Atiyah–Bott fixed-point theorem: Difference between revisions

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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 ''M''&times;''M'', 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 ''M''&times;''M'', 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 ''E''<submath>''j''E_j</submath>, 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 bundle|sections]] give rise to an [[endomorphism of an elliptic complex|endomorphism of the elliptic complex]] ''T''. Such a ''T'' 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.