Atiyah–Bott fixed-point theorem: Difference between revisions

Revision as of 17:27, 28 June 2019

Revision as of 17:28, 28 June 2019

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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'' 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 |dead-url=yes |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.