Atiyah–Bott fixed-point theorem: Difference between revisions

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{{otheruseother uses|Atiyah–Bott formula}}

{{short description|Fixed-point theorem for smooth manifolds}}

{{otheruse|Atiyah–Bott formula}}

In [[mathematics]], the '''Atiyah–Bott fixed-point theorem''', proven by [[Michael Atiyah]] and [[Raoul Bott]] in the 1960s, is a general form of the [[Lefschetz fixed-point theorem]] for [[smooth manifold]]s ''M'', which uses an [[elliptic complex]] on ''M''. This is a system of [[elliptic differential operator]]s on [[vector bundle]]s, generalizing the [[de Rham complex]] constructed from smooth [[differential form]]s which appears in the original Lefschetz fixed-point theorem.

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:<math>\varphi_j \colon f^{-1}(E_j) \to E_j</math>

for each ''j'', such that the resulting maps on [[section (fiber bundle)|sections]] give rise to an [[endomorphism]] of an [[elliptic complex]] <math>T</math>. Such an endomorphism <math>T</math> has ''Lefschetz number''

:<math>L(T),</math>

which by definition is the [[alternating sum]] of its [[trace of an endomorphism|traces]] on each graded part of the homology of the elliptic complex.

The form of the theorem is then

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==History==

The early history of this result is entangled with that of the [[Atiyah–Singer index theorem]]. There was other input, as is suggested by the alternate name ''Woods Hole fixed-point theorem'' that was used in the past (referring properly to the case of isolated fixed points).<ref>{{cite web |title=Report on the Meeting to Celebrate the 35th Anniversary of the Atiyah-Bott Theorem |publisher=[[Woods Hole Oceanographic Institution]] |url=http://www.whoi.edu/mpcweb/meetings/atiyah_bott_35.html |url-status=dead |archivedate=April 30, 2001 |archiveurl=https://web.archive.org/web/20010430161636/http://www.whoi.edu/mpcweb/meetings/atiyah_bott_35.html }}</ref> A 1964 meeting at [[Woods Hole]] brought together a varied group:

<blockquote>[[Martin Eichler|Eichler]] started the interaction between fixed-point theorems and [[automorphic form]]s. [[Goro Shimura|Shimura]] played an important part in this development by explaining this to Bott at the Woods Hole conference in 1964.<ref>{{cite web |title=The work of Robert MacPherson |url=https://www.math.ubc.ca/~cass/macpherson/talk.pdf }}</ref></blockquote>

As Atiyah puts it:<ref>''Collected Papers'' III p.2.</ref>

<blockquote>[at the conference]...Bott and I learnt of a conjecture of Shimura concerning a generalization of the Lefschetz formula for holomorphic maps. After much effort we convinced ourselves that there should be a general formula of this type [...]; . </blockquote>

and they were led to a version for elliptic complexes.

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==Proofs==

In the context of [[algebraic geometry]], the statement applies for smooth and proper varieties over an algebraically closed field. This variant of the Atiyah–Bott fixed point formula was proved by {{harvtxt|Kondyrev|Prikhodko|2018}} by expressing both sides of the formula as appropriately chosen [[categorical trace]]s.

==See also==

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==References==

*{{Citation|first1=Michael F.|last1= Atiyah|author1-link=Michael Atiyah| first2= Raoul|last2= Bott | author2-link=Raoul Bott|title=A Lefschetz Fixed Point Formula for Elliptic Differential Operators|journal= [[Bulletin of the American Mathematical Society]] |volume=72 |year=1966|pages= 245–50

|url=httphttps://www.amsprojecteuclid.org/bulljournals/1966bulletin-of-the-american-mathematical-society/volume-72/issue-022/S0002A-9904Lefschetz-1966fixed-11483point-0formula-for-elliptic-differential-operators/bams/home1183527784.htmlpdf| doi=10.1090/S0002-9904-1966-11483-0|issue=2 |doi-access=free}}. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.

*{{Citation|first1=Michael F.|last1= Atiyah|author1-link=Michael Atiyah|first2= Raoul|last2= Bott |author2-link=Raoul Bott| title=A Lefschetz Fixed Point Formula for Elliptic Complexes: I |journal=[[Annals of Mathematics]] | series = Second Series|volume= 86|issue=2 |year= 1967|pages= 374–407|doi=10.2307/1970694|jstor=1970694}} and {{citation|first1=Michael F.|last1= Atiyah|author1-link=Michael Atiyah|first2= Raoul|last2= Bott |author2-link=Raoul Bott| |title=A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications

|journal=[[Annals of Mathematics]] | series = Second Series|volume=88|issue=3|year= 1968|pages=451–491|doi=10.2307/1970721|jstor=1970721}}. These gives the proofs and some applications of the results announced in the previous paper.

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==External links==

* {{cite web |first=Loring W. |last=Tu |title=The Atiyah-Bott fixed point theorem |work=The life and works of Raoul Bott |date=December 21, 2005 |url=http://brauer.math.harvard.edu/history/bott/bottbio/node18.html }}

* {{Cite web | last1 = Tu | first1 = Loring W. | title = On the Genesis of the Woods Hole Fixed Point Theorem | journal = [[Notices of the American Mathematical Society]] | volume = 62 | issue = 10 | pages = 1200–1206 | publisher = American Mathematical Society | ___location = Providence, RI | date = November 2015 | url = httphttps://www.ams.org/notices/201510/rnoti-p1200.pdf |format=PDF}}

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