nLab
Reidemeister trace

Idea

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?.

Definition

Suppose $M$ is a closed manifold and $f:M\to M$ a self-map. Deform $f$ so that it has isolated fixed points. We say that two fixed points $x$ and $y$ are in the same fixed-point class? if there is a path $\gamma$ from $x$ to $y$ such that $f(\gamma )$ is homotopic to $\gamma$ rel the endpoints ($x$ and $y$). Let $\mathbb{Z}[{\pi }_1(M{)}_f]$ denote the free abelian group on the set of fixed-point classes. Then the Reidemeister trace of $f$ is the formal sum

$$R(f)≔\sum _{f(x)=x}{\mathrm{ind}}_f(x)\cdot [x]\in \mathbb{Z}[{\pi }_1(M{)}_f]$$R(f) \coloneqq \sum_{f(x)=x} ind_f(x) \cdot [x] \in \mathbb{Z}[\pi_1(M)_f]

where ${\mathrm{ind}}_f(x)$ is the index of the fixed point $x$ of $f$. This definition is homotopy invariant.

An equivalent definition can be obtained algebraically, or category-theoretically using the bicategorical trace.

Properties

• The sum of all the coefficients in the Reidemeister trace is the Lefschetz number $L(f)$.

• The number of nonzero coefficients in the Reidemeister trace is the Nielsen number? $N(f)$.

• If $M$ is a closed manifold of dimension at least 3, and $R(f)=0$, then $f$ is homotopic to a map without fixed points. Thus, the Reidemeister trace supports a converse to the Lefschetz fixed-point theorem?.

References

The Reidemeister trace was introduced in

• Kurt Reidemeister Automorphismen von Homotopiekettenringen, Mathematische Annalen, 112:586–593 (1936)

A modern treatment is in

• S. Husseini, Generalized Lefschetz numbers, Transactions of the American Mathematical Society, 272:247–274 (1982)

See also

• 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

• Kate Ponto, Michael Shulman, The multiplicativity of fixed point invariants (arXiv:1203.0950)