Axioms for a local Reidemeister trace in fixed
point and coincidence theory on differentiable
manifolds
P. Christopher Staecker
August 24, 2021
Abstract
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 exis-
tence results for the local Reidemeister trace in coincidence theory.
1 Introduction
The Reidemeister trace is a fundamental invariant in topological fixed point
theory, generalizing both the Lefschetz and Nielsen numbers. It was originally
defined by Reidemeister in [11]. A more modern treatment, under the name
“generalized Lefschetz number,” was given by Husseini in [9].
If X is a finite connected CW-complex with universal covering space X̃
and fundamental group π, then the cellular chain complex Cq(X̃) is a free Zπ-
module. If f : X → X is a cellular map and f̃ : X̃ → X̃ is a lift of f , then the
induced map f̃q : Cq(X̃) → Cq(X̃) can be viewed as a matrix with entries in Zπ
(with respect to some chosen Zπ basis for Cq(X̃)). We then define
RT (f, f̃) =
(−1)qρ(tr(f̃q)),
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
RT (f, f̃) =
[α]∈R
ind([α]) [α],
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
http://arxiv.org/abs/0704.1891v2
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
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:
• (Additivity) If U1 and U2 are disjoint open subsets of U whose union con-
tains all coincidence points of f and g on U , then
ind(f, g, U) = ind(f, g, U1) + ind(f, g, U2).
• (Homotopy) If f and g are “admissably homotopic” to f ′ and g′, then
ind(f, g, U) = ind(f ′, g′, U)
• (Normalization) If L(f, g) denotes the coincidence Lefschetz number of f
and g, then
ind(f, g,X) = L(f, g).
In the spirit of the above theorem, we demonstrate the existence and unique-
ness 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.
2 The Axioms
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 X̃ and Ỹ with projection maps pX : X̃ → X and pY : Ỹ → Y .
A lift of some map f : X → Y is a map f̃ : X̃ → Ỹ with pY ◦ f̃ = f ◦ pX .
Let f, g : X → Y be maps, with induced homomorphisms φ, ψ : π1(X) →
π1(Y ) respectively. We will view elements of π1(X) and π1(Y ) as covering
transformations, so that for any x̃ ∈ X̃ and σ ∈ π1(X), we have f̃(σx̃) =
φ(σ)f̃ (x̃) and g̃(σx̃) = ψ(σ)g̃(x̃).
We will partition the elements of π1(Y ) into equivalence classes defined by
the “doubly twisted conjugacy” relation:
α ∼ β ⇐⇒ α = ψ(σ)−1βφ(σ).
The equivalence classes with respect to this relation (denoted e.g. [α]) are called
Reidemeister classes. The set of Reidemeister classes is denoted R[f, g].
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 si ∈ S and ki ∈ Z, and i ranging over a finite set.
For some maps f, g : X → Y and an open subset U ⊂ X , let
Coin(f, g, U) = {x ∈ U | f(x) = g(x)}.
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 Ft, Gt : X × [0, 1] → X of f, g to f
′, g′ with {(x, t) ∈ U × [0, 1] |
Ft(x) = Gt(x)} compact.
Let C(X,Y ) be the set of admissable tuples, all tuples of the form (f, f̃ , g, g̃, U)
where f, g : X → Y are maps, (f, g, U) is an admissable triple, and f̃ and g̃ are
lifts of f and g.
Let (f, f̃ , g, g̃, U), (f ′, f̃ ′, g′, g̃′, U) ∈ C(X,Y ) with (f, g, U) admissably ho-
motopic to (f ′, g′, U) by homotopies Ft, Gt. By the homotopy lifting property,
there are unique lifted homotopies F̃t, G̃t : X̃ × [0, 1] → Ỹ with F̃0 = f̃ and
G̃0 = g̃. If we additionaly have F̃1 = f̃
′ and G̃1 = g̃
′, then we say that the
tuples (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admisssably homotopic.
Throughout the following, let RT be any function which to an admissable
tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g]. Our first three
axioms for the local Reidemeister trace are modeled after the axioms of Theorem
Axiom 1 (Additivity). Given (f, f̃ , g, g̃, U) ∈ C(X,Y ), if U1 and U2 are disjoint
open subsets of U with Coin(f, g, U) ⊂ U1 ∪ U2, then
RT(f, f̃ , g, g̃, U) = RT(f, f̃ , g, g̃, U1) + RT(f, f̃ , g, g̃, U2).
Axiom 2 (Homotopy). If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably ho-
motopic admissable tuples, then
RT(f, f̃ , g, g̃, U) = RT(f ′, f̃ ′, g′, g̃′, U).
Axiom 3 (Normalization). If (f, f̃ , g, g̃, X) ∈ C(X,Y ), then
c(RT(f, f̃ , g, g̃, X)) = L(f, g),
where L(f, g) is the Lefschetz number of f and g.
We will require one additional axiom to make some connections with Nielsen
theory, based on a well-known property of the Reidemeister trace:
Axiom 4 (Lift invariance). For any (f, f̃ , g, g̃, U) ∈ C(X,Y ), and any α, β ∈
π1(Y ) we have
c(RT(f, f̃ , g, g̃, U)) = c(RT(f, αf̃ , g, βg̃, U)).
The four axioms above are enough to demonstrate some relationships be-
tween RT and the coincidence index.
Proposition 1. If RT satisfies the homotopy, additivity, normalization, and
lift invariance axioms, then
c(RT(f, f̃ , g, g̃, U)) = ind(f, g, U)
for any (f, f̃ , g, g̃, U) ∈ C(X,Y ), where ind denotes the coincidence index (see
[7]).
Proof. Let ω = c ◦ RT : C(X,Y ) → Z. By the lift invariance axiom, ω is
independent of the choice of lifts. Thus ω can be viewed as a function from
the set of all admissable triples to Z. It is clear that ω satisfies the three
axioms of Theorem 1, since they are implied by our additivity, homotopy, and
normalization axioms for RT (disregarding the lift parameters). Thus ω is the
coincidence index.
Proposition 2. If RT satisfies the additivity, homotopy, normalization, and lift
invariance axioms and c(RT(f, f̃ , g, g̃, U)) 6= 0, then there is some σ ∈ π1(Y )
such that σf̃ and g̃ have a coincidence on p−1X (U).
Proof. By Proposition 1, if c(RT (f, f̃ , g, g̃, U)) 6= 0 then ind(f, g, U) 6= 0, and
so f and g have a coincidence on U . Let x ∈ U be this coincidence point,
and choose x̃ ∈ p−1X (x). Then since f̃ and g̃ are lifts, the points f̃(x̃) and
g̃(x̃) will project to the same point of Y by pY . Thus there is some covering
transformation σ with σf̃(x̃) = g̃(x̃).
The four axioms given above are not sufficient to uniquely characterize the
Reidemeister trace in fixed point or coincidence theory. For instance, the func-
tion defined by
T (f, f̃ , g, g̃, U) = ind(f, g, U)[1],
where [1] is the Reidemeister class of the trivial element 1 ∈ π1(Y ), satisfies all
of the axioms above, but provides none of the expected data concerning R[f, g],
and so that function cannot be the Reidemeister trace.
An additional axiom is needed, one which somehow indicates the elements
of R[f, g] which are to appear in the Reidemeister trace. Our final axiom is
a sort of strengthening of Proposition 2, which specifies the Reidemeister data
associated to the coincidence points.
Axiom 5 (Coincidence of lifts). If [α] appears with nonzero coefficient in RT(f, f̃ , g, g̃, U),
then αf̃ and g̃ have a coincidence on p−1
Any function RT which to a tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an
element of ZR[f, g], and satisfies the additivity, homotopy, normalization, lift
invariance, and coincidence of lifts axioms we will call a local Reidemeister trace.
Our main result (Theorem 3) states that there is a unique such function.
3 Uniqueness
Let (f, f̃ , g, g̃, U) ∈ C(X,Y ), let Ũ = p−1
(U), and let
C(f̃ , g̃, Ũ , [α]) = pX(Coin(αf̃ , g̃, Ũ)).
For each α we have C(f̃ , g̃, Ũ , [α]) ⊂ Coin(f, g, U), and such coincidence sets are
called coincidence classes. That these classes are well defined is a consequence
of the following lemma, which appears in slightly different language as Lemma
2.3 of [4].
Lemma 1. Let α, β ∈ π1(Y ), maps f, g : X → Y , and an open subset U ⊂ X
be given. Then:
• [α] = [β] if and only if
pX Coin(αf̃ , g̃, Ũ) = pX Coin(βf̃ , g̃, Ũ)
for any lifts f̃ , g̃.
• If [α] 6= [β], then pX Coin(αf̃ , g̃, Ũ) and pX Coin(αf̃ , g̃, Ũ) are disjoint for
any lifts f̃ , g̃.
Given the above notation, the coincidence of lifts axiom could be restated
as follows: If [α] appears with nonzero coefficient in RT (f, f̃ , g, g̃, U), then
C(f̃ , g̃, Ũ , [α]) is nonempty. For each coincidence point x in U , define [xef,eg] ∈
R[f, g] as that class [α] for which x ∈ C(f̃ , g̃, Ũ , [α]).
Theorem 2. If RT is a local Reidemeister trace and Coin(f, g, U) is a set of
isolated points, then
RT(f, f̃ , g, g̃, U) =
x∈Coin(f,g,U)
ind(f, g, Ux)[xef,eg],
where Ux is an isolating neighborhood for the coincidence point x.
Proof. By the additivity property, we need only show that
RT (f, f̃ , g, g̃, Ux) = ind(f, g, Ux)[xef,eg].
First, we observe that no element of R[f, g] other than [xef,eg] appears as a
term with nonzero coefficient in RT (f, f̃ , g, g̃, Ux): If some [β] does appear with
nonzero coefficient, then we know by the coincidence of lifts axiom that βf̃ and
g̃ have a coincidence on Ũx = p
(Ux). Projection of this coincidence point
gives a coincidence point in Ux which necessarily must be x, since x is the
only coincidence point in Ux. Thus x ∈ pX Coin(βf̃ , g̃, Ũx), which means that
[β] = [xef,eg].
Since [xef,eg] is the only element of R[f, g] appearing in RT(f, f̃ , g, g̃, U), we
RT(f, f̃ , g, g̃, Ux) = k[xef,eg]
for some k ∈ Z (possibly k = 0). Proposition 1 says that the coefficient sum
must equal the index, and so k = ind(f, g, Ux) as desired.
The above is a strong result for maps whose coincidence sets are isolated. In
order to leverage this result for arbitrary maps, we will make use of a technical
lemma, a combination of Lemmas 13 and 15 from [12].
Lemma 2. Let (f, g, U) be an admissable triple, and let V ⊂ U be an open
subset containing Coin(f, g, U) with compact closure V̄ ⊂ U . Then (f, g, V ) is
admissably homotopic to an admissable triple (f ′, g′, V ), where f ′ and g′ have
isolated coincidence points in V .
The above lemma is used to approximate any maps by maps having isolated
coincidence points, and we obtain our uniqueness theorem:
Theorem 3. There is at most one local Reidemeister trace defined on C(X,Y ).
Proof. Let RT be local Reidemeister trace, and take (f, f̃ , g, g̃, U) ∈ C(X,Y ).
Then by Lemma 2 there is an open subset V ⊂ U with Coin(f, U) ⊂ V and
maps f ′, g′ with isolated coincidence points with (f, g, V ) admissably homotopic
to (f ′, g′, V ). Then by the homotopy axiom there are lifts f̃ ′, g̃′ of f and g with
RT (f, f̃ , g, g̃, U) = RT (f ′, f̃ ′, g′, g̃′, V ).
The coincidence points of f ′ and g′ in V are isolated, so we have
RT (f, f̃ , g, g̃, U) =
x∈Coin(f ′,g′,V )
ind(f ′, g′, Vx)[xef ′,eg′ ],
where Vx is an isolating neighborhood of the coincidence point x. This gives
an explicit formula for the computation of RT (f, f̃ , g, g̃, U). The only choice
made in the computation is of the admissable homotopy to (f ′, g′, V ), but any
alternative choice must give the same local Reidemeister trace by the homotopy
axiom. Thus all local Reidemeister traces must be computed in the same way,
giving the same result, which means that there can be only one.
4 Uniqueness in fixed point theory
In the special case where Y = X and g is taken to be the identity map id :
X → X , the above method can be used with slight modifications to prove a
uniqueness result for the local Reidemeister trace in the fixed point theory of
possibly nonorientable manifolds.
We have not in this paper made explicit use of the orientability hypothesis,
but it is a necessary hypothesis for the uniqueness of the coincidence index in
Theorem 1, which was used in Proposition 1. An accounting of orientations
is needed in coincidence theory to distinguish between points of index +1 and
index−1 (though see [4] for an approach to an index for nonorientable manifolds,
which does not always give an integer). Orientability is not needed in local fixed
point theory, since the notion of an orientation preserving selfmap is well-defined
locally, even on a manifold with no global orientation. Thus the uniqueness of
the fixed point index in [6] does not require orientability, and we will not require
it here.
Let C(X) be the set of all tuples of the form (f, f̃ , ı̃, U), where f : X → X
is a selfmap, f̃ : X̃ → X̃ is a lift of f , the map ı̃ : X̃ → X̃ is a lift of
the identity map, and U is an open subset of X with compact fixed point set
Fix(f, U) = Coin(f, id, U). Let R[f ] = R[f, id]. Two tuples (f, f̃ , ı̃, U) and
(f ′, f̃ ′, ı̃, U) are said to be admissably homotopic if there is some homotopy Ft
of f to f ′ with {(x, t) | Ft(x) = x} compact, and Ft lifts to a homotopy of f̃ to
f̃ ′.
Our uniqueness theorem is then:
Theorem 4. If X is a (possibly nonorientable) differentiable manifold, then
there is a unique function taking an admissable tuple (f, f̃ , ı̃, U) to an element
of ZR[f ] satisfying the following axioms:
• (Additivity) If U1 and U2 are disjoint open subsets of U with Fix(f, U) ⊂
U1 ∪ U2, then
RT(f, f̃ , ı̃, U) = RT(f, f̃ , ı̃, U1) + RT(f, f̃ , ı̃, U2)
• (Homotopy) If (f, f̃ , ı̃, U) is admissably homotopic to (f ′, f̃ ′, ı̃, U), then
RT(f, f̃ , ı̃, U) = RT(f ′, f̃ ′, ı̃, U)
• (Weak normalization) If f is a constant map, then
c(RT(f, f̃ , ı̃, U)) = 1
• (Lift invariance) For any α, β ∈ π1(X), we have
c(RT(f, f̃ , ı̃, U)) = c(RT(f, αf̃ , βı̃, U))
• (Coincidence of lifts) If [α] appears with nonzero coefficient in RT(f, f̃ , ı̃, U),
then αf̃ and ı̃ have a coincidence point on p−1X (U).
Proof. First we note that a result analagous to Proposition 1 can be obtained
in the fixed point setting using only the weak normalization axiom: Using the
three axioms of [6], which make use of an appropriately weakened normalization
axiom, we see that c ◦ RT is the fixed point index.
Then letting g = id in the proof of Theorem 2, we have that, if f has isolated
fixed points,
RT(f, f̃ , ı̃, U) =
x∈Fix(f,Ux)
ind(f, Ux)[xef,eı],
where ind denotes the fixed point index, and Ux is an isolating neighborhood
for the fixed point x.
A fixed point version of Lemma 2 can be found in Lemmas 4.1 and 3.3 of
[6], and the proof of Theorem 3 can be mimicked to obtain our uniqueness
result.
Note that the uniqueness in fixed point theory requires only a weakened
version of the normalization axiom. A uniqueness result for coincidence theory
using only the weak normalization axiom can be obtained if we restrict ourselves
to self-maps of a particular (not necessarily orientable) manifold. This would
use a proof similar to the above, using results from Section 5 of [12].
5 Existence
The existence of a local Reidemeister trace in fixed point theory for connected
finite dimensional locally compact polyhedra is established by Fares and Hart in
[5]. There, the slightly more general localH-Reidemeister trace is defined, called
“the local generalized H-Lefschetz number”. An extension of this paper to the
mod H theory would not be difficult. The fact that the mod H Reidemeister
classes are unions of ordinary Reidemeister classes allows the same results to be
obtained without substantial modifications.
In [5], the additivity and homotopy axioms are proved in Proposition 3.2.9
and Proposition 3.2.8, respectively. A strong version of the lift invariance ax-
iom (see our Theorem 6) is proved in Proposition 3.2.4. The coincidence of lifts
axiom is not stated explicitly by Fares and Hart, but is a straightforward con-
sequence of their trace-like definition (if some [α] has nonzero coefficient in the
Reidemeister trace, it neccesarily comes from some simplex in the covering space
containing a fixed point of αf̃). A result analogous to the Wecken Trace Theo-
rem (which trivially implies the normalization and weak normalization axioms)
is given in Theorem 3.3.1.
No Reidemeister trace for coincidence theory, either local or global, has
appeared previously in the literature. The proof of Theorem 3 furnishes the
appropriate definition, as follows: Given an admissable tuple (f, f̃ , g, g̃, U), we
find (by Lemma 2) an admissably homotopic tuple (f ′, f̃ ′, g′, g̃′, V ) with isolated
coincidence points, and we define
RT (f, f̃ , g, g̃, U) =
x∈Coin(f ′,g′,V )
ind(f ′, g′, Vx)[xef ′,eg′ ],
where Vx is an isolating neighborhood for the coincidence point x. The above
is well defined provided that it is independent of the choice of the admissably
homotopic tuple. This is ensured by the following lemma:
Lemma 3. If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably homotopic tu-
ples with isolated coincidence points, then
x∈Coin(f,g,U)
ind(f, g, Ux)[xef,eg] =
x′∈Coin(f ′,g′,U)
ind(f ′, g′, Ux′)[x
ef ′,eg′
where Ux is an isolating neighborhood for the coincidence point x ∈ Coin(f, g, U),
and Ux′ is an isolating neighborhood of the coincidence point x
′ ∈ Coin(f ′, g′, U).
Proof. We define the index of a coincidence class C of f and g as follows:
indC =
ind(f, g, Ux).
A class is called essential if its index is nonzero.
Since f and g are homotopic to f ′ and g′, we have R[f, g] = R[f ′, g′]. Call
this common set of Reidemeister classes R. Letting Ũ = p−1X (U), the statement
of the Lemma is equivalent to
[α]∈R
indC(f̃ , g̃, Ũ , [α])[α] =
[α]∈R
indC(f̃ ′, g̃′, Ũ , [α])[α],
and we need only show that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]) for any [α].
We will prove this using Brooks’s notion of homotopy-relatedness of coincidence
classes, exposited in detail in [2] and briefly in [3].
Let Ft, F̃t, Gt, G̃t be homotopies realizing the admissable homotopy of (f, f̃ , g, g̃, U)
and (f ′, f̃ ′, g′, g̃′, U). Two coincidence points x ∈ Coin(f, g, U) and x′ ∈ Coin(f ′, g′, U)
are (Ft, Gt)–related if there is some path γ(t) in X connecting x to x
′ such that
the paths Ft(γ(t)) and Gt(γ(t)) are homotopic in Y as paths with fixed end-
points. Two coincidence classes are related if at least one point of one is related
to at least one point of the other. Theorem II.22 of [2] shows that the notion of
(Ft, Gt)-relatedness gives a bijective correspondence between the essential coin-
cidence classes of (f, g) and those of (f ′, g′). Theorem IV.24 of [2] further shows
that any two such related classes will have the same index.
What remains is an elementary argument using covering-space theory. Let
C = C(f̃ , g̃, Ũ , [α]), and let C′ be the unique coincidence class of (f ′, g′) which
is (Ft, Gt)-related to C. We need only show that C
′ = C(f̃ ′, g̃′, Ũ , [α]), and
thus (since homotopy-relatedness preserves the index) that indC(f̃ , g̃, Ũ , [α]) =
indC(f̃ ′, g̃′, Ũ , [α]).
Choose a point x ∈ C, and let x′ be a point in C′ which is (Ft, Gt) related
to x. Then there is some path γ in X from x to x′ with Ft(γ(t)) homotopic to
Gt(γ(t)). Let x̃ be some point with pX(x̃) = x and αf̃(x̃) = g̃(x̃). We can lift
γ to a path γ̃ in X̃ starting at x̃. Since Ft(γ(t)) is homotopic to Gt(γ(t)), we
will have F̃t(γ̃(t)) homotopic to G̃t(γ̃(t)), which in particular means that they
will have the same endpoint. This common endpoint is αf̃ ′(γ̃(1)) = g̃′(γ̃(1)),
which must project by pX to the point x
′. Thus x′ ∈ pX(Coin(αf̃
′, g̃′, Ũ)), and
so C′ = C(f̃ ′, g̃′, Ũ , [α]), as desired.
We have thus produced a meaningful definition of a local coincidence Reide-
meister trace on orientable differentiable manifolds of the same dimension, and
the proof above suffices to give:
Theorem 5 (Wecken Coincidence Trace Theorem). Let RT be the unique lo-
cal coincidence Reidemeister trace satisfying our five axioms. Then for any
(f, f̃ , g, g̃, U) ∈ C(X,Y ) with Ũ = p−1X (U), we have
RT(f, f̃ , g, g̃, U) =
[α]∈R[f,g]
indC(f̃ , g̃, Ũ , [α])[α].
In conclusion we prove a stronger form of the lift invariance axiom, a coin-
cidence version of a well-known property of the Reidemeister trace.
Theorem 6. Let RT be the unique local coincidence Reidemeister trace satis-
fying our five axioms. If
RT(f, f̃ , g, g̃, U) =
[σ]∈R[f,g]
k[σ][σ]
for k[σ] ∈ Z, then for any α, β ∈ π1(Y ), we have
RT(f, αf̃ , g, βg̃, U) =
[σ]∈R[f,g]
k[σ][βσα
Proof. Letting Ũ = p−1
(U), by Theorem 5 we know that k[σ] = indC(f̃ , g̃, Ũ , [σ]).
Then we have
C(αf̃ , βg̃, [σ]) = pX Coin(σαf̃ , βg̃, Ũ) = pX Coin(β
−1σαf̃ , g̃, Ũ) = C(f̃ , g̃, [β−1σα]),
and thus indC(αf̃ , βg̃, Ũ , [σ]) = k[β−1σα]. Now by Theorem 5 again, we have
RT(f, αf̃ , g, βg̃, Ũ) =
[σ]∈R[f,g]
indC(αf̃ , βg̃, [σ])[σ] =
[σ]∈R[f,g]
k[β−1σα][σ]
[γ]∈R[f,g]
k[γ][βγα
as desired.
References
[1] M. Arkowitz and R. Brown. The Lefschetz-Hopf theorem and axioms for
the Lefschetz number. Fixed Point Theory and Applications, 1:1–11, 2004.
[2] R. Brooks. Coincidences, Roots, and Fixed Points. Doctoral dissertation,
University of California, Los Angeles, 1967.
[3] R. Brooks and R. Brown. A lower bound on the ∆-Nielsen number. Trans-
actions of the American Mathematical Society, 143:555–564, 1969.
[4] R. Dobreńko and J. Jezierski. The coincidence Nielsen number on nonori-
entable manifolds. The Rocky Mountain Journal of Mathematics, 23:67–85,
1993.
[5] J. Fares and E. Hart. A generalized Lefschetz number for local Nielsen
fixed point theory. Topology and Its Applications, 59:1–23, 1994.
[6] M. Furi, M. P. Pera, and M. Spadini. On the uniqueness of the fixed point
index on differentiable manifolds. Fixed Point Theory and Applications,
4:251–259, 2004.
[7] D. L. Gonçalves. Coincidence theory. In R.F. Brown, editor, The Handbook
of Topological Fixed Point Theory, pages 3–42. Springer, 2005.
[8] D. L. Gonçalves and J. Weber. Axioms for the equivariant Lefschetz num-
ber and for the Reidemeister trace. Journal of Fixed Point Theory and
Applications, 2:55–72, 2007.
[9] S. Husseini. Generalized Lefschetz numbers. Transactions of the American
Mathematical Society, 272:247–274, 1982.
[10] B. Jiang. Lectures on Nielsen fixed point theory. Contemporary Mathemat-
ics 14, American Mathematical Society, 1983.
[11] K. Reidemeister. Automorphismen von Homotopiekettenringen. Mathema-
tische Annalen, 112:586–593, 1936.
[12] P. C. Staecker. On the uniqueness of the coincidence index on orientable dif-
ferentiable manifolds. Topology and Its Applications, 154:1961–1970, 2007,
arxiv eprint math.GN/0607751.
http://arxiv.org/abs/math/0607751
[13] F. Wecken. Fixpunktklassen I, II, III. Mathematische Annalen, 117,
118:659–671, 216–234, 544–577, 1941, 1942.
	Introduction
	The Axioms
	Uniqueness
	Uniqueness in fixed point theory
	Existence

  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.

Introduction
The Reidemeister trace is a fundamental invariant in topological fixed point
theory, generalizing both the Lefschetz and Nielsen numbers. It was originally
defined by Reidemeister in [11]. A more modern treatment, under the name
“generalized Lefschetz number,” was given by Husseini in [9].
If X is a finite connected CW-complex with universal covering space X̃
and fundamental group π, then the cellular chain complex Cq(X̃) is a free Zπ-
module. If f : X → X is a cellular map and f̃ : X̃ → X̃ is a lift of f , then the
induced map f̃q : Cq(X̃) → Cq(X̃) can be viewed as a matrix with entries in Zπ
(with respect to some chosen Zπ basis for Cq(X̃)). We then define
RT (f, f̃) =
(−1)qρ(tr(f̃q)),
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
RT (f, f̃) =
[α]∈R
ind([α]) [α],
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
http://arxiv.org/abs/0704.1891v2
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
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:
• (Additivity) If U1 and U2 are disjoint open subsets of U whose union con-
tains all coincidence points of f and g on U , then
ind(f, g, U) = ind(f, g, U1) + ind(f, g, U2).
• (Homotopy) If f and g are “admissably homotopic” to f ′ and g′, then
ind(f, g, U) = ind(f ′, g′, U)
• (Normalization) If L(f, g) denotes the coincidence Lefschetz number of f
and g, then
ind(f, g,X) = L(f, g).
In the spirit of the above theorem, we demonstrate the existence and unique-
ness 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.
2 The Axioms
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 X̃ and Ỹ with projection maps pX : X̃ → X and pY : Ỹ → Y .
A lift of some map f : X → Y is a map f̃ : X̃ → Ỹ with pY ◦ f̃ = f ◦ pX .
Let f, g : X → Y be maps, with induced homomorphisms φ, ψ : π1(X) →
π1(Y ) respectively. We will view elements of π1(X) and π1(Y ) as covering
transformations, so that for any x̃ ∈ X̃ and σ ∈ π1(X), we have f̃(σx̃) =
φ(σ)f̃ (x̃) and g̃(σx̃) = ψ(σ)g̃(x̃).
We will partition the elements of π1(Y ) into equivalence classes defined by
the “doubly twisted conjugacy” relation:
α ∼ β ⇐⇒ α = ψ(σ)−1βφ(σ).
The equivalence classes with respect to this relation (denoted e.g. [α]) are called
Reidemeister classes. The set of Reidemeister classes is denoted R[f, g].
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 si ∈ S and ki ∈ Z, and i ranging over a finite set.
For some maps f, g : X → Y and an open subset U ⊂ X , let
Coin(f, g, U) = {x ∈ U | f(x) = g(x)}.
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 Ft, Gt : X × [0, 1] → X of f, g to f
′, g′ with {(x, t) ∈ U × [0, 1] |
Ft(x) = Gt(x)} compact.
Let C(X,Y ) be the set of admissable tuples, all tuples of the form (f, f̃ , g, g̃, U)
where f, g : X → Y are maps, (f, g, U) is an admissable triple, and f̃ and g̃ are
lifts of f and g.
Let (f, f̃ , g, g̃, U), (f ′, f̃ ′, g′, g̃′, U) ∈ C(X,Y ) with (f, g, U) admissably ho-
motopic to (f ′, g′, U) by homotopies Ft, Gt. By the homotopy lifting property,
there are unique lifted homotopies F̃t, G̃t : X̃ × [0, 1] → Ỹ with F̃0 = f̃ and
G̃0 = g̃. If we additionaly have F̃1 = f̃
′ and G̃1 = g̃
′, then we say that the
tuples (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admisssably homotopic.
Throughout the following, let RT be any function which to an admissable
tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an element of ZR[f, g]. Our first three
axioms for the local Reidemeister trace are modeled after the axioms of Theorem
Axiom 1 (Additivity). Given (f, f̃ , g, g̃, U) ∈ C(X,Y ), if U1 and U2 are disjoint
open subsets of U with Coin(f, g, U) ⊂ U1 ∪ U2, then
RT(f, f̃ , g, g̃, U) = RT(f, f̃ , g, g̃, U1) + RT(f, f̃ , g, g̃, U2).
Axiom 2 (Homotopy). If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably ho-
motopic admissable tuples, then
RT(f, f̃ , g, g̃, U) = RT(f ′, f̃ ′, g′, g̃′, U).
Axiom 3 (Normalization). If (f, f̃ , g, g̃, X) ∈ C(X,Y ), then
c(RT(f, f̃ , g, g̃, X)) = L(f, g),
where L(f, g) is the Lefschetz number of f and g.
We will require one additional axiom to make some connections with Nielsen
theory, based on a well-known property of the Reidemeister trace:
Axiom 4 (Lift invariance). For any (f, f̃ , g, g̃, U) ∈ C(X,Y ), and any α, β ∈
π1(Y ) we have
c(RT(f, f̃ , g, g̃, U)) = c(RT(f, αf̃ , g, βg̃, U)).
The four axioms above are enough to demonstrate some relationships be-
tween RT and the coincidence index.
Proposition 1. If RT satisfies the homotopy, additivity, normalization, and
lift invariance axioms, then
c(RT(f, f̃ , g, g̃, U)) = ind(f, g, U)
for any (f, f̃ , g, g̃, U) ∈ C(X,Y ), where ind denotes the coincidence index (see
[7]).
Proof. Let ω = c ◦ RT : C(X,Y ) → Z. By the lift invariance axiom, ω is
independent of the choice of lifts. Thus ω can be viewed as a function from
the set of all admissable triples to Z. It is clear that ω satisfies the three
axioms of Theorem 1, since they are implied by our additivity, homotopy, and
normalization axioms for RT (disregarding the lift parameters). Thus ω is the
coincidence index.
Proposition 2. If RT satisfies the additivity, homotopy, normalization, and lift
invariance axioms and c(RT(f, f̃ , g, g̃, U)) 6= 0, then there is some σ ∈ π1(Y )
such that σf̃ and g̃ have a coincidence on p−1X (U).
Proof. By Proposition 1, if c(RT (f, f̃ , g, g̃, U)) 6= 0 then ind(f, g, U) 6= 0, and
so f and g have a coincidence on U . Let x ∈ U be this coincidence point,
and choose x̃ ∈ p−1X (x). Then since f̃ and g̃ are lifts, the points f̃(x̃) and
g̃(x̃) will project to the same point of Y by pY . Thus there is some covering
transformation σ with σf̃(x̃) = g̃(x̃).
The four axioms given above are not sufficient to uniquely characterize the
Reidemeister trace in fixed point or coincidence theory. For instance, the func-
tion defined by
T (f, f̃ , g, g̃, U) = ind(f, g, U)[1],
where [1] is the Reidemeister class of the trivial element 1 ∈ π1(Y ), satisfies all
of the axioms above, but provides none of the expected data concerning R[f, g],
and so that function cannot be the Reidemeister trace.
An additional axiom is needed, one which somehow indicates the elements
of R[f, g] which are to appear in the Reidemeister trace. Our final axiom is
a sort of strengthening of Proposition 2, which specifies the Reidemeister data
associated to the coincidence points.
Axiom 5 (Coincidence of lifts). If [α] appears with nonzero coefficient in RT(f, f̃ , g, g̃, U),
then αf̃ and g̃ have a coincidence on p−1
Any function RT which to a tuple (f, f̃ , g, g̃, U) ∈ C(X,Y ) associates an
element of ZR[f, g], and satisfies the additivity, homotopy, normalization, lift
invariance, and coincidence of lifts axioms we will call a local Reidemeister trace.
Our main result (Theorem 3) states that there is a unique such function.
3 Uniqueness
Let (f, f̃ , g, g̃, U) ∈ C(X,Y ), let Ũ = p−1
(U), and let
C(f̃ , g̃, Ũ , [α]) = pX(Coin(αf̃ , g̃, Ũ)).
For each α we have C(f̃ , g̃, Ũ , [α]) ⊂ Coin(f, g, U), and such coincidence sets are
called coincidence classes. That these classes are well defined is a consequence
of the following lemma, which appears in slightly different language as Lemma
2.3 of [4].
Lemma 1. Let α, β ∈ π1(Y ), maps f, g : X → Y , and an open subset U ⊂ X
be given. Then:
• [α] = [β] if and only if
pX Coin(αf̃ , g̃, Ũ) = pX Coin(βf̃ , g̃, Ũ)
for any lifts f̃ , g̃.
• If [α] 6= [β], then pX Coin(αf̃ , g̃, Ũ) and pX Coin(αf̃ , g̃, Ũ) are disjoint for
any lifts f̃ , g̃.
Given the above notation, the coincidence of lifts axiom could be restated
as follows: If [α] appears with nonzero coefficient in RT (f, f̃ , g, g̃, U), then
C(f̃ , g̃, Ũ , [α]) is nonempty. For each coincidence point x in U , define [xef,eg] ∈
R[f, g] as that class [α] for which x ∈ C(f̃ , g̃, Ũ , [α]).
Theorem 2. If RT is a local Reidemeister trace and Coin(f, g, U) is a set of
isolated points, then
RT(f, f̃ , g, g̃, U) =
x∈Coin(f,g,U)
ind(f, g, Ux)[xef,eg],
where Ux is an isolating neighborhood for the coincidence point x.
Proof. By the additivity property, we need only show that
RT (f, f̃ , g, g̃, Ux) = ind(f, g, Ux)[xef,eg].
First, we observe that no element of R[f, g] other than [xef,eg] appears as a
term with nonzero coefficient in RT (f, f̃ , g, g̃, Ux): If some [β] does appear with
nonzero coefficient, then we know by the coincidence of lifts axiom that βf̃ and
g̃ have a coincidence on Ũx = p
(Ux). Projection of this coincidence point
gives a coincidence point in Ux which necessarily must be x, since x is the
only coincidence point in Ux. Thus x ∈ pX Coin(βf̃ , g̃, Ũx), which means that
[β] = [xef,eg].
Since [xef,eg] is the only element of R[f, g] appearing in RT(f, f̃ , g, g̃, U), we
RT(f, f̃ , g, g̃, Ux) = k[xef,eg]
for some k ∈ Z (possibly k = 0). Proposition 1 says that the coefficient sum
must equal the index, and so k = ind(f, g, Ux) as desired.
The above is a strong result for maps whose coincidence sets are isolated. In
order to leverage this result for arbitrary maps, we will make use of a technical
lemma, a combination of Lemmas 13 and 15 from [12].
Lemma 2. Let (f, g, U) be an admissable triple, and let V ⊂ U be an open
subset containing Coin(f, g, U) with compact closure V̄ ⊂ U . Then (f, g, V ) is
admissably homotopic to an admissable triple (f ′, g′, V ), where f ′ and g′ have
isolated coincidence points in V .
The above lemma is used to approximate any maps by maps having isolated
coincidence points, and we obtain our uniqueness theorem:
Theorem 3. There is at most one local Reidemeister trace defined on C(X,Y ).
Proof. Let RT be local Reidemeister trace, and take (f, f̃ , g, g̃, U) ∈ C(X,Y ).
Then by Lemma 2 there is an open subset V ⊂ U with Coin(f, U) ⊂ V and
maps f ′, g′ with isolated coincidence points with (f, g, V ) admissably homotopic
to (f ′, g′, V ). Then by the homotopy axiom there are lifts f̃ ′, g̃′ of f and g with
RT (f, f̃ , g, g̃, U) = RT (f ′, f̃ ′, g′, g̃′, V ).
The coincidence points of f ′ and g′ in V are isolated, so we have
RT (f, f̃ , g, g̃, U) =
x∈Coin(f ′,g′,V )
ind(f ′, g′, Vx)[xef ′,eg′ ],
where Vx is an isolating neighborhood of the coincidence point x. This gives
an explicit formula for the computation of RT (f, f̃ , g, g̃, U). The only choice
made in the computation is of the admissable homotopy to (f ′, g′, V ), but any
alternative choice must give the same local Reidemeister trace by the homotopy
axiom. Thus all local Reidemeister traces must be computed in the same way,
giving the same result, which means that there can be only one.
4 Uniqueness in fixed point theory
In the special case where Y = X and g is taken to be the identity map id :
X → X , the above method can be used with slight modifications to prove a
uniqueness result for the local Reidemeister trace in the fixed point theory of
possibly nonorientable manifolds.
We have not in this paper made explicit use of the orientability hypothesis,
but it is a necessary hypothesis for the uniqueness of the coincidence index in
Theorem 1, which was used in Proposition 1. An accounting of orientations
is needed in coincidence theory to distinguish between points of index +1 and
index−1 (though see [4] for an approach to an index for nonorientable manifolds,
which does not always give an integer). Orientability is not needed in local fixed
point theory, since the notion of an orientation preserving selfmap is well-defined
locally, even on a manifold with no global orientation. Thus the uniqueness of
the fixed point index in [6] does not require orientability, and we will not require
it here.
Let C(X) be the set of all tuples of the form (f, f̃ , ı̃, U), where f : X → X
is a selfmap, f̃ : X̃ → X̃ is a lift of f , the map ı̃ : X̃ → X̃ is a lift of
the identity map, and U is an open subset of X with compact fixed point set
Fix(f, U) = Coin(f, id, U). Let R[f ] = R[f, id]. Two tuples (f, f̃ , ı̃, U) and
(f ′, f̃ ′, ı̃, U) are said to be admissably homotopic if there is some homotopy Ft
of f to f ′ with {(x, t) | Ft(x) = x} compact, and Ft lifts to a homotopy of f̃ to
f̃ ′.
Our uniqueness theorem is then:
Theorem 4. If X is a (possibly nonorientable) differentiable manifold, then
there is a unique function taking an admissable tuple (f, f̃ , ı̃, U) to an element
of ZR[f ] satisfying the following axioms:
• (Additivity) If U1 and U2 are disjoint open subsets of U with Fix(f, U) ⊂
U1 ∪ U2, then
RT(f, f̃ , ı̃, U) = RT(f, f̃ , ı̃, U1) + RT(f, f̃ , ı̃, U2)
• (Homotopy) If (f, f̃ , ı̃, U) is admissably homotopic to (f ′, f̃ ′, ı̃, U), then
RT(f, f̃ , ı̃, U) = RT(f ′, f̃ ′, ı̃, U)
• (Weak normalization) If f is a constant map, then
c(RT(f, f̃ , ı̃, U)) = 1
• (Lift invariance) For any α, β ∈ π1(X), we have
c(RT(f, f̃ , ı̃, U)) = c(RT(f, αf̃ , βı̃, U))
• (Coincidence of lifts) If [α] appears with nonzero coefficient in RT(f, f̃ , ı̃, U),
then αf̃ and ı̃ have a coincidence point on p−1X (U).
Proof. First we note that a result analagous to Proposition 1 can be obtained
in the fixed point setting using only the weak normalization axiom: Using the
three axioms of [6], which make use of an appropriately weakened normalization
axiom, we see that c ◦ RT is the fixed point index.
Then letting g = id in the proof of Theorem 2, we have that, if f has isolated
fixed points,
RT(f, f̃ , ı̃, U) =
x∈Fix(f,Ux)
ind(f, Ux)[xef,eı],
where ind denotes the fixed point index, and Ux is an isolating neighborhood
for the fixed point x.
A fixed point version of Lemma 2 can be found in Lemmas 4.1 and 3.3 of
[6], and the proof of Theorem 3 can be mimicked to obtain our uniqueness
result.
Note that the uniqueness in fixed point theory requires only a weakened
version of the normalization axiom. A uniqueness result for coincidence theory
using only the weak normalization axiom can be obtained if we restrict ourselves
to self-maps of a particular (not necessarily orientable) manifold. This would
use a proof similar to the above, using results from Section 5 of [12].
5 Existence
The existence of a local Reidemeister trace in fixed point theory for connected
finite dimensional locally compact polyhedra is established by Fares and Hart in
[5]. There, the slightly more general localH-Reidemeister trace is defined, called
“the local generalized H-Lefschetz number”. An extension of this paper to the
mod H theory would not be difficult. The fact that the mod H Reidemeister
classes are unions of ordinary Reidemeister classes allows the same results to be
obtained without substantial modifications.
In [5], the additivity and homotopy axioms are proved in Proposition 3.2.9
and Proposition 3.2.8, respectively. A strong version of the lift invariance ax-
iom (see our Theorem 6) is proved in Proposition 3.2.4. The coincidence of lifts
axiom is not stated explicitly by Fares and Hart, but is a straightforward con-
sequence of their trace-like definition (if some [α] has nonzero coefficient in the
Reidemeister trace, it neccesarily comes from some simplex in the covering space
containing a fixed point of αf̃). A result analogous to the Wecken Trace Theo-
rem (which trivially implies the normalization and weak normalization axioms)
is given in Theorem 3.3.1.
No Reidemeister trace for coincidence theory, either local or global, has
appeared previously in the literature. The proof of Theorem 3 furnishes the
appropriate definition, as follows: Given an admissable tuple (f, f̃ , g, g̃, U), we
find (by Lemma 2) an admissably homotopic tuple (f ′, f̃ ′, g′, g̃′, V ) with isolated
coincidence points, and we define
RT (f, f̃ , g, g̃, U) =
x∈Coin(f ′,g′,V )
ind(f ′, g′, Vx)[xef ′,eg′ ],
where Vx is an isolating neighborhood for the coincidence point x. The above
is well defined provided that it is independent of the choice of the admissably
homotopic tuple. This is ensured by the following lemma:
Lemma 3. If (f, f̃ , g, g̃, U) and (f ′, f̃ ′, g′, g̃′, U) are admissably homotopic tu-
ples with isolated coincidence points, then
x∈Coin(f,g,U)
ind(f, g, Ux)[xef,eg] =
x′∈Coin(f ′,g′,U)
ind(f ′, g′, Ux′)[x
ef ′,eg′
where Ux is an isolating neighborhood for the coincidence point x ∈ Coin(f, g, U),
and Ux′ is an isolating neighborhood of the coincidence point x
′ ∈ Coin(f ′, g′, U).
Proof. We define the index of a coincidence class C of f and g as follows:
indC =
ind(f, g, Ux).
A class is called essential if its index is nonzero.
Since f and g are homotopic to f ′ and g′, we have R[f, g] = R[f ′, g′]. Call
this common set of Reidemeister classes R. Letting Ũ = p−1X (U), the statement
of the Lemma is equivalent to
[α]∈R
indC(f̃ , g̃, Ũ , [α])[α] =
[α]∈R
indC(f̃ ′, g̃′, Ũ , [α])[α],
and we need only show that indC(f̃ , g̃, Ũ , [α]) = indC(f̃ ′, g̃′, Ũ , [α]) for any [α].
We will prove this using Brooks’s notion of homotopy-relatedness of coincidence
classes, exposited in detail in [2] and briefly in [3].
Let Ft, F̃t, Gt, G̃t be homotopies realizing the admissable homotopy of (f, f̃ , g, g̃, U)
and (f ′, f̃ ′, g′, g̃′, U). Two coincidence points x ∈ Coin(f, g, U) and x′ ∈ Coin(f ′, g′, U)
are (Ft, Gt)–related if there is some path γ(t) in X connecting x to x
′ such that
the paths Ft(γ(t)) and Gt(γ(t)) are homotopic in Y as paths with fixed end-
points. Two coincidence classes are related if at least one point of one is related
to at least one point of the other. Theorem II.22 of [2] shows that the notion of
(Ft, Gt)-relatedness gives a bijective correspondence between the essential coin-
cidence classes of (f, g) and those of (f ′, g′). Theorem IV.24 of [2] further shows
that any two such related classes will have the same index.
What remains is an elementary argument using covering-space theory. Let
C = C(f̃ , g̃, Ũ , [α]), and let C′ be the unique coincidence class of (f ′, g′) which
is (Ft, Gt)-related to C. We need only show that C
′ = C(f̃ ′, g̃′, Ũ , [α]), and
thus (since homotopy-relatedness preserves the index) that indC(f̃ , g̃, Ũ , [α]) =
indC(f̃ ′, g̃′, Ũ , [α]).
Choose a point x ∈ C, and let x′ be a point in C′ which is (Ft, Gt) related
to x. Then there is some path γ in X from x to x′ with Ft(γ(t)) homotopic to
Gt(γ(t)). Let x̃ be some point with pX(x̃) = x and αf̃(x̃) = g̃(x̃). We can lift
γ to a path γ̃ in X̃ starting at x̃. Since Ft(γ(t)) is homotopic to Gt(γ(t)), we
will have F̃t(γ̃(t)) homotopic to G̃t(γ̃(t)), which in particular means that they
will have the same endpoint. This common endpoint is αf̃ ′(γ̃(1)) = g̃′(γ̃(1)),
which must project by pX to the point x
′. Thus x′ ∈ pX(Coin(αf̃
′, g̃′, Ũ)), and
so C′ = C(f̃ ′, g̃′, Ũ , [α]), as desired.
We have thus produced a meaningful definition of a local coincidence Reide-
meister trace on orientable differentiable manifolds of the same dimension, and
the proof above suffices to give:
Theorem 5 (Wecken Coincidence Trace Theorem). Let RT be the unique lo-
cal coincidence Reidemeister trace satisfying our five axioms. Then for any
(f, f̃ , g, g̃, U) ∈ C(X,Y ) with Ũ = p−1X (U), we have
RT(f, f̃ , g, g̃, U) =
[α]∈R[f,g]
indC(f̃ , g̃, Ũ , [α])[α].
In conclusion we prove a stronger form of the lift invariance axiom, a coin-
cidence version of a well-known property of the Reidemeister trace.
Theorem 6. Let RT be the unique local coincidence Reidemeister trace satis-
fying our five axioms. If
RT(f, f̃ , g, g̃, U) =
[σ]∈R[f,g]
k[σ][σ]
for k[σ] ∈ Z, then for any α, β ∈ π1(Y ), we have
RT(f, αf̃ , g, βg̃, U) =
[σ]∈R[f,g]
k[σ][βσα
Proof. Letting Ũ = p−1
(U), by Theorem 5 we know that k[σ] = indC(f̃ , g̃, Ũ , [σ]).
Then we have
C(αf̃ , βg̃, [σ]) = pX Coin(σαf̃ , βg̃, Ũ) = pX Coin(β
−1σαf̃ , g̃, Ũ) = C(f̃ , g̃, [β−1σα]),
and thus indC(αf̃ , βg̃, Ũ , [σ]) = k[β−1σα]. Now by Theorem 5 again, we have
RT(f, αf̃ , g, βg̃, Ũ) =
[σ]∈R[f,g]
indC(αf̃ , βg̃, [σ])[σ] =
[σ]∈R[f,g]
k[β−1σα][σ]
[γ]∈R[f,g]
k[γ][βγα
as desired.
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	Introduction
	The Axioms
	Uniqueness
	Uniqueness in fixed point theory
	Existence
