Research
I'm interested in
topological fixed point theory, especially the Lefschetz fixed point theorem and it converse. I study generalizations of Nielsen theory and look for ways to describe these invariants that make connections between invariants or to other areas of algebraic topology. I primarily approach these invariants though duality in category
theory.
- Fixed point
theory and trace for bicategories
The Lefschetz fixed point theorem follows easily from the identification
of the Lefschetz number with the fixed point index. This identification
is a consequence of the functoriality of the trace in symmetric monoidal
categories.
There are refinements of
the Lefschetz number and the fixed point index that
give a converse to the Lefschetz fixed point theorem.
An important part of this theorem is the identification of these
different invariants.
We define a generalization of the trace in symmetric monoidal
categories to a trace in bicategories with shadows. We show the
invariants used in the converse of the Lefschetz fixed point theorem
are examples of this trace and that the functoriality of the trace provides
some of the necessary identifications.
The methods used here do not use simplicial techniques and so generalize
readily to other contexts.
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