Atiyah–Bott fixed-point theorem: Difference between revisions

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{{other uses|Atiyah–Bott formula}}

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

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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The idea is to find the correct replacement for the [[Lefschetz number]], which in the classical result is an integer counting the correct contribution of a [[Fixed point (mathematics)|fixed point]] of a smooth mapping

<math> f\colon M \to M.</math>

:''f'':''M'' → ''M''.

Intuitively, the fixed points are the points of intersection of the [[graph of a function|graph]] of ''f'' with the diagonal (graph of the identity mapping) in ''<math>M''&\times;'' M''</math>, and the Lefschetz number thereby becomes an [[intersection number]]. The Atiyah–Bott theorem is an equation in which the [[Sides of an equation|LHS]] must be the outcome of a global topological (homological) calculation, and the [[Sides of an equation|RHS]] a sum of the local contributions at fixed points of ''f''.

Counting [[codimension]]s in ''<math>M''&\times;'' M''</math>, a [[Transversality (mathematics)|transversality]] assumption for the graph of ''f'' and the diagonal should ensure that the fixed point set is zero-dimensional. Assuming ''M'' a [[closed manifold]] should ensure then that the set of intersections is finite, yielding a finite summation as the RHS of the expected formula. Further data needed relates to the elliptic complex of vector bundles <math>E_j</math>, namely a [[bundle map]]

Further data needed relates to the elliptic complex of vector bundles ''E''_''j'', namely a [[bundle map]] from

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

:φ_''j'':''f''^&minus;1 ''E''_''j'' → ''E''_''j''

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

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

:''L''(''T'')

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

:''<math>L''(''T'') = Σ\sum_x \left(Σ\sum_j (&minus;-1)^{''^j''} \mathrm{trace}\, φ_{''\varphi_{j'',''x''}}\right)/δ\delta(''x'').</math>

Here trace φ<submath>''\varphi_{j'',''x''}</submath> means the trace of φ<submath>''\varphi_{j'',}</submath> at a fixed point ''x'' of ''f'', and δ<math>\delta(''x'')</math> is the [[determinant]] of the endomorphism <math>I &minus; ''-Df''</math> at ''x'', with ''<math>Df''</math> the derivative of ''f'' (the non-vanishing of this is a consequence of transversality). The outer summation is over the fixed points ''x'', and the inner summation over the index ''j'' in the elliptic complex.

Specializing the Atiyah–Bott theorem to the de Rham complex of smooth differential forms yields the original Lefschetz fixed-point formula. A famous application of the Atiyah–Bott theorem is a simple proof of the [[Weyl character formula]] in the theory of [[Lie groups]].{{clarify|date=May 2012}}

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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'' [http://www.whoi.edu/mpcweb/meetings/atiyah_bott_35.html] 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>http{{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.

In the recollection of [[William Fulton (mathematician)|William Fulton]], who was also present at the conference, the first to produce a proof was [[Jean-Louis Verdier]].

==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==

*M.{{Citation|first1=Michael F.|last1= Atiyah;|author1-link=Michael R.Atiyah| first2= Raoul|last2= Bott ''| author2-link=Raoul Bott|title=A Lefschetz Fixed Point Formula for Elliptic Differential Operators.''|journal= Bull.[[Bulletin Am.of Math.the Soc.American Mathematical Society]] |volume=72 (|year=1966),|pages= 245–50. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.

|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-72/issue-2/A-Lefschetz-fixed-point-formula-for-elliptic-differential-operators/bams/1183527784.pdf| 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.

*M. F. Atiyah; R. Bott ''A Lefschetz Fixed Point Formula for Elliptic Complexes:'' [http://links.jstor.org/sici?sici=0003-486X%28196709%292%3A86%3A2%3C374%3AALFPFF%3E2.0.CO%3B2-N ''A Lefschetz Fixed Point Formula for Elliptic Complexes: I''] [http://links.jstor.org/sici?sici=0003-486X%28196811%292%3A88%3A3%3C451%3AALFPFF%3E2.0.CO%3B2-B ''II. Applications''] The Annals of Mathematics 2nd Ser., Vol. 86, No. 2 (Sep., 1967), pp.&nbsp;374–407 and Vol. 88, No. 3 (Nov., 1968), pp.&nbsp;451–491. These gives the proofs and some applications of the results announced in the previous paper.

*{{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.

*{{Citation|title=Categorical Proof of Holomorphic Atiyah–Bott Formula|last1=Kondyrev|first1=Grigory|last2=Prikhodko|first2=Artem|journal=J. Inst. Math. Jussieu|year=2018|pages=1–25|doi=10.1017/S1474748018000543|arxiv=1607.06345}}

==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 = https://www.ams.org/notices/201510/rnoti-p1200.pdf |format=PDF}}

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