Atiyah–Bott fixed-point theorem: Difference between revisions

Revision as of 19:53, 5 June 2012

Revision as of 20:09, 24 September 2012

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Here trace φ_''j'',''x'' means the trace of φ_''j'', at a fixed point ''x'' of ''f'', and δ(''x'') is the [[determinant]] of the endomorphism I − ''Df'' at ''x'', with ''Df'' 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 me|date=May 2012}}

==History==

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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). A 1964 meeting at [[Woods Hole]] brought together a varied group:

<blockquote>''[[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://www.math.ubc.ca/~cass/macpherson/talk.pdf</ref>. </blockquote>

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