Reidemeister Trace Research Articles

Isovariant homotopy theory and fixed point invariants

An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein–Williams equivariant intersection theory for a finite group G. We prove that under certain reasonable dimension and codimension conditions on H-fixed subspaces (for H≤G), the fixed points of a self-map of a compact smooth G-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.

Poincaré complex diagonals and the Bass trace conjecture

For a finitely dominated Poincaré duality space $M$ , we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$ . We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

The Reidemeister trace of an n-valued map

In topological fixed point theory, the Reidemeister trace is an invariant associated to a selfmap of a polyhedron which combines information from the Lefschetz and Nielsen numbers. In this paper we define the Reidemeister trace in the context of n-valued selfmaps of compact polyhedra. We prove several properties of the Reidemeister trace which generalize properties from the single-valued theory, and prove an averaging formula.

Rational Reidemeister trace of an outer automorphism of finite order

Let H be a group of type FP over the rationals and let ϕ:H→H be an automorphism whose class in Out(H) has finite order and such that the Lefschetz number of some iterate ϕi is nonzero. We show that under a suitable Bass conjecture assumption, this iterate ϕi is conjugate to a finite order automorphism, and the intersection of the fixed point set of ϕ with the center of H is of bounded exponent. In the torsion-free case it follows that the center of the HNN extension 〈H,t|∀h∈H:tht−1=ϕ(h)〉 is infinite cyclic. These results are related to Geoghegan and Nicas' first order Euler characteristic of a group G with a finite Eilenberg-MacLane complex; this is a morphism χ1(G) from the center of G to H1(G;k) where k is a ring of coefficients. We give a complete determination of χ1(G) with rational coefficients.

Preimage homomorphism indices of preimage classes

Under finite regular covering spaces, each preimage class of the base spaces is covered by some preimage classes of the covering spaces. We find an explicit relationship between those preimage classes in terms of covering transformation groups. Thereby we obtain a relationship of the homomorphism indices between them in terms of covering transformation groups. This is a local averaging formula and is the main result of our paper. The local formula gives rise to a global formula for the preimage homomorphism index and a formula for the preimage Reidemeister trace.

Periodic Points and Topological Restriction Homology

Abstract We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.

More general averaging formulae for preimage classes

Similar to the Reidemeister trace in Nielsen fixed point theory, we define some Reidemeister invariants for preimage classes. Some general averaging formulae are obtained, which imply all existed averaging formulae.

Topological Hochschild homology and higher characteristics

We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.

An averaging formula for the coincidence Reidemeister trace

In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.

The equivalence of two methods: finding representatives of non--empty Nielsen classes

Let $f:X\to X$ be a self--map with $X$ a wedge of circles or a compact surface with boundary, so that the fundamental group of $X$ is finitely generated and free. In [3], Wagner presents an algorithm for extracting information from the homomorphism induced by $f$ on the fundamental group. This information involves the fixed point index of $f$ and the Nielsen classes of fixed points of $f$. The step in which the representatives of Nielsen classes, Wagner tails, are calculated is equivalent to a step in the method presented by Fadell and Husseini in [1]. The Fadell--Husseini method was designed for closed two dimensional CW--complexes, but the step in which they use the Fox calculus, to determine terms in the unreduced Reidemeister trace, produces the Wagner tails and their indices. The equivalence of these steps was stated in [2] without proof. Further developments in this area have caused continued interest in the techniques, and a clarification of the equivalence is needed. Here we provide the proof and an example.

Coincidence invariants and higher Reidemeister traces

The Lefschetz number and fixed point index can be thought of as two different descriptions of the same invariant. The Lefschetz number is algebraic and defined using homology. The index is defined more directly from the topology and is a stable homotopy class. Both the Lefschetz number and index admit generalizations to coincidences and the comparison of these invariants retains its central role. In this paper, we show that the identification of the Lefschetz number and index using formal properties of the symmetric monoidal trace extends to coincidence invariants. This perspective on the coincidence index and Lefschetz number also suggests difficulties for generalizations to a coincidence Reidemeister trace.

Equivariant fixed-point theory

We reexamine equivariant generalizations of the Lefschetz number and Reidemeister trace using categorical traces. This gives simple, conceptual descriptions of the invariants as well as direct comparisons to previously defined generalizations. These comparisons are illuminating applications of the additivity and multiplicativity of the categorical trace.

The multiplicativity of fixed point invariants

We prove two general factorization theorems for fixed-point invariants of fibrations: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar multiplicativity results for the Lefschetz and Nielsen numbers of a fibration. Moreover, the proofs of these theorems are essentially formal, taking place in the abstract context of bicategorical traces. This makes generalizations to other contexts straightforward.

Axioms for the coincidence index of maps between manifolds of the same dimension

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in Z⊕Z2. We also show in each setting that the group of values for the index (either Z or Z⊕Z2) is determined by the axioms.Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which characterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds.

A Reidemeister trace for fibred maps

A Reidemeister trace for fibred maps is defined as the alternating sum of suitable (elementary) traces for linear morphisms of fibred cellular free modules with local coefficients. This invariant extends in a natural way the classical construction of the generalized Lefschetz number—Reidemeister trace—to the category of fibred CW-complexes.

A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles

We give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups.

Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.

Equivariant fixed-point indices of iterated maps

We describe an equivariant version (for actions of a finite group G) of Dold’s index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered.

Axioms for the equivariant Lefschetz number and for the Reidemeister trace

We characterize the Reidemeister trace, the equivariant Lefschetz number and the equivariant Reidemeister trace in terms of certain axioms.

The Reidemeister trace and the calculation of the Nielsen number

The Reidemeister trace and the calculation of the Nielsen number