Topology and its Applications 229 (2017) 213–225
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Topology and its Applications

www.elsevier.com/locate/topol

Obstruction theory for coincidences of multiple maps ✩

Thaís F.M. Monis a, Peter Wong b,∗

a São Paulo State University (UNESP), Institute of Geosciences and Exact Sciences (IGCE), Rio Claro, 
Av. 24A, 1515 Bela Vista, Rio Claro-SP, Brazil
b Department of Mathematics, Bates College, Lewiston, ME 04240, USA

a r t i c l e i n f o a b s t r a c t

Article history:
Received 24 April 2017
Received in revised form 14 July 
2017
Accepted 28 July 2017
Available online 2 August 2017

MSC:
primary 55M20
secondary 55S35

Keywords:
Obstruction theory
Lefschetz coincidence theory
Local coefficients

Let f1, ..., fk : X → N be maps from a complex X to a compact manifold N , 
k ≥ 2. In previous works [1,12], a Lefschetz type theorem was established so that the 
non-vanishing of a Lefschetz type coincidence class L(f1, ..., fk) implies the existence 
of a coincidence x ∈ X such that f1(x) = ... = fk(x). In this paper, we investigate 
the converse of the Lefschetz coincidence theorem for multiple maps. In particular, 
we study the obstruction to deforming the maps f1, ..., fk to be coincidence free. We 
construct an example of two maps f1, f2 : M → T from a sympletic 4-manifold M
to the 2-torus T such that f1 and f2 cannot be homotopic to coincidence free maps 
but for any f : M → T , the maps f1, f2, f are deformable to be coincidence free.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

The celebrated Lefschetz coincidence theorem states that for any two maps f, g : M → N between closed 
connected oriented triangulated n-manifolds, if the Lefschetz coincidence number (trace) L(f, g) is non-zero 
then the coincidence set C(f, g) = {x ∈ M | f(x) = g(x)} must be non-empty. However, the converse does 
not hold in general. In this direction, E. Fadell showed [4] that if N is simply-connected then the vanishing 
of L(f, g) is sufficient to deform the maps f ∼ f ′, g ∼ g′ so that C(f ′, g′) = ∅. For non-simply connected N , 
the vanishing of the Nielsen number N(f, g) often provides the converse. Following [4] and [5], it was shown 
in [7] that the (primary) obstruction on(f, g) (n ≥ 3) to deforming f and g to be coincidence free is Poincaré 
dual to the twisted Thom class of the normal bundle of the diagonal Δ(N) in N ×N with appropriate local 
coefficients.

✩ This work was initiated during the second author’s visit to the Department of Mathematics at UNESP-Rio Claro, May 18–
June 17, 2014 and completed during his visit to the department Feb. 21–26, 2016. The second author would like to thank the 
Mathematics Department for the invitation and financial support. The first author was supported by FAPESP of Brazil Grant 
number 2014/17609-0.
* Corresponding author.

E-mail addresses: tfmonis@rc.unesp.br (T.F.M. Monis), pwong@bates.edu (P. Wong).
http://dx.doi.org/10.1016/j.topol.2017.07.017
0166-8641/© 2017 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.topol.2017.07.017
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mailto:tfmonis@rc.unesp.br
mailto:pwong@bates.edu
http://dx.doi.org/10.1016/j.topol.2017.07.017
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214 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
Suppose M is a compact topological space, N is a closed connected oriented manifold, and f1, ..., fk :
M → N are maps. A Lefschetz type coincidence class L(f1, ..., fk) was defined in [1] and it was shown that 
L(f1, ..., fk) �= 0 implies C(f1, ..., fk) := {x ∈ M | f1(x) = ... = fk(x)} is non-empty. This Lefschetz type 
result has been extended to non-orientable N in [12].

The purpose of this paper is to examine the converse of this Lefschetz type theorem, that is, the problem 
when

L(f1, ..., fk) = 0 ⇒ f1 ∼ f ′
1, ..., fk ∼ f ′

k such that C(f ′
1, ..., f

′
k) = ∅.

The approach here is via obstruction theory, following [4,5,3,7]. We examine the primary obstruction to 
deforming f1, ..., fk to be coincidence free on the (k − 1)n-th skeleton where n = dimN . We prove an 
analogous converse of the Lefschetz type coincidence theorem when N is simply-connected and dimM =
(k − 1)n. We give further examples of Jiang-type spaces N for which the converse theorem holds. This 
paper is organized as follows. In section 2, we generalize the Lefschetz coincidence classes defined in [1]
and in [12] to homomorphisms Λ(f1, ..., fk; RNk) with arbitrary local coefficients. This homomorphism is 
similar to a certain homomorphism φ of [8], relating to the preimage of a map. In section 3, we study the 
primary obstruction to deforming f1, ..., fk to be coincidence free using the appropriate local coefficient 
system π∗

j (F ). The calculation of the local system π∗
j (F ) is carried out in section 4. We then compute the 

obstruction to deformation in section 5 and prove the converse of the Lefschetz coincidence theorem for 
multiple maps in section 6. We also compare our result to similar Nielsen type results of P. Staecker [14]. 
In section 7, we illustrate our results with examples. In particular, we construct two maps f1, f2 : M → N , 
dimM > dimN such that f1, f2 are not deformable to be coincidence free but for any f : M → N , f1, f2, f
are homotopic to be coincidence free (see Examples 7.1 and 7.2).

2. Lefschetz coincidence homomorphism with local coefficients

In this section, we review the Lefschetz coincidence classes introduced in [1] and in [12] and compare 
them with a more general homomorphism studied in [8].

Let X be a compact topological space and N be a connected closed n-manifold. For now, let us assume 
N is oriented. Suppose f1, ..., fk : X → N are maps, μ ∈ Hn(N ×N, N ×N \ Δ(N)) be the Thom class of 
the normal bundle of the diagonal Δ(N) in N ×N . Then, in [1], the authors defined

L1(f1, ..., fk) = [(j × ...× j) ◦ (h1, ..., hk−1)]∗(μ× ...× μ) (2.1)

where hi : X → N ×N is given by hi(x) = (fi(x), fi+1(x)) and j : N ×N ↪→ (N ×N, N ×N \Δ(N)) is the 
inclusion. In [12], a similar Lefschetz type coincidence class was defined as follows. Let Δk(N) = {(x, ..., x) ∈
Nk | x ∈ N} be the diagonal of N in Nk and μk be the Thom class of the normal bundle of Δk(N) in Nk

in H(k−1)n(Nk, Nk \Δk(N); R×Γ∗
N × ... ×Γ∗

N ). Here R is a principal ideal domain and Γ∗
N = Hom(ΓN , R)

for the R-orientation system ΓN on N . Define h̃i : X → N by h̃i(x) = (f1(x), fi+1(x)). Then

L2(f1, ..., fk) = [i ◦ (f1, ..., fk)]∗(μk)

= [(j × ...× j) ◦ (h̃1, ..., h̃k−1)]∗(μ× ...× μ)
(2.2)

where i : Nk → (Nk, Nk \ Δk(N)) is the inclusion. Note that

(j × ...× j) ◦ (h1, ..., hk−1) = σ ◦ i ◦ (f1, ..., fk)

where σ(x1, ..., xk) = ((x1, x2), (x2, x3), ..., (xk−1, xk)). The induced homomorphism σ∗ : ⊗k−1[Hn(N2, N2 \
Δ(N))] → H(k−1)n(Nk, Nk\Δk(N)) is an isomorphism when N is orientable. In this case, σ∗(μ ×... ×μ) = μk

and hence L1(f1, ..., fk) = L2(f1, ..., fk).



T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225 215
A close inspection indicates that L2(f1, ..., fk) is in fact a special case of the coincidence index homo-
morphism studied in [8] which we recall as follows. Let X, Y be closed manifolds of dimension a and b, 
respectively. Suppose B ⊂ Y is a closed submanifold of dimension �. Let F : X → Y be transverse to B so 
that C = F−1(B) is a closed submanifold of dimension a − b + �. Suppose RY is a local system on Y with 
typical group R, RX = F ∗(RY ), R′

X = RX × ΓX and R′
C = i∗1(R′

X) where i1 : C = F−1(B) ↪→ X and ΓX

is the R-orientation system on X. For any 0 ≤ j, define φ(j) to be the composite homomorphism given by

Hj(Y, Y \B;RY ) F∗
−→ Hj(X,X \ C;RX) A−1

−→ Ha+b−j(C;R′
C)

where A represents the Alexander Duality isomorphism (see e.g. [13]). It was shown in [8, Theorem 2.4]
that for any r ∈ Hj(Y, Y \B; RY ),

i1∗φ(j)(r) = F ∗(j∗2 (r)) ∩ [zX ]

where j2 : Y ↪→ (Y, Y \ B) and [zX ] is the (twisted) fundamental class of X with coefficients in RX . It 
follows from sections 4.3 and 4.5 of [8] that φ((k − 1)n) �= 0 ⇒ C �= ∅. Now let a = (k − 1)n, b = kn, Y =
Nk, B = Δk(N), F = (f1, ..., fk), C = C(f1, ..., fk) and j = (k − 1)n. If RY = R × Γ∗

N × ... × Γ∗
N then it is 

easy to see that

L2(f1, ..., fk) = I∗ ◦A ◦ φ((k − 1)n)(μk) (2.3)

where I : X ↪→ (X, X \ C) is the inclusion. Now L2(f1, ..., fk) �= 0 implies that φ((k − 1)n) �= 0. Thus, the 
Lefschetz type coincidence theorems of [12] and of [1] follow immediately.

Next, we define a Lefschetz coincidence homomorphism with any arbitrary local coefficient system RNk .
Let X be a compact topological space and N a closed manifold of dimension n. Suppose f1, ..., fk : X → N

are maps. Let F = (f1, ..., fk) : (X, X \ C(f1, ..., fk)) → (Nk, Nk \ Δk(N)), I : X ↪→ (X, X − C(f1, ..., fk)). 
Define the Lefschetz coincidence homomorphism of f1, ..., fk with local coefficients RNk to be

Λ(f1, ..., fk;RNk) := I∗ ◦ F ∗ : H(k−1)n(Nk, Nk \ Δk(N);RNk) → H(k−1)n(X;F ∗(RNk)). (2.4)

Thus, we have

Λ(f1, ..., fk;R× Γ∗
N × ...× Γ∗

N )(μk) := i∗ ◦ (f1, ..., fk)∗(μk) = L2(f1, ..., fk) (2.5)

where i : Nk ↪→ (Nk, Nk \Δk(N)). Now, the following result, which generalizes the main theorems of [1,12], 
is immediate.

Theorem 2.1. Let X be a compact topological space, N a closed manifold of dimension n and k ≥ 3. For 
any maps f1, ..., fk : X → N and local coefficient RNk , if Λ(f1, ..., fk; RNk) �= 0 then C(f1, ..., fk) �= ∅.

In section 5, we will generalize the Lefschetz coincidence classes of [1,12] to Λ(f1, ..., fk; RNk)(μ̃k) for an 
appropriate local coefficient and cohomology class μ̃k which will be called the twisted Thom class of Nk.

3. Obstructions and local coefficients

In [4], E. Fadell showed that the classical Lefschetz coincidence class L(f, g) coincides with the obstruc-
tion to deforming f and g to be coincidence free provided the target manifold is simply-connected. For 
non-simply connected manifolds, the obstruction class [5] necessarily involves certain local coefficient sys-
tem. We now recall, following the treatment in [5] or in [3], the local coefficients employed in the obstruction 
to deformation.



216 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
Let f : X → Y be a map from a finite connected complex X to a closed connected manifold Y and 
B ⊂ Y be a closed submanifold. First, we replace the inclusion i : Y \ B ↪→ Y by a fiber map p : E → Y

where E = {(x, ω) | x ∈ Y \ B, ω ∈ Y [0,1], ω(0) = x} and p(x, ω) = ω(1). Let the typical fiber be 
F = p−1(x0) = {(x, ω) | ω(1) = x0} for some x0 ∈ Y \ B. It follows that πj(F ; x0) ∼= πj+1(Y, Y \ B; x0)
where x0 denotes the constant path at x0, i.e., x0(t) = x0 for all t.

It is straightforward to verify that f is deformable into Y \B, i.e., f ∼ f ′ so that f ′(X) ⊆ Y \B, if and 
only if f can be lifted to a map f̃ : X → E such that p ◦ f̃ = f . Furthermore, f is deformable into Y \B if 
and only if the pullback fibration q : f∗(p) → X induced by f has a (global) section.

If (Y, Y \B) is (m −1) connected. It follows from the classical obstruction theory for lifting (see e.g. [17]) 
that the primary obstruction to finding a section to q is given by a class om(f) ∈ Hm(X; π∗

m−1(F )) where 
π∗
m−1(F ) is the local system on X induced by f : X → Y from the local system πm−1(F ) on Y . This class 

is the obstruction to deforming f to f ′ with f ′(X(m)) ⊂ Y \B where X(m) is the m-skeleton of X.
Now let X = M, Y = Nk, B = Δk(N) and f = (f1, ..., fk) where f1, ..., fk : M → N are maps from a 

finite connected CW-complex M to a closed connected manifold N of dimension n, where k ≥ 2. Next, we 
will study the obstruction to deforming the maps f1, ..., fk to be coincidence free but first we must calculate 
the local coefficient system π∗

m−1(F ).
In this setting, we set

E = E(Nk, Nk \ Δk(N)) = {(y, ω) | y ∈ Nk \ Δk(N) and ω : I → Nk is a path with ω(0) = y}.

Let p : E → Nk be the fiber map given by p(y, ω) = ω(1). If we take a point y0 = (y1
0 , . . . , y

k
0 ) ∈ Nk the 

fiber over y0, F = p−1(y0), is given by

F = {(ω(0), ω) | ω(1) = y0}.

In other words, the fiber is the space of paths in Nk which begins in Nk \ Δk(N) and end at y0. Let 
y0 ∈ Nk \ Δk(N) and there is an isomorphism

πj(F ; y0) � πj+1(Nk, Nk \ Δk(N); y0).

The above identification can be seen as the following: the element [α] ∈ πj(F ; y0), where α : (Sj , e) → (F, y0), 
is mapped to [β] ∈ πj+1(Nk, Nk \ Δk(N); y0), where β : (Dj+1, Sj , e) → (Nk, Nk \ Δk(N), y0) is given by

β(e + t(v − e)) = α(z)(1 − t),

where e = (1, 0, . . . , 0) and v ∈ Sj .
Now, since (Nk, Nk \ Δk(N)) is (n(k − 1) − 1)-connected, F is (n(k − 1) − 2)-connected. Therefore, if 

3 ≤ n(k−1) then π(n(k−1)−1)(F ) forms a local system on Nk. Our next step is to describe such local system 
π(n(k−1)−1)(F ) on Nk.

4. The local system π(n(k−1)−1)(F )

As we saw in the last section, if we choose a base point y0 = (y1
0 , . . . , y

k
0 ) ∈ Nk \ Δk(N), we have a 

natural identification

ψy0 : πn(k−1)(Nk, Nk \ Δk(N); y0) � πn(k−1)−1(F ; y0).

In fact, ψy0 is an isomorphism of local systems on Nk \ Δk(N). We now determine the action of



T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225 217
π1(Nk \ Δk(N); y0) = π1(N ; y1
0) × · · · × π1(N ; yk0 ) = π × · · · × π︸ ︷︷ ︸

k−times

on

πn(k−1)(Nk, Nk \ Δk(N); y0).

Let η : Ñ → N be the universal cover of N . We assume that y0 = (y1
0 , . . . , y

k
0 ) lies inside a small tubular 

neighborhood of Δk(N) and we consider the diagram

Ñ

η

(Ñk, Ñk \ ξ−1(Δk(N)))

ξ

N (Nk, Nk \ Δk(N))

where ξ = η × · · · × η︸ ︷︷ ︸
k−times

and the horizontal maps are projections on the first coordinate. These horizontal 

maps are fibered pairs with fibers (Nk−1, Nk−1 \ (y1
0 , y

1
0 , . . . , y

1
0)︸ ︷︷ ︸

(k−1)−times

) (lower horizontal map) and (Ñk−1, Ñk−1 \

η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0)︸ ︷︷ ︸

(k−1)−times

) (upper horizontal map). Then, choosing ỹ0 = (ỹ1
0 , . . . , ỹ

k
0 ) ∈ Ñk, we 

have the isomorphisms

πn(k−1)(Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0); (ỹ2

0 , . . . , ỹ
k
0 ))

� πn(k−1)(Ñk, Ñk \ ξ−1(Δk(N)); ỹ0)

and

πn(k−1)(Nk−1, Nk−1 \ (y1
0 , y

1
0 , . . . , y

1
0); (y2

0 , . . . , y
k
0 ))

� πn(k−1)(Nk, Nk \ Δk(N); y0).

Moreover, because ξ : (Ñk, Ñk \ ξ−1(Δk(N))) → (Nk, Nk \ Δk(N)) is a covering map, we have

πn(k−1)(Ñk, Ñk \ ξ−1(Δk(N)); ỹ0) � πn(k−1)(Nk, Nk \ Δk(N); y0).

Therefore

πn(k−1)(Nk, Nk \ Δk(N); y0)

� πn(k−1)(Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0); (ỹ2

0 , . . . , ỹ
k
0 )).

Furthermore, by simple connectivity, it follows from the relative Hurewicz theorem that

πn(k−1)(Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0); (ỹ2

0 , . . . , ỹ
k
0 ))

� Hn(k−1)(Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0)).

Note that



218 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
(Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0)) = (Ñ , Ñ \ η−1(y1

0)) × · · · × (Ñ , Ñ \ η−1(y1
0))︸ ︷︷ ︸

(k−1)−times

and, therefore,

πn(k−1)(Nk, Nk \ Δk(N); y0)

� Hn(Ñ , Ñ \ η−1(y1
0)) ⊗ · · · ⊗Hn(Ñ , Ñ \ η−1(y1

0))︸ ︷︷ ︸
(k−1)−times

.

Let η−1(y1
0) = {ỹi}i with the convention that ỹ1 = ỹ1

0 . Let V be an euclidean neighborhood of y1
0 in N

and Ṽi an euclidean neighborhood of ỹi such that η−1(V ) = �iṼi and η : Ṽi → V is a homeomorphism. By 
excision,

Hn(Ñ , Ñ \ η−1(y1
0)) �

∑
i

Hn(Ṽi, Ṽi \ ỹi).

Identifying π with the covering transformations of Ñ , given σ ∈ π, if σỹ1
0 = ỹi then σṼ1 = Ṽi.

Choose a local orientation of V at y1
0 , which determines a generator γ1 ∈ Hn(Ṽ1, Ṽ1 \ ỹ1

0). Let α ∈ π and 
set γα = αγ1. Thus, γα generates Hn(Ṽi, Ṽi \ ỹi) if αỹ1

0 = ỹi.
Since (Ñk−1, Ñk−1 \ η−1(y1

0) × η−1(y1
0) × · · · × η−1(y1

0)) is the fiber of the fiber pair map

Ñ ← (Ñk, Ñk \ ξ−1(Δk(N)))

over each ỹi, for each ỹi we have a fiber inclusion

θi : (Ñk−1, Ñk−1 \ η−1(y1
0) × η−1(y1

0) × · · · × η−1(y1
0)) ↪→ (Ñk, Ñk \ ξ−1(Δk(N)))

given by

θi(u) = (ỹi, u), u ∈ Ñk−1.

Now, we identify Z[π × · · · × π︸ ︷︷ ︸
(k−1)−times

] with the image of

Hn(Ñ , Ñ \ η−1(y1
0)) ⊗ · · · ⊗Hn(Ñ , Ñ \ η−1(y1

0))︸ ︷︷ ︸
(k−1)−times

under θ1∗ via the correspondence

(α1, . . . , αk−1) �→ θ1∗(γα1 ⊗ · · · ⊗ γαk−1) = θ1∗(α1γ1 ⊗ · · · ⊗ αk−1γ1).

Observe that θ1∗(γ1 ⊗ · · · ⊗ γ1) can be represented by an n(k− 1)-cell {ỹ1
0} × (Dn

1 )k−1 in Ñk transverse to 
ξ−1(Δk(N)) at (ỹ1

0 , ỹ
1
0 , . . . , ỹ

1
0). Thus, for σ ∈ π, the diagonal element

(σ, σ, . . . , σ) ∈ π × π × · · · × π︸ ︷︷ ︸
k−times

sends {ỹ1
0} × (Dn

1 )k−1 to an n(k − 1)-cell {ỹi} × (Dn
i )k−1 transverse to ξ−1(Δk(N)) at (ỹi, ỹi, . . . , ỹi) if 

σỹ1
0 = ỹi. Thus, one can see that



T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225 219
(σ, σ, . . . , σ)θ1∗(γ1 ⊗ γ1 ⊗ · · · ⊗ γ1) = (sgnσ)k−1θ1∗(γ1 ⊗ γ1 ⊗ · · · ⊗ γ1).

Now, we can compute the action of πk on Z[πk−1] via the above identifications:

(σ1, . . . , σk) ◦ (α1, . . . , αk−1) ≡ (σ1, . . . , σk)θ1∗(α1γ1 ⊗ · · · ⊗ αk−1γ1)

= (σ1, . . . , σk)(1 × α1 × · · · × αk−1)θ1∗(γ1 ⊗ · · · ⊗ γ1)

= (σ1, σ2α1, . . . , σkαk−1)θ1∗(γ1 ⊗ · · · ⊗ γ1)

= (1, σ2α1σ
−1
1 , . . . , σkαk−1σ

−1
1 )(σ1, . . . , σ1)θ1∗(γ1 ⊗ · · · ⊗ γ1)

= (sgnσ1)k−1(1, σ2α1σ
−1
1 , . . . , σkαk−1σ

−1
1 )θ1∗(γ1 ⊗ · · · ⊗ γ1)

= (sgnσ1)k−1θ1∗(σ2α1σ
−1
1 γ1 ⊗ · · · ⊗ σkαk−1σ

−1
1 γ1)

≡ (sgnσ1)k−1(σ2α1σ
−1
1 , . . . , σkαk−1σ

−1
1 ).

(4.1)

By pulling back the local system by the map f : M → Nk, the action of π1(M) on Z[πk−1] is given by

γ ◦ (α1, . . . , αk−1) = (ϕ1(γ), ..., ϕk(γ)) ◦ (α1, . . . , αk−1)

= (sgnϕ1(γ))k−1(ϕ2(γ)α1ϕ1(γ)−1, . . . , ϕk(γ)αk−1ϕ1(γ)−1).
(4.2)

Here, ϕi is the homomorphism induced by fi.

5. Lefschetz coincidence class as primary obstruction

Following [5], the primary obstruction to deforming f off a subspace B was defined as a universal element
in [3]. In our setting, first we triangulate N and hence Nk so that the (k − 1)n skeleton (Nk)((k−1)n) ⊂
Nk \Δk(N) for k ≥ 3. Then there is the (only) primary obstruction μ̃k ∈ H(k−1)n(Nk, Nk \Δk(N); Z[πk−1])
to deforming the identity map 1Nk off the subspace Δk(N). We call this element μ̃k the twisted Thom class
of the normal bundle of Δk(N) in Nk or simply the twisted Thom class of Nk.

For any i, j, consider the map

e : (N i+j−1, N i+j−1 \ Δi+j−1(N)) → (N i, N i \ Δi(N)) × (N j , N j \ Δj(N))

given by e(x1, x2, ..., xi+j−1) = ((x1, x2, ..., xi), (x1, xi+1, xi+2, ..., xi+j−1)).

Proposition 5.1. The map e induces a homomorphism

H(i−1)n(N×i ;Z[πi−1]) ⊗H(j−1)n(N×j ;Z[πj−1]) → H(i+j−2)n(N×i+j−1 ;Z[πi+j−2])

such that e∗(μ̃i ⊗ μ̃j) = μ̃i+j−1. Here N×k denotes the pair (Nk, Nk \ Δk(N)) and π = π1(N).

Proof. First there is an Eilenberg–Zilber map

EZ : C∗(N×i ;Z[πi−1]) ⊗ C∗(N×j ;Z[πj−1]) → C∗(N×i+j ;Z[πi−1] ⊗ Z[πj−1])

where the action of πi+j = πi × πj on Z[πi−1] ⊗ Z[πj−1] is the diagonal action. More precisely, for 
(σ1, ..., σi+j) ∈ πi+j , we have

(σ1, ..., σi+j) ◦ (α1, ..., αi−1, β1, ..., βj−1)

=(sgnσ )i−1(σ α σ−1, ..., σ α σ−1) ⊗ (sgnσ )j−1(σ β σ−1 , ..., σ β σ−1 ).
(5.1)
1 2 1 1 i i−1 1 i+1 i+2 1 i+1 i+j j−1 i+1



220 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
Now the map e induces a local system e∗(Z[πi−1] ⊗ Z[πj−1]) on N×i+j−1 and this system is given by the 
following action:

(σ1, ..., σi+j−1) ∗ (α1, ..., αi−1, β1, ..., βj−1)

=e#(σ1, ..., σi+j−1) ◦ (α1, ..., αi−1, β1, ..., βj−1)

=(σ1, ..., σi, σ1, σi+1, ..., σi+j−1) ◦ (α1, ..., αi−1, β1, ..., βj−1)

=(sgnσ1)i−1(σ2α1σ
−1
1 , ..., σiαi−1σ

−1
1 ) · (sgnσ1)j−1(σi+1β1σ

−1
1 , ..., σi+j−1βj−1σ

−1
1 ).

This action coincides with that of (4.1) so that e∗(Z[πi−1] ⊗ Z[πj−1]) coincides with the local system 
Z[πi+j−2] discussed above. Now the obstruction to deforming the identity map 1Nq off the subspace Δq(N)
has a simple cochain representation given by c(q−1)n(1Nq)(σ) = [σ] ∈ π(q−1)n(N×q ). Let σ1×σ2×... ×σi+j−1
be an (i + j − 1)n simplex where each σi is an n-simplex in N and c̃p denote the cochain representing the 
twisted Thom class μ̃p. Then

〈c̃i+j−1, σ1 × σ2 × ...× σi+j−1〉 = [σ1 × σ2 × ...× σi × σi+1 × ...× σi+j−1] ∈ Z[πi+j−2]

= e∗([σ1 × σ2 × ...× σi] ⊗ [σ1 × σi+1 × ...× σi+j−1])

= e∗(〈c̃i, σ1 × σ2 × ...× σi〉 ⊗ 〈c̃j , σ1 × σi+1 × ...× σi+j−1]〉)

= e∗(〈c̃i ⊗ c̃j , (σ1 × σ2 × ...× σi) ⊗ (σ1 × σi+1 × ...× σi+j−1)〉)

= 〈c̃i ⊗ c̃j , e∗((σ1 × σ2 × ...× σi) ⊗ (σ1 × σi+1 × ...× σi+j−1))〉.

It follows that e∗(μ̃i ⊗ μ̃j) = μ̃i+j−1. �
As an immediate corollary, we have the following useful result.

Corollary 5.2. Let e′ : N×k → N×2 × ...×N×2︸ ︷︷ ︸
k−1

be defined by

e′(x1, ..., xk) = ((x1, x2), (x1, x3), ..., (x1, xk)).

Suppose n = dimN ≥ 3. Then e′ induces a homomorphism e′ ∗ such that e′ ∗(μ̃2 ⊗ ...⊗ μ̃2︸ ︷︷ ︸
k−1

) = μ̃k.

Using the twisted Thom class as an element in the cohomology of N×k with local coefficients 
Z[πk−1], we define the twisted Lefschetz coincidence class of f1, ..., fk to be the element L(f1, ..., fk) :=
Λ(f1, ..., fk; Z[πk−1])(μ̃k) as defined in (2.4). Thus,

L(f1, ..., fk) = I∗ ◦ F ∗ : H(k−1)n(Nk, Nk \ Δk(N);Z[πk−1]) → H(k−1)n(X;F ∗(RZ[πk−1])).

Since μ̃k is the obstruction to deforming the identity off the subspace Δk(N), it follows from [3] that 
L(f1, ..., fk) = o(k−1)n(f1, ..., fk) the primary obstruction to deforming f1, ..., fk to be coincidence free on 
the (k − 1)n skeleton of X.

In the case when N is oriented, it was already shown in [7] that the primary obstruction to deforming f and 
g is mapped to the classical Lefschetz coincidence number L(f, g) under the augmentation homomorphism 
Z[π] → Z. Next, we show an analogous result, that is, there is a natural homomorphism η̃ induced by the 
augmentation map such that η̃(L(f1, ..., fk)) = L2(f1, ..., fk). Here, we assume the principal ideal domain 
R is Z.



T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225 221
Theorem 5.3. The augmentation map εk−1 : Z[πk−1] → Γ∗
N × ... ×Γ∗

N induces a homomorphism η̃ such that 
η̃(L(f1, ..., fk)) = L2(f1, ..., fk).

Proof. By the Alexander duality isomorphism (see [13, Thm. 6.4]), we have the following commutative 
diagram.

H(k−1)n(N×k ;Z[πk−1]) A−1
1−−−−→ Hn(Δk(N); ΓNk ⊗ Z[πk−1])

η

⏐⏐� ⏐⏐�1⊗εk−1

H(k−1)n(N×k ;Z× Γ∗
N × ...× Γ∗

N ) A2←−−−− Hn(Δk(N); ΓNk ⊗ Z× Γ∗
N × ...× Γ∗

N )

Here, A1 and A2 are the corresponding duality isomorphisms so that η = A2 ◦ (1 ⊗ εk−1) ◦A−1
1 where εk−1

is the augmentation map. For any σ ∈ π, the action of (σ, ..., σ) ∈ π1(Δk(N)) on Z × Γ∗
N × ... × Γ∗

N is the 
same as the action on Z[πk−1] followed by εk−1. This implies that the twisted fundamental class of Δk(N)
with coefficients in ΓNk ⊗ Z[πk−1] is mapped under 1 ⊗ εk−1 to the twisted fundamental class of Δk(N)
with coefficients in ΓNk ⊗ Z × Γ∗

N × ... × Γ∗
N . By duality, the twisted Thom class μ̃k is mapped under η to 

μk. Define η̃ by 〈η̃(m), σ〉 := η(〈m, σ〉). Pulling back the classes μ̃k, μk yields the assertion. �
Theorem 5.4. Let f1, ..., fk : X → N be maps from a finite complex X to a closed connected manifold N of 
dimension n, n ≥ 3. Then

o(k−1)n(f1, ..., fk) = on(f1, f2) ∪ on(f1, f3) ∪ ... ∪ on(f1, fk).

Proof. In Corollary 5.2 we proved that if e′ : N×k → N×2 × ...×N×2︸ ︷︷ ︸
k−1

is the map defined by e′(x1, ..., xk) =

((x1, x2), (x1, x3), ..., (x1, xk)) then e′ ∗(μ̃2 ⊗ ...⊗ μ̃2︸ ︷︷ ︸
k−1

) = μ̃k. Let us consider the inclusions I : X → (X, X \

C(f1, . . . , fk)) and ji : X → (X, X \ C(f1, fi)), i = 2, . . . , k. Then,

o(k−1)n(f1, ..., fk) = I∗(f1, . . . , fk)∗(μ̃k)

and

on(f1, fi) = j∗i (f1, fi)∗(μ̃2), i = 2, . . . , k.

Note that e′ ◦ (f1, . . . , fk) ◦ I = ((f1, f2) × . . .× (f1, fk)) ◦ (j1, . . . , jk−1). Therefore

o(k−1)n(f1, ..., fk) = I∗(f1, . . . , fk)∗(μ̃k)

= I∗(f1, . . . , fk)∗(e′ ∗(μ̃2 ⊗ ...⊗ μ̃2))

= (j1, . . . , jk−1)∗ ◦ ((f1, f2) × . . .× (f1, fk))∗(μ̃2 ⊗ ...⊗ μ̃2)

= j∗1(f1, f2)∗(μ̃2) ∪ j∗2 (f1, f3)∗(μ̃2) ∪ ... ∪ j∗k−1(f1, fk)∗(μ̃2)

= on(f1, f2) ∪ on(f1, f3) ∪ ... ∪ on(f1, fk) �
Remark 5.5. Since o(k−1)n(f1, ..., fk) = L(f1, ..., fk) and on(f1, fi) = L(f1, fi), Theorem 5.4 states that for 
n ≥ 3, we have

L(f1, ..., fk) = L(f1, f2) ∪ L(f1, f3) ∪ ... ∪ L(f1, fk). (5.2)



222 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
Combining with the homomorphism η̃ as in Theorem 5.3, it is easy to establish the following product

L2(f1, ..., fk) = L2(f1, f2) ∪ L2(f1, f3) ∪ ... ∪ L2(f1, fk)

which is obtained in [12, Thm. 4.2].

6. Converse of Lefschetz coincidence theorem

The converse of the Lefschetz coincidence theorem in our setting amounts to studying the problem 
when the maps f1, ..., fk can be deformed to be coincidence free. For instance in the classical case, it has 
been shown [4] that when N is simply-connected then the vanishing of the Lefschetz coincidence number 
provides a converse to the Lefschetz coincidence theorem. To study this problem for multiple maps, we use 
the obstruction theory developed in the previous sections.

Next, we need to calculate the obstruction cocycle, following [3, pp. 20–21]. Suppose M is a closed (k −
1)n-dimensional manifold and N is a closed n-manifold. By general position, we may assume that the map 
f = (f1, ..., fk) : M → Nk is transverse to the submanifold Δk(N) so that C = f−1(Δk(N)) = C(f1, ..., fk)
is a finite set of coincidence points of f1, . . . , fk. Let C = {s1, ..., sr}. For each isolated coincidence point si, 
we may assume that there is a maximal (k−1)n-simplex σi containing si such that C∩σi = {si}. Following 
[5,3], the primary obstruction cocycle c(k−1)n(f) to deforming f into the subspace Nk \Δk(N) is given by

c(k−1)n(f)(σ) =
{

0 if σ ∩ C = ∅

λ(f, si)[f |σ] if σ = σi.
(6.1)

Here, f |σi
represents the restriction map from (σi, ∂σi) to (Vi × ... × Vi, Vi × ... × Vi \ {(yi, ..., yi)}) where 

Vi is an open neighborhood of yi = f1(si) = ... = fk(si) in N . Thus, [f |σi
] ∈ π(k−1)n(Nk, Nk \ Δk(N)) ∼=

π(k−1)n−1(F ). Furthermore the coefficient λ(f, si) = f |∗σi
(μ̃k) is the local coincidence index defined by the 

restriction f |σi
using the homomorphism

f |∗σi
: H(k−1)n(Nk, Nk \ Δk(N)) → H(k−1)n(σi, σi \ {si}).

Since (M, M − C) = (M, M − {s1, ..., sr}), it follows from excision and additivity that

I∗ ◦ f∗(μ̃k) =
r∑

i=1
λ(f, si)

where I : M ↪→ (M, M − C). It follows that

c(k−1)n(f) =
r∑
i

λ(f, si)[f |σi
] (6.2)

and

L(f1, ..., fk) =
r∑

i=1
λ(f, si). (6.3)

Theorem 6.1. Let f1, . . . , fk : M → N from a closed connected (k − 1)n-manifold M to a closed connected 
n-manifold N . Then f1, . . . , fk are deformable to be coincidence free if, and only if, o(k−1)n(f1, ..., fk) = 0
where the obstruction cocycle is given by (6.2).



T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225 223
In the case where N is simply-connected, o(k−1)n(f1, ..., fk) = L(f1, ..., fk) = L2(f1, ..., fk) = L1(f1, ..., fk)
since N is orientable and the coefficients are simple. In particular, when N = Sn is the n-sphere, the formula 
of [1, Prop. 3.2] becomes

o(k−1)n(f1, ..., fn) =
k−1∑
i=0

(−1)(i−1)nf̂k−i

∗
(e)

where e = en × ...× en︸ ︷︷ ︸
(k−1)−times

, for en the cohomology fundamental class of Sn and, f̂j = (f1, ..., fj−1, fj+1, ..., fk). 

This formula generalizes a similar formula in [7, Theorem 3.4]. Furthermore, the equality o(k−1)n(f1, ..., fk) =
L2(f1, . . . , fk) generalizes [4, Theorem 1.1] and we have the following converse theorem of the Lefschetz 
coincidence theorem when N is simply connected.

Theorem 6.2. Let f1, . . . , fk : M → N from a closed connected (k − 1)n-manifold M to a closed connected 
and simply-connected n-manifold N . Then f1, . . . , fk are deformable to be coincidence free if, and only if, 
L2(f1, ..., fk) = 0.

To obtain the converse Lefschetz theorem, Nielsen coincidence theory is often employed. In [14], P. 
Staecker independently investigated the coincidence problem for multiple maps from the Nielsen coincidence 
theory point of view. More precisely, Staecker considered the following two maps F = (f1, ..., f1), G =
(f2, ..., fk) : M → Nk−1 and employed the usual Nielsen coincidence theory by defining the Lefschetz and 
Nielsen coincidence numbers L(F, G) and N(F, G). When N is non-orientable, an appropriate (semi)index 
was used to define essentiality of the coincidence classes. We should point out that when N is orientable, 
|L2(f1, ..., fk)| = |L(F, G)| and o(k−1)n(f1, ..., fk) = 0 ⇔ N(F, G) = 0. In this case, since C = C(f1, ..., fk) =
{s1, ..., sr} is a finite set, C is a disjoint union of N(F, G) coincidence classes. Thus, the coincidence classes 
will have index of the same sign, if N is a Jiang space; a nilmanifold [9], an orientable coset space G/K of 
a compact connected Lie group G by a closed subgroup K [16]; or a C-nilpotent space whose fundamental 
group has a finite index center [10] where C denotes the class of finite groups. For these spaces, L(F, G) =
0 ⇒ N(F, G) = 0. Thus, we have the following converse theorem.

Theorem 6.3. Let f1, . . . , fk : M → N from a closed connected (k − 1)n-manifold M to a closed connected 
orientable n-manifold N . Suppose N is a Jiang space; a nilmanifold, an orientable coset space G/K of a 
compact connected Lie group G by a closed subgroup K; or a C-nilpotent space whose fundamental group has 
a finite index center where C is the class of finite groups. Then f1, . . . , fk are deformable to be coincidence 
free if, and only if, L2(f1, ..., fk) = 0.

7. Examples

When n = 2 and k ≥ 3, (k − 1)n ≥ 3 so that π(k−1)n−1(F ) ∼= π(k−1)n(Nk, Nk \ Δk(N)) is an abelian 
group so it is a local coefficient system on Nk even when N is a surface. Thus, Theorem 6.2 and Theorem 6.3
are valid when n = 2 provided k ≥ 3. Next we illustrate by an example in which f1, f2, f3 : X → Sg are 
maps from a closed 4-manifold to a surface Sg such that f1, fi cannot be deformed to be coincidence free 
for i = 2, 3. However, f1, f2, f3 are deformable to be coincidence free.

Example 7.1. Let X = S1 ×S3, f1 : X → S2 be given by f1(x1, x2) = h(x2) where x2 ∈ S3 and h : S3 → S2

is the Hopf map. Let f2 = c be the constant map at some c ∈ S2. Note that f1 : X → S2 is a fibration 
and f∗

1 ([μ]) = 0 where [μ] is the cohomology fundamental group of the base space S2. This follows from 
the fact that H2(X) = 0. Thus, we conclude that L2(f1, f2) = 0. Note that f1 and f2 cannot be deformed 



224 T.F.M. Monis, P. Wong / Topology and its Applications 229 (2017) 213–225
to be coincidence free. Now let f3 : X → S2 be any map. Since L2(f1, f2) = 0, by [12, Thm. 4.2], we have 
L2(f1, f2, f3) = 0. The 2-sphere S2 is simply connected so by Theorem 6.2, f1, f2, f3 are deformable to be 
coincidence free. In particular, if we choose f3 = f2 then f1, fi are not deformable to be coincidence free 
for i = 2, 3. In fact, any maps f1, f2, f3 : X → S2 are deformable to be coincidence free. Note that X is not 
symplectic so by the classical condition of Thurston [15], the fiber S1 × S1 is null homologous in X. As it 
turns out, Thurston’s condition is only necessary for X to be symplectic as we use that to construct the 
next example.

Example 7.2. According to R. Geiges [6], there exist symplectic 4-manifolds X that is a principal T 2-bundle 
over T 2 but the fibration is not symplectic so that i∗([F ]) = 0 in H2(X) where [F ] is the fundamental class 
of the fiber F = T 2. According to a result of Gottlieb (see its generalization in [8]), if p : X → T 2 is such 
a bundle then p∗([T 2]) = 0. It follows that L2(p, c) = 0 for any c ∈ T 2 (base). On the other hand, T 2 is 
aspherical, so it follows from the main result of [11] that the (abelianized) obstruction o2(p, c) �= 0. Again, 
for any f : X → T 2, L2(p, c, f) = 0. Since T 2 is a Jiang space, it follows from Theorem 6.3 that p, c, f are 
deformable to be coincidence free.

We can now generalize the examples above as follows.

Example 7.3. Let M = M (k−1)n be an even dimensional closed connected oriented manifold and p : M → N

is a fibration over a closed connected oriented n-manifold N with a typical fiber F , a closed connected 
oriented manifold. Assume that M is not symplectic, N is aspherical, and is a Jiang-type space as in 
Theorem 6.3. For any c ∈ N and any maps f3, ..., fk : M → N , L2(p, c, f3, ..., fk) = 0 so that p, c, f3, ..., fk
are deformable to be coincidence free but p, c cannot be homotopic to coincidence free maps.

Remark 7.4. Following an observation of F.B. Fuller, Brooks [2] showed that if C(f ′, g′) denotes the co-
incidence set for f ′ ∼ f, g′ ∼ g then there exists g′′ ∼ g such that C(f ′, g′) = C(f, g′′). In other words, 
deforming both maps can be achieved by deforming only one of the two maps. In this multiple map setting, 
we can ask the same question: Suppose f1 ∼ f ′

1, ..., fk ∼ f ′
k with C(f ′

1, ..., f
′
k). Can we obtain the same 

coincidence set by fixing more than one of the k-maps? Using the last example with k = 3, we take the 
three maps to be either (1) p, p, c or (2) c, c, p where c denotes the constant map. If we fix the first two 
maps in each case, deforming the third map will always yield coincidences. This means that one can fix 
at most one map without changing the coincidence set so that the main result of [2] cannot be improved. 
Furthermore, since the approach of Staecker [14] is to consider the (codimension zero) coincidence problem 
for the maps F = (f1, ..., f1) and G = (f2, ..., fk), our discussion above shows that if f1, ..., fk are deformable 
to be coincidence free then G cannot be kept fixed, in other words, one must deform G to some G′ with 
C(F, G′) = ∅.

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	Obstruction theory for coincidences of multiple maps
	1 Introduction
	2 Lefschetz coincidence homomorphism with local coefficients
	3 Obstructions and local coefficients
	4 The local system π(n(k-1) -1)(F)
	5 Lefschetz coincidence class as primary obstruction
	6 Converse of Lefschetz coincidence theorem
	7 Examples
	References