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

If X is a finite connected CW-complex with universal covering space X and fundamental group π, then the cellular chain complex C q ( X) is a free Zπmodule. If f : X → X is a cellular map and f : X → X is a lift of f , then the induced map f q : C q ( X) → C q ( X) can be viewed as a matrix with entries in Zπ (with respect to some chosen Zπ basis for C q ( X)). We then define

where tr is the sum of the diagonal entries of the matrix, and ρ is the projection into the “Reidemeister classes” of π. The Reidemeister trace, then, is an element of ZR, where R is the set of Reidemeister classes. Wecken, in [13], proved what we will refer to as the Wecken Trace Theorem, that

where ind ([α]) is the index of the Nielsen fixed point class associated to [α] (see e.g. [10]). Thus the number of terms appearing in the Reidemeister trace with nonzero coefficient is equal to the Nielsen number of f , and by the Lefschetz-Hopf Theorem, the sum of the coefficients is equal to the Lefschetz number of f .

Recent work of Furi, Pera, and Spadini in [6] has given a new proof of the uniqueness of the fixed point index on orientable manifolds with respect to three natural axioms. In [12] their approach was extended to the coincidence index. The result is the following theorem: Theorem 1. Let X and Y be oriented differentiable manifolds of the same dimension. The coincidence index ind(f, g, U ) of two mappings f, g : X → Y over some open set U ⊂ X is the unique integer-valued function satisfying the following axioms:

disjoint open subsets of U whose union contains all coincidence points of f and g on U , then

• (Homotopy) If f and g are “admissably homotopic” to f ′ and g ′ , then

In the spirit of the above theorem, we demonstrate the existence and uniqueness of a local Reidemeister trace in coincidence theory subject to five axioms. A local Reidemeister trace for fixed point theory was given by Fares and Hart in [5], but no Reidemeister trace (local or otherwise) has appeared in the literature for coincidence theory.

We note that recent work by Gonçalves and Weber in [8] gives axioms for the Reidemeister trace in fixed point theory using entirely different methods. Their work uses no locality properties, and is based on axioms for the Lefschetz number by Arkowitz and Brown in [1].

In Section 2 we present our axiom set, and we prove the uniqueness in coincidence theory in Section 3. In the special case of local fixed point theory, we can obtain a slightly stronger uniqueness result which we discuss in Section 4. Section 5 is a demonstration of the existence in the setting of coincidence theory.

This paper contains pieces of the author’s doctoral dissertation. The author would like to thank his dissertation advisor Robert F. Brown for assistance with both the dissertation work and with this paper. The author would also like to thank Peter Wong, who guided the early dissertation work and interested him in the coincidence Reidemeister trace.

Throughout the paper, unless otherwise stated, let X and Y denote connected orientable differentiable manifolds of the same dimension. All maps f, g : X → Y will be assumed to be continuous. The universal covering spaces of X and Y will be denoted X and Y with projection maps p X : X → X and p Y :

Let f, g : X → Y be maps, with induced homomorphisms φ, ψ : π 1 (X) → π 1 (Y ) respectively. We will view elements of π 1 (X) and π 1 (Y ) as covering transformations, so that for any x ∈ X and σ ∈ π 1 (X), we have f (σ x) = φ(σ) f ( x) and g(σ x) = ψ(σ) g( x).

We will partition the elements of π 1 (Y ) into equivalence classes defined by the “doubly twisted conjugacy” relation: For any set S, let ZS denote the free abelian group generated by S, whose elements we write as sums of elements of S with integer coefficients. For any such abelian group, there is a homomorphism c : ZS → Z defined as the sum of the coefficients:

for s i ∈ S and k i ∈ Z, and i ranging over a finite set.

For some maps f, g : X → Y and an open subset U ⊂ X, let

We say that the triple (f, g, U ) is admissable if Coin(f, g, U ) is compact. Two triples (f, g, U ) and (f ′ , g ′ , U ) are admissably homotopic if there is some pair of homotopies

Let C(X, Y ) be the set of admissable tuples, all tuples of the form (f, f , g, g, U ) where f, g : X → Y are maps, (f, g, U ) is an admissable triple, and f and g are lifts of f and g.

Let (f, f , g, g, U ), (f ′ , f ′ , g ′ , g ′ , U ) ∈ C(X, Y ) with (f, g, U ) admissably homotopic to (f ′ , g ′ , U ) by homotopies F t , G t . By the homotopy lifting property, there are unique lifted homotopies

then we say that the tuples (f, f , g, g, U ) and (f ′ , f ′ , g ′ , g ′ , U ) are admisssably homo

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