history

The early history is linked to the Atiyah-Singer index rate . In a narrower sense, the first ideas arose at a conference in Woods Hole , Massachusetts in 1964 (therefore also called Woods Hole Fixed Point Theorem). Apparently the original reason comes from a remark by Martin Eichler about the connection between fixed point sentences and automorphic forms, which Gorō Shimura explained at the Raoul Bott conference . He assumed the existence of a Lefschetz fixed point theorem for holomorphic mappings.

literature

• Michael F. Atiyah, Raoul Bott: A Lefschetz Fixed Point Formula for Elliptic Differential Operators. In: Bulletin of the American Mathematical Society . Vol. 72, No. 2, 1966, pp. 245-250, ( online ).
• Michael F. Atiyah, Raoul Bott: A Lefschetz Fixed Point Formula for Elliptic Complexes: I. In: Annals of Mathematics . Series 2, Vol. 86, No. 2, Sept. 1967 pp. 374-407, doi : 10.2307 / 1970694 .
• Michael F. Atiyah, Raoul Bott: A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications. In: Annals of Mathematics. Series 2, Vol. 88, No. 3, Nov. 1968, pp. 451-491, doi : 10.2307 / 1970721 , (Evidence and Applications).
• Nicole Berline , Ezra Getzler , Michèle Vergne : Heat Kernels and Dirac Operators. Springer, Berlin et al. 2004, ISBN 3-540-20062-2 , chap. 6. 2.

Web links

• Section in essay on Bott's work by Tu, English
• Meeting on the 35th birthday of the theorem in Woods Hole, English ( Memento from January 6, 2004 in the Internet Archive )
• McPherson's lecture at the Woods Hole Conference, English (PDF file; 560 kB)