The Reidemeister trace, developed by Reidemeister and Wecken, is an algebraic invariant of a self-map of a “finite” topological space. It gives information about the existence or nonexistence of fixed points, and refines both the Lefschetz number and Nielsen number?.
Suppose is a closed manifold and a self-map. Deform so that it has isolated fixed points. We say that two fixed points and are in the same fixed-point class? if there is a path from to such that is homotopic to rel the endpoints ( and ). Let denote the free abelian group on the set of fixed-point classes. Then the Reidemeister trace of is the formal sum
where is the index of the fixed point of . This definition is homotopy invariant.
An equivalent definition can be obtained algebraically, or category-theoretically using the bicategorical trace.
The sum of all the coefficients in the Reidemeister trace is the Lefschetz number .
The number of nonzero coefficients in the Reidemeister trace is the Nielsen number? .
If is a closed manifold of dimension at least 3, and , then is homotopic to a map without fixed points. Thus, the Reidemeister trace supports a converse to the Lefschetz fixed-point theorem?.
The Reidemeister trace was introduced in
A modern treatment is in
Peter Staecker, The Reidemeister trace: computation by nilpotentization and extension to coincidence theory (PhD thesis)
Peter Staecker, Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds, (arXiv:0704.1891v2)
A reformulation of the Reidemeister trace in terms of bicategorical trace is in