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Reidemeister trace

Contents

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 MM is a closed manifold and f:MMf\colon M\to M a self-map. Deform ff so that it has isolated fixed points. We say that two fixed points xx and yy are in the same fixed-point class? if there is a path γ\gamma from xx to yy such that f(γ)f(\gamma) is homotopic to γ\gamma rel the endpoints (xx and yy). Let [π 1(M) f]\mathbb{Z}[\pi_1(M)_f] denote the free abelian group on the set of fixed-point classes. Then the Reidemeister trace of ff is the formal sum

R(f) f(x)=xind f(x)[x][π 1(M) f] R(f) \coloneqq \sum_{f(x)=x} ind_f(x) \cdot [x] \in \mathbb{Z}[\pi_1(M)_f]

where ind f(x)ind_f(x) is the index of the fixed point xx of ff. 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)L(f).

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

  • If MM is a closed manifold of dimension at least 3, and R(f)=0R(f)=0, then ff 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

Revised on October 12, 2013 20:52:03 by Toby Bartels (98.19.41.253)