Atiyah–Bott fixed-point theorem: Difference between revisions

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{{other uses|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.

{{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.

==Formulation==

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

==History==

The early history of this result is entangled with that of the [[Atiyah-SingerAtiyah–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==

*[[Bott residue formula]]

==Notes==

Line 46 ⟶ 54:

==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. Math. Soc. 72 (1966), 245-50. This states a theorem calculatingof the LefschetzAmerican numberMathematical ofSociety]] an|volume=72 endomorphism of an elliptic complex.|year=1966|pages= 245–50

|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. 374-407 and Vol. 88, No. 3 (Nov., 1968), pp. 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}}

{{DEFAULTSORT:Atiyah-Bott Fixed-Point Theorem}}

[[Category:Differential topology]]

[[Category:Fixed-point pointstheorems]]

[[Category:MathematicalTheorems theoremsin differential topology]]