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University Park
Altoona

Thursday, November 20, 2008
4:00 pm, 210 LRC

Kate Ponto
University of Notre Dame

Traces and fixed points.

Abstract: We can associate an integer called the Lefschetz number to each endomorphism of a closed manifold. The Lefschetz fixed point theorem states that the Lefschetz number of an endomorphism with no fixed points is zero. This theorem is a consequence of the identification of the Lefschetz number, a global invariant, with a local invariant. There are many proofs of this identification, one uses the trace in a symmetric monoidal category.

For closed manifolds of dimension at least three, the Lefschetz number of an endomorphism can be refined to an invariant that is zero if and only if the endomorphism has no fixed points. A major step in the proof of this result is
the identification of a local invariant with a global invariant. I will explain how to prove this identification using the trace in a bicategory. I will also explain how to use this perspective to define equivariant and fiberwise fixed
point invariants.

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We meet weekly on Thursday afternoons, alternating between Penn State Altoona and Penn State University Park. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Fall 2008 semester: Wojtek Dorabiala and Aissa Wade.

 

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