Atiyah–Bott fixed-point theorem: Difference between revisions

Revision as of 19:35, 12 May 2011

Revision as of 19:35, 12 May 2011

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:''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 ''M''×''M'', 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 ''M''×''M'', a [[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.

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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-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]].

==History==

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==References==

*M. F. Atiyah; R. Bott ''A Lefschetz Fixed Point Formula for Elliptic Differential Operators.'' Bull. Am. Math. Soc. 72 (1966), 245-50245–50. 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.