Atiyah–Bott fixed-point theorem: Difference between revisions

Revision as of 05:24, 6 May 2010

Revision as of 00:22, 22 July 2010

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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-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.