A Lefschetz type homomorphism for coincidence of several maps

Abstract

Given $p$-maps $f_{1},\cdots,f_{p}:X\to M,$ $p\geq 2,$ from an arbitrary topological space to an orientable closed connected $m$-manifold, in this paper we define a graded homomorphism $\Lambda_{f_{1}\cdots f_{p}}:H(X)\to H(M^{p-1})$ of degree $-m(p-1)$ called by Lefschetz homomorphism. If the Lefschetz homomorphism is nontrivial then there is a point $x\in X$ such that $f_{1}(x)=\cdots=f_{p}(x).$ The Lefschetz homomorphism $\Lambda_{f_{1}\cdots f_{p}}$ can be represented as a Knill-like trace.

Key words: Coincidence theory, Lefschetz homomorphism, Knill trace

1 Introduction

Let $f,g:X\to M$ be two maps from a topological space $X$ into an oriented compact $m$-manifold $M.$ In [7] was developed a Lefschetz coincidence theory to study the set $\operatorname{Coin}(f,g)=\{x\in X\mid f(x)=g(x)\}.$ The author defined a graded homomorphism $\Lambda_{fg}:H(X)\to H(M)$ of degree $(-m)$ and proved that if $\Lambda_{fg}$ is nontrivial then there exist a point $x\in X$ such that $f(x)=g(x).$

A natural generalization of the coincidence problem is to study the coincidence problem for several maps, that is, given $f_{1},f_{2},\cdots,f_{p}:X\to M$ is there $x\in X$ such that $f_{1}(x)=\cdots=f_{p}(x)$ ? Several authors have been working on this problem, see for example [1, 8, 4, 5, 6].

In [1] the authors defined a cohomology class $L(f_{1},\cdots,f_{p})$ called by Lefschetz class such that $L(f_{1},\cdots,f_{p})\neq 0$ then $\operatorname{Coin}(f_{1},\cdots,f_{p})\neq\emptyset.$ This theory was generalized in [5] in the case where $M$ is not necessarily orientable.

In [6] the obstruction theory was used to study when $(f_{1},\cdots,f_{p})$ can be deformed to $(f^{\prime}_{1},\cdots,f^{\prime}_{p})$ such that $\operatorname{Coin}(f^{\prime}_{1},\cdots,f^{\prime}_{p})=\emptyset.$ This theory was generalized for fiber bundles in [4].

In this work we study $\operatorname{Coin}(f_{1},\cdots,f_{p})$ using a “combinatorial” way and we extend the theory developed in [7] for $p$-maps $f_{1},\cdots,f_{p}:X\to M,$ where $p\geq 2.$ We have defined a graded homomorphism $\Lambda_{f_{1}\cdots f_{p}}:H(X)\to H(M^{p-1}),$ called by Lefschetz homomorphism, and prove that if $\Lambda_{f_{1}\cdots f_{p}}$ is not trivial then the set $\operatorname{Coin}(f_{1},\cdots,f_{p})$ is not empty. We have proved in Theorem 4.6 that

$$\Lambda_{f_{1}\cdots f_{p}}=\pm\eta^{\prime}\circ(\Lambda_{f_{1}f_{2}}\otimes\cdots\otimes\Lambda_{f_{1}f_{p}})\circ\kappa_{X}\circ\Delta_{X}^{(p-1)},$$

where $\Delta_{X}^{(p-1)}$ is the homological coproduct induced by the $(p-1)$-fold diagonal and $\kappa_{X}$ is the sign homomorphism defined in Section 4.

When $p=2,$ $X=M\times Y$ and $f_{1}$ or $f_{2}$ is the projection map $\pi:M\times Y\to M$ then was defined in [3] a trace called by the Knill trace to study fixed points of parametrized maps. The Lefschetz homomorphism $\Lambda_{f_{1}\cdots f_{p}}$ can be represented as a Knill-like trace, see Remark 4.2.

This work is organized into four sections besides this one. In Section 2 we present the definitions of the Knill trace for a parametrized map, the Lefschetz and the coincidence homomorphims. In Section 3 we define the coincidence homomorphism for $p$-maps and present a representation formula, Theorem 3.8. In section 4 we define the Lefschetz homomorphism for $p$-maps and present a result relating that Lefschetz homomorphism for coincidence of two maps, see Theorem 4.6.

In section 5 we compute the Lefschetz homomorphism for some special cases.

2 Preliminaries

2.1 Notations

Given a $m$-manifold $M$ we will denote

$$M^{\times}=(M\times M,M\times M\setminus\Delta_{M}),$$

where $\Delta_{M}=\{(x,x)\mid x\in M\}$ and

$$M^{\times p}=(M^{\times})^{p}=M^{\times}\times\cdots\times M^{\times},\quad(p\text{-times}).$$

We will also denote by $\Delta_{M^{p}}=\{((z_{1},\cdots,z_{p}),(z_{1},\cdots,z_{p}))\mid z_{i}\in M\}$ the diagonal in $M^{p}\times M^{p}$ and

$$M^{\times}_{p}=(M^{p}\times M^{p},M^{p}\times M^{p}\setminus\Delta_{M^{p}}).$$

We recall that the product of two pairs of topological spaces $(Z_{1},B_{1})$ and $(Z_{2},B_{2})$ is defined by

$$(Z_{1},B_{1})\times(Z_{2},B_{2}):=(Z_{1}\times Z_{2},Z_{1}\times B_{2}\cup B_{1}\times Z_{2}).$$

More generally given $(Z_{i},B_{i}),$ $i=1,\cdots,k$ $k$-pairs of topological spaces we define

$$\prod_{i=1}^{k}(Z_{i},B_{i}):=\left(\prod_{i=1}^{k}Z_{i},\,\,\bigcup_{i=1}^{k}Y_{i}\right),$$

where $Y_{i}=Z_{1}\times\cdots\times Z_{i-1}\times B_{i}\times Z_{i+1}\times\cdots\times Z_{k}.$

Lemma 2.1.

Let $X_{i}$ be topological spaces, $u_{i}\in H^{r_{i}}(X_{i},R)$ and $z_{i}\in H_{s_{i}}(X_{i},R)$ for each $i=1,\cdots,n.$ Then

$$(u_{1}\times u_{2}\times\cdots\times u_{n})\smallfrown(z_{1}\times z_{2}\times\cdots\times z_{n})=(-1)^{A_{n}}(u_{1}\smallfrown z_{1})\times\cdots\times(u_{n}\smallfrown z_{n}),$$

where

$$A_{n}=\sum_{j=2}^{n}r_{j}\left(\sum_{i=1}^{j-1}(s_{i}-r_{i})\right).$$

Proof.

The case $n=2$ follows from [10, Chapter 5, pg 142, exercise 14]. The general case follows by a simple induction. ∎

Corollary 2.2.

If $r_{i}=r$ and $s_{i}=s$ for all $i=1,\cdots,n$ then

$$A_{n}=\frac{n(n-1)r(s-r)}{2}$$

If $\pi_{1}:M\times M\to M$ is the projection on the first coordinate then the Thom isomorphism $\varphi_{M}:H_{k}(M^{\times})\to H_{k-m}(M)$ is given by $\varphi_{M}(x)={\pi_{1}}_{\#}(\tau_{M}\smallfrown x),$ where $\tau_{M}\in H^{m}(M^{\times})$ is the Thom class of the fiber bundle pair $\xi_{2}(M)\equiv M^{\times}\stackrel{{\scriptstyle{\scriptstyle\pi_{1}}}}{{{\longrightarrow}}}M.$

The $p$-times product of $\xi_{2}(M)$ we will produce the fiber bundle pair $\xi_{p}(M)\equiv M^{\times p}\stackrel{{\scriptstyle{\scriptstyle\pi}}}{{{\longrightarrow}}}M^{p}$ where $\pi=\underbrace{\pi_{1}\times\cdots\times\pi_{1}}_{p\text{-times}}.$ The Thom class of $\xi_{p}(M)$ will be denoted by $\tau_{M^{p}}.$

The Thom class of the fiber bundle pair $M^{\times}_{p}\stackrel{{\scriptstyle{\scriptstyle\pi^{\prime}}}}{{{\longrightarrow}}}M^{p}$ will be denoted by $\tau_{M^{\times}_{p}},$ where $\pi^{\prime}$ is the projection on the first coordinate.

2.2 Traces

The following theory of the Lefschetz class of a graded homomorphism presented in [2, Section 1] or [7, Section 2] will be useful in this work.

Let $E_{\ast}$ be a finite dimensional graded vector space over a field $\mathbb{F}.$ Consider the dual graded vector space $E^{\ast},$ where $E^{j}=\operatorname{Hom}_{\mathbb{F}}(E_{j},\mathbb{F}).$ Let $C_{\ast}$ be another graded vector space over $\mathbb{F}$ and suppose that a cap product with values in $C_{\ast}$ is given, that is, a collection of homomorphism;

$$E^{r}\otimes E_{s}\stackrel{{\scriptstyle{\scriptstyle\smallfrown}}}{{{\longrightarrow}}}C_{s-r}.$$

There is a natural isomorphism $\Theta:E^{k}\otimes E_{k+n}\to\operatorname{Hom}_{\mathbb{F}}(E_{k},E_{k+n})$ given by

$$\Theta(h\otimes y)(x)=(-1)^{|y||x|}\langle h,x\rangle y,$$

where $|w|$ denotes the degree of $w,$ $\langle,\rangle:E^{j}\otimes E_{j}\to\mathbb{F}$ is the dual paring, $h\in E^{k},$ $x\in E_{k}$ and $y\in E_{k+n}.$

Remark 2.3.

Here there is a small difference in the definition of $\Theta$ compared to those given in [2, Section 1]. There was defined $\Theta(h\otimes y)(x)=\langle h,x\rangle y,$ that is, there is a difference in the signal. In this work, to present the definition of Lefschetz class homomorphism for the $p$-maps $f_{1},\cdots,f_{p}$, we will use the definition above.

Definition 2.4.

The trace of a homomorphism $f\in\operatorname{Hom}_{\mathbb{F}}(E_{k},E_{k+n})$ of degree $n$ is given by

$$\operatorname{trace}(f)=\smallfrown\Theta^{-1}(f)\in C_{n}.$$

Proposition 2.5.

[2, Proposition 2] If $\{a_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ is a basis for $E_{k}$ with corresponding dual basis $\{\bar{a}_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ of $E^{k}$ then

$$\operatorname{trace}(f)=\sum_{j=1}^{\alpha_{k}}\bar{a}_{j}^{k}\smallfrown f(a_{j}^{k}).$$

Definition 2.6.

Let $E_{\ast}$ be a finite dimensional graded vector space equipped with a cap product taking values in the graded vector space $C_{\ast}.$ If $h:E_{\ast}\to E_{\ast}$ is a homomorphism of degree $n$ then the Lefschetz class of $h,$ $L_{n}(h)\in C_{n},$ is;

$$L_{n}(h)=\sum_{k}(-1)^{k(k+n)}\operatorname{trace}(h_{k}).$$

From [2, Proposition 1.2] we obtain the following expression of the Lefschetz class $L_{n}(h)$ as a Knill-like trace;

Proposition 2.7.

If $h:E_{\ast}\to E_{\ast}$ is a homomorphism of degree $n$ then

$$L_{n}(h)=\sum_{k\geq 0}(-1)^{k(k+n)}\sum_{j=1}^{\alpha_{k}}\bar{a}_{j}^{k}\smallfrown h(a_{j}^{k})$$

where $\{a_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ is a basis for $E_{k}$ with corresponding dual basis $\{\bar{a}_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ of $E^{k},$ for each $k.$

We have analogous definitions considering the dual graded vector space $E^{\ast}$ see [2, Definitions 1.5 and 1.7].

2.3 The Knill trace for a parametrized map

Let $F:(X,A)\times Y\to(X,A)$ be a continuous parametrized map, where $Y$ is the parameter space, $\mathbb{F}$ a field and $(X,A)$ a topological pair of spaces such that $H_{\ast}(X,A)$ is finite dimensional of $\mathbb{F}.$ We say that $(x,y)$ is a fixed point of $F$ if $F(x,y)=x.$

Let $u\in H_{n}(Y)$ and $F_{u}:H(X,A)\to H(X,A)$ be the homomorphism of degree $n$ given by $F_{u}(v)=F_{\ast}(u\times v).$ Let $H^{j}(X,A)\otimes H_{j+n}(X,A)\stackrel{{\scriptstyle{\scriptstyle\smallfrown}}}{{{\to}}}H_{n}(X)$ the cap product. In [3] was given the following definition of a Lefschetz class of $F,$ $L_{n}(F_{u})\in H_{n}(X).$

Definition 2.8.

The Knill trace of $F:(X,A)\times Y\to(X,A)$ is the homomorphism of degree 0 $L(F):H_{\ast}(Y)\to H_{\ast}(X)$ given by $L(F)(u)=L_{n}(F_{u})\in H_{n}(X),$ where $u\in H_{n}(Y).$

Follows from [3, Theorem 1] which if $L(F)$ is the non trivial homomorphism then $F$ has a fixed point. Note that $L(F)$ depends only the homotopy class of $F,$ and from [2, Proposition 2] we have;

$$L(F)(u)=\sum_{k\geq 0}(-1)^{k(k+n)}\sum_{j=1}^{\beta_{k}}\bar{b}_{j}^{k}\smallfrown F_{\ast}(u\times b_{j}^{k})$$

where $u\in H_{n}(Y)$ and $\{b_{j}^{k}\mid j=1,\dots,\beta_{k}\}$ is a basis for $H_{k}(X,A)$ with corresponding dual basis $\{\bar{b}_{j}^{k}\mid j=1,\dots,\beta_{k}\}$ of $H^{k}(X,A),$ for each $k.$ For more details and some calculations of this trace see [2].

2.4 The Lefschetz and coincidence homomorphisms

Substantial part of this subsection come from [7]. From now on the homology and cohomology groups will be with rational coefficients.

Remark 2.9.

The theory developed in [7] consider the manifold $M$ possibly with boundary. Thus only in this section we will suppose which $M$ may have boundary.

Let $f:(X,A)\to(M,\partial M)$ and $g:X\to M$ be two maps, where $X$ is a topological space, with $A\subset X,$ and $M$ is an oriented connected compact $m$-manifold with boundary $\partial M.$

We suppose that $\operatorname{Coin}(f,g)\cap A=\emptyset.$ The map $(f\times g)\circ\delta:(X,A)\to M^{\times}$ is well defined.

Definition 2.10.

The coincidence homomorphism of the pair $(f,g)$ is the homomorphism $I_{fg}:H(X,A)\to H(M^{\times})$ of degree $0$ defined by

$$I_{fg}=(f\times g)_{\#}\circ\delta_{\#}.$$

Is not difficult to see that $I_{fg}\neq 0\Longrightarrow\operatorname{Coin}(f,g)\neq\emptyset.$ There is a representation formula for $I_{fg}$ in terms of the fundamental class $O_{M}$ of $M,$ see [7, Theorem 4.5].

We consider the transfer of $f$ with respect to a fixed element $z\in H_{n+m}(X,A),$ that is, the homomorphism $f_{!}^{z}:H_{\ast}(M)\to H_{\ast+n}(X)$ of degree $n$ defined by;

$$f_{!}^{z}=(f^{\#}\circ D^{-1})\smallfrown z,$$

where $D:H^{\ast}(M,\partial M)\to H_{m-\ast}(M)$ is the Poincaré-Lefschetz duality isomorphism with

$$D(x)=x\smallfrown O_{M},$$

and $O_{M}\in H_{m}(M,\partial M)$ is the fundamental class of $(M,\partial M).$ Therefore the homomorphism $g_{\#}\circ f_{!}^{z}:H(M)\to H(M)$ has degree $n.$

Definition 2.11.

The Lefschetz homomorphism $\Lambda_{fg}:H_{\ast}(X,A)\to H_{\ast-m}(M)$ of the pair $(f,g)$ is the homomorphism of degree $(-m)$ given by

$$\Lambda_{fg}(z)=L(g_{\#}\circ f_{!}^{z}),\qquad z\in H_{n+m}(X,A).$$

Note that the degree of $g_{\#}\circ f_{!}^{z}$ is $|z|-m.$ Thus we can represent the Lefschetz homomorphism $\Lambda_{fg}$ as a Knill like trace by

$$\Lambda_{fg}(z)=\sum_{k}(-1)^{k(k+|z|-m)}\sum_{j}\bar{a}_{j}^{k}\smallfrown g_{\#}\circ f_{!}^{z}(a_{j}^{k}),$$

where $\{a_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ is a basis for $H_{k}(M)$ with corresponding dual basis $\{\bar{a}_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ of $H^{k}(M),$ for each $k.$ Follows from [7, Theorem 6.1] that $\Lambda_{fg}(z)\neq 0$ then the pair $(f,g)$ has a coincidence. The next result gives a relation between the coincidence homomorphism and the Lefschetz homomorphism.

Theorem 2.12.

[7, Theorem 6.1] The coincidence homomorphism is equal to the Lefschetz homomorphism:

$$\varphi_{M}\circ I_{fg}=\Lambda_{fg}.$$

Moreover, if $\Lambda_{fg}\neq 0$ then $(f,g)$ has a coincidence.

The Lefschetz homomorphism for coincidence generalizes the Knill trace of a parametrized map defined in [2, Definition 2.1].

Corollary 2.13.

[7, Corollary 5.7] Let $F,p:M\times Y\to M$ be maps where $p$ is the projection. We have $\operatorname{Fix}(F)=\operatorname{Coin}(p,F)$ and

$$\Lambda_{pF}(O_{M}\times u)=L({F_{u}}_{\#}),\qquad u\in H(Y),$$

where $F_{u}:H(M)\to H(M)$ is given by $F_{u}(x)=(-1)^{(m-|x|)|u|}F_{\#}(x\times u).$

3 The coincidence homomorphism for $p$-maps

Let $f_{1},\cdots,f_{p}:X\to M$ continuous maps, where $p\geq 2,$ $X$ is a topological space and $M$ is an oriented connected compact $m$-manifold without boundary. We define $F,G:X\to M^{p-1},$ the $(p-1)$-fold product of $M,$ by

$$F=(f_{1},\cdots,f_{1})\qquad\text{and}\qquad G=(f_{2},\cdots,f_{p}).$$

Note that $\operatorname{Coin}(F,G)=\operatorname{Coin}(f_{1},\cdots,f_{p}).$

Proposition 3.1.

The map $f:X\to M^{p}$ given by $f=(f_{1},\cdots,f_{p})$ can be deformed to a map $g=(g_{1},\cdots,g_{p})$ such that $\operatorname{Coin}(g_{1},\cdots,g_{p})=\emptyset$ if and only if the pair $(F,G)$ can be deformed to a pair of maps $(F^{\prime},G^{\prime})$ such that $\operatorname{Coin}(F^{\prime},G^{\prime})=\emptyset.$

Proof.

Follows from [4, Proposition 2.2] in the particular case when $B$ is a point. ∎

From Proposition 3.1 is reasonable to define a Lefschetz homomorphism for $(F,G)$ to study the coincidence set of the $p$-maps $f_{1},\cdots,f_{p}.$

Let $A\subset X$ such that $\operatorname{Coin}(F,G)\cap A=\emptyset.$ The map $(F\times G)\circ\delta:(X,A)\to M^{\times}_{p-1}$ is well defined, where $\delta:X\to X\times X$ is given by $\delta(x)=(x,x).$

Definition 3.2.

The coincidence homomorphism of the $p$-maps $(f_{1},\cdots,f_{p}),$ is the homomorphism $I_{FG}:H(X,A)\to H(M^{\times}_{p-1})$ of degree $0$ defined by

$$I_{FG}=(F\times G)_{\#}\circ\delta_{\#}.$$

We will denote $I_{f_{1}\cdots f_{p}}=I_{FG}.$

Is not difficult to see that $I_{FG}\neq 0\Longrightarrow\operatorname{Coin}(F,G)\neq\emptyset\Longrightarrow\operatorname{Coin}(f_{1},\cdots,f_{p})\neq\emptyset.$

Proposition 3.3.

From dimensional reasons we have, if $z\in H_{j}(X,A)$ for $j<m(p-1)$ or $j>2m(p-1)$ then $I_{FG}(z)=0.$

Proposition 3.4.

The map $\psi:M^{\times(p-1)}\to M^{\times}_{p-1}$ define by;

$$\psi((x_{1},x_{2}),(x_{3},x_{4}),\cdots,(x_{2(p-1)-1},x_{2(p-1)}))=((x_{1},x_{3},\cdots,x_{2(p-1)-1}),(x_{2},x_{4},\cdots,x_{2(p-1)}))$$

is a homeomorphism.

Proof.

We remember that

$$M^{\times(p-1)}=\prod_{i=1}^{p-1}(M\times M,M\times M\setminus\Delta_{M})=\left(\prod_{i=1}^{p-1}M^{2},\,\,\bigcup_{i=1}^{p-1}Y_{i}\right),$$

where $Y_{i}=M^{2}\times\cdots\times M^{2}\times\underbrace{(M\times M\setminus\Delta_{M})}_{i\text{-position}}\times M^{2}\times\cdots\times M^{2}.$ Now is clear that $\psi$ is well defined. The inverse $\psi^{-1}:M^{\times}_{p-1}\to M^{\times(p-1)}$ is naturally given by

$$\psi^{-1}((z_{1},\cdots,z_{p-1}),(w_{1},\cdots,w_{p-1}))=((z_{1},w_{1}),\cdots,(z_{p-1},w_{p-1})).$$

∎

The map $\psi$ can be viewed as a product of projections. In fact, denote by $\pi_{1}^{i}:M^{2}=M\times M\to M$ and $\pi_{2}^{i}:M\times M\to M$ the projections on the first and the second coordinate, respectively, where $M^{2}$ is in the $i$-position on the products in $M^{\times(p-1)},$ $i=1,\cdots,p-1.$ Thus

$$\psi=\widetilde{\pi_{1}}\times\widetilde{\pi_{2}}$$

where $\widetilde{\pi_{1}},\widetilde{\pi_{2}}:\prod_{i=1}^{p-1}M^{2}\to M^{p-1}$ are given by

$$\widetilde{\pi_{1}}=\pi_{1}^{1}\times\pi_{1}^{2}\times\cdots\times\pi_{1}^{p-1}\qquad\text{and}\qquad\widetilde{\pi_{2}}=\pi_{2}^{1}\times\pi_{2}^{2}\times\cdots\times\pi_{2}^{p-1}.$$

For the next result we consider $A=\emptyset.$ Let $\delta_{p-1}:X\to X^{p-1}$ be the $(p-1)$-fold diagonal map and let

$$\eta_{X}:H(X)\otimes\cdots\otimes H(X)\longrightarrow H(X^{p-1})$$

be the K$\ddot{u}$nneth isomorphism. We define

$$\Delta_{X}^{(p-1)}:=\eta_{X}^{-1}\circ(\delta_{p-1})_{\#}:H(X)\longrightarrow H(X)^{\otimes(p-1)}.$$

Let also

$$\eta:H(M^{\times})\otimes\cdots\otimes H(M^{\times})\longrightarrow H(M^{\times(p-1)})$$

be the K$\ddot{u}$nneth isomorphism. We have the following commutative diagram;

Theorem 3.5.

From Diagram (2) we obtain

$$\psi_{\#}\circ\eta\circ(I_{f_{1}f_{2}}\otimes\cdots\otimes I_{f_{1}f_{p}})\circ\Delta_{X}^{(p-1)}=I_{FG}.$$

Remark 3.6.

There is another way to define a type of coincidence homomorphism for the $p$-maps $f_{1},\cdots,f_{p}.$ Consider the homomorphism

$$\overline{I_{f_{1}\cdots f_{p}}}:=(f_{1}\times\cdots\times f_{p})_{\#}\circ(\delta_{p})_{\#}:H(X,A)\to H(M^{p},M^{p}\setminus\Delta_{p}),$$

where $\delta_{p}:X\to X^{p}$ is the $p$-fold diagonal and $\Delta_{p}=\{(x,\cdots,x)\mid x\in M\}.$ In this way is not possible to use the theory developed in [7]. Despite that, the map $e:(M^{p},M^{p}\setminus\Delta_{p})\to M^{\times(p-1)}$ defined by

$$e(x_{1},\cdots,x_{p})=((x_{1},x_{2}),(x_{1},x_{3}),\cdots,(x_{1},x_{p})),$$

gives the following relation;

$$\psi_{\#}\circ e_{\#}\circ\overline{I_{f_{1}\cdots f_{p}}}=I_{FG}.$$

From now on in the definition of the transfer and of the Lefschetz homomorphism we take $A=\emptyset.$ Analogous to [7, Definition 4.4] we can consider the transfer of $F.$

Definition 3.7.

The transfer of $F$ with respect to $z\in H_{m(p-1)+n}(X)$ is the homomorphism $F_{!}^{z}:H_{\ast}(M^{p-1})\to H_{\ast+n}(X)$ of degree $n$ given by

$$F_{!}^{z}=(F^{\#}\circ D^{-1})\smallfrown z,$$

where $D:H^{k}(M^{p-1})\to H_{m(p-1)-k}(M^{p-1})$ is the Poincaré duality isomorphism given by $D(x)=x\smallfrown O_{M^{p-1}},$ and $\smallfrown$ is the usual cap-product.

By [7, Theorem 4.5] we have the following;

Theorem 3.8 (Representation formula).

$$I_{FG}(z)=I_{\#}(\operatorname{Id}\otimes G_{\#}\circ F_{!}^{z})\delta_{\#}(O_{M^{p-1}}),$$

where $O_{M^{p-1}}$ is the fundamental class of $M^{p-1},$ $I:M^{p-1}\times M^{p-1}\to M^{\times}_{p-1}$ is the inclusion and $G_{\#}\circ F_{!}^{z}:H(M^{p-1})\to H(M^{p-1})$ is the homomorphism of degree $n.$

4 The Lefschetz homomorphism for $p$-maps

We have that for $z\in H_{m(p-1)+n}(X)$ the homomorphism $G_{\#}\circ F_{!}^{z}:H_{\ast}(M^{p-1})\to H_{\ast+n}(M^{p-1})$ has degree $n.$ Thus from Definition 2.6 we obtain $L(G_{\#}\circ F_{!}^{z})\in H_{n}(M^{p-1}).$

Definition 4.1.

The Lefschetz homomorphism for the $p$-maps $f_{1},\cdots,f_{p}:X\to M$ is the homomorphism $\Lambda_{FG}:H_{\ast}(X)\to H_{\ast-m(p-1)}(M^{p-1})$ of degree $-m(p-1)$ given by;

$$\Lambda_{FG}(z)=L(G_{\#}\circ F_{!}^{z}),$$

where $z\in H_{m(p-1)+n}(X).$ We will denote $\Lambda_{f_{1}\cdots f_{p}}=\Lambda_{FG}.$

Remark 4.2.

By Proposition 2.7 the Lefschetz homomorphism $\Lambda_{FG}$ has a representation as a Knill-like trace given by;

$$\Lambda_{FG}(z)=\sum_{k}(-1)^{k(k+|z|-m(p-1))}\sum_{j}\bar{a}_{j}^{k}\smallfrown G_{\#}\circ F_{!}^{z}(a_{j}^{k}),$$

where $\{a_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ is a basis for $H_{k}(M^{p-1})$ with corresponding dual basis $\{\bar{a}_{j}^{k}\mid j=1,\dots,\alpha_{k}\}$ of $H^{k}(M^{p-1}),$ for each $k.$

Let $\tau_{M^{\times}_{p-1}}$ be the Thom class of the fiber bundle pair $M^{\times}_{p-1}\stackrel{{\scriptstyle{\scriptstyle\pi^{\prime}}}}{{{\longrightarrow}}}M^{p-1}.$ From [7, Theorem 6.1] we have;

Theorem 4.3 (Lefschetz-type coincidence theorem for $p$-maps).

$$\varphi_{M^{\times}_{p-1}}\circ I_{FG}=\Lambda_{FG},$$

where $\varphi_{M^{\times}_{p-1}}:H(M^{\times}_{p-1})\to H(M^{p-1})$ is the Thom isomorphism given by

$$\varphi_{M^{\times}_{p-1}}(w)={\pi^{\prime}}_{\#}(\tau_{M^{\times}_{p-1}}\smallfrown w).$$

Moreover, if $\Lambda_{FG}\neq 0$ then $\operatorname{Coin}(f_{1},\cdots,f_{p})\neq\emptyset.$

Given the fiber bundle pair $\xi=M^{\times}\stackrel{{\scriptstyle{\scriptstyle\pi_{1}}}}{{{\longrightarrow}}}M$ we can consider the $q$-product, $q\geq 1,$ of that fiber bundle pair by itself; $\xi_{q}=M^{\times q}\stackrel{{\scriptstyle{\scriptstyle\pi}}}{{{\longrightarrow}}}M^{q},$ where the map $\pi$ is given by $\pi=\underbrace{\pi_{1}\times\cdots\times\pi_{1}}_{q\text{-times}}.$

Since $M$ is orientable then choose an orientation for $M,$ with the correspond fundamental class $O_{M},$ the Thom class $\tau_{M}$ for $\xi$ is unique, see [9]. Analogously we have that $\tau_{M^{\times}_{q}}$ is unique.

Proposition 4.4.

Let $\tau_{M^{q}}$ be the Thom class of the fiber bundle pair $M^{\times q}\stackrel{{\scriptstyle{\scriptstyle\pi}}}{{{\longrightarrow}}}M^{q},$ and let $\psi_{q}:M^{\times q}\to M^{\times}_{q}$ be the homeomorphism defined as in Proposition 3.4. Then

$$\psi_{q}^{\#}(\tau_{M^{\times}_{q}})=\tau_{M^{q}}=\pm\underbrace{\tau_{M}\times\cdots\times\tau_{M}}_{q\text{-times}}.$$

Proof.

The first equality follows from fact that $\psi_{q}$ is an isomorphism of orientable fiber bundles and satisfies $\pi^{\prime}\circ\psi_{q}=\pi.$ The second equality follows from the characterization of the Thom class and from the product orientation. In fact, the restriction of $\tau_{M}\times\cdots\times\tau_{M}$ to every fiber is the product of the orientation generators, and therefore it is the Thom class of the product bundle up to the choice of orientation. ∎

We define $\overline{\varphi}:H(M^{\times})\otimes\cdots\otimes H(M^{\times})\to H(M)\otimes\cdots\otimes H(M)$, with $(p-1)$ factors, by;

$$\overline{\varphi}=\varphi_{M}\otimes\cdots\otimes\varphi_{M}.$$

Note that $\overline{\varphi}$ has an inverse given by $\overline{\varphi}^{-1}=\varphi_{M}^{-1}\otimes\cdots\otimes\varphi_{M}^{-1}.$

Let $r=p-1.$ For a homogeneous element $x=x_{1}\otimes\cdots\otimes x_{r}\in H(M^{\times})^{\otimes r}$, where $|x_{i}|=s_{i},$ we define

$$\kappa_{M}(x)=(-1)^{A_{r}(x)}x,\qquad A_{r}(x)=\sum_{j=2}^{r}m\left(\sum_{i=1}^{j-1}(s_{i}-m)\right).$$

Analogously, for a homogeneous element $z=z_{1}\otimes\cdots\otimes z_{r}\in H(X)^{\otimes r}$, we define

$$\kappa_{X}(z)=(-1)^{A_{r}(z)}z,\qquad A_{r}(z)=\sum_{j=2}^{r}m\left(\sum_{i=1}^{j-1}(|z_{i}|-m)\right).$$

Proposition 4.5.

The following relation holds up to signal:

$$\varphi_{M^{r}}\circ\eta=\pm\eta^{\prime}\circ\overline{\varphi}\circ\kappa_{M},$$

where $\eta^{\prime}$ is also a K$\ddot{u}$nneth isomorphism. Moreover,

$$\varphi_{M^{r}}=\varphi_{M^{\times}_{r}}\circ\psi_{r\#}.$$

Proof.

The map $\psi_{r}$ satisfies $\pi^{\prime}\circ\psi_{r}=\pi_{1}\times\cdots\times\pi_{1}=\pi.$ From Proposition 4.4 and by naturality of the cap product we have

$$\begin{array}[]{lll}\varphi_{M^{r}}(w)&=&\pi_{\#}(\tau_{M^{r}}\smallfrown w)\\ &=&{\pi^{\prime}}_{\#}(\psi_{r\#}(\psi_{r}^{\#}(\tau_{M^{\times}_{r}})\smallfrown w))\\ &=&{\pi^{\prime}}_{\#}(\tau_{M^{\times}_{r}}\smallfrown\psi_{r\#}(w))\\ &=&\varphi_{M^{\times}_{r}}(\psi_{r\#}(w)).\end{array}$$

Therefore $\varphi_{M^{r}}=\varphi_{M^{\times}_{r}}\circ\psi_{r\#}.$ Let $x=x_{1}\otimes\cdots\otimes x_{r}\in H(M^{\times})^{\otimes r}$ be homogeneous. We have

$$\begin{array}[]{lll}\varphi_{M^{r}}(\eta(x))&=&\varphi_{M^{r}}(x_{1}\times\cdots\times x_{r})\\ &=&\pm\pi_{\#}((\tau_{M}\times\cdots\times\tau_{M})\smallfrown(x_{1}\times\cdots\times x_{r}))\\ &=&\pm(-1)^{A_{r}(x)}\pi_{\#}((\tau_{M}\smallfrown x_{1})\times\cdots\times(\tau_{M}\smallfrown x_{r}))\\ &=&\pm(-1)^{A_{r}(x)}(\varphi_{M}(x_{1})\times\cdots\times\varphi_{M}(x_{r}))\\ &=&\pm\eta^{\prime}\circ\overline{\varphi}\circ\kappa_{M}(x),\end{array}$$

where $A_{r}(x)$ is given by Lemma 2.1. Therefore $\varphi_{M^{r}}\circ\eta=\pm\eta^{\prime}\circ\overline{\varphi}\circ\kappa_{M}.$ ∎

Theorem 4.6.

We have;

$$\Lambda_{f_{1}\cdots f_{p}}=\Lambda_{FG}=\pm\eta^{\prime}\circ(\Lambda_{f_{1}f_{2}}\otimes\cdots\otimes\Lambda_{f_{1}f_{p}})\circ\kappa_{X}\circ\Delta_{X}^{(p-1)}.$$

Equivalently, if

$$\Delta_{X}^{(p-1)}(z)=\sum_{\nu}z_{\nu,1}\otimes\cdots\otimes z_{\nu,p-1}$$

with homogeneous terms, then

$$\Lambda_{f_{1}\cdots f_{p}}(z)=\pm\sum_{\nu}(-1)^{A_{p-1}(z_{\nu})}\Lambda_{f_{1}f_{2}}(z_{\nu,1})\times\cdots\times\Lambda_{f_{1}f_{p}}(z_{\nu,p-1}).$$

Proof.

By Theorems 3.5, 4.3 and Proposition 4.5 we obtain;

$$\begin{array}[]{lll}\Lambda_{f_{1}\cdots f_{p}}=\Lambda_{FG}&=&\varphi_{M^{\times}_{p-1}}\circ I_{FG}\\ &=&\varphi_{M^{\times}_{p-1}}\circ\psi_{\#}\circ\eta\circ(I_{f_{1}f_{2}}\otimes\cdots\otimes I_{f_{1}f_{p}})\circ\Delta_{X}^{(p-1)}\\ &=&\varphi_{M^{p-1}}\circ\eta\circ(I_{f_{1}f_{2}}\otimes\cdots\otimes I_{f_{1}f_{p}})\circ\Delta_{X}^{(p-1)}\\ &=&\pm\eta^{\prime}\circ\overline{\varphi}\circ\kappa_{M}\circ(I_{f_{1}f_{2}}\otimes\cdots\otimes I_{f_{1}f_{p}})\circ\Delta_{X}^{(p-1)}\\ &=&\pm\eta^{\prime}\circ(\Lambda_{f_{1}f_{2}}\otimes\cdots\otimes\Lambda_{f_{1}f_{p}})\circ\kappa_{X}\circ\Delta_{X}^{(p-1)}.\end{array}$$

The last equality follows because each $I_{f_{1}f_{j}}$ has degree $0$ and $\varphi_{M}\circ I_{f_{1}f_{j}}=\Lambda_{f_{1}f_{j}}.$ ∎

Remark 4.7.

Similarly [7], is possible to define the coincidence and Lefschetz homomophisms for $p$-maps in the case where $M$ have boundary to study coincidences of maps $f_{1},\cdots,f_{p}$ in the following situation; $f_{1}:(X,A)\to(M,\partial M)$ and $f_{2},\cdots,f_{p}:X\to M.$

5 Examples

(I) Let $r=p-1,$ $X=M^{r},$ fix a point $a\in M$ and consider the maps

$$f_{1}(x_{1},\cdots,x_{r})=a,\qquad f_{j}(x_{1},\cdots,x_{r})=x_{j-1},\quad j=2,\cdots,p.$$

Then $F:X\to M^{r}$ is the constant map with value $(a,\cdots,a)$ and $G:X\to M^{r}$ is the identity map. Thus

$$\operatorname{Coin}(f_{1},\cdots,f_{p})=\{(a,\cdots,a)\}.$$

Let $O_{X}\in H_{mr}(X)$ be the fundamental class. Since $F$ is constant, the transfer $F_{!}^{O_{X}}:H(X)\to H(X)$ is zero in every degree except $mr,$ and on the fundamental class it is the identity up to the choice of orientation. Therefore

$$\Lambda_{f_{1}\cdots f_{p}}(O_{X})=\pm[pt]\neq 0\in H_{0}(M^{r}).$$

This gives a non trivial Lefschetz homomorphism for every $p\geq 2.$

(II) Let $f,g:X\to M$ be maps where $X=M\times S^{1}$ and $M$ is an orientable connected $m$-dimensional suitable manifold. For the definition of suitable, see [11, Section 4]. We define $\psi_{fg}:X\to M$ by

$$\psi_{fg}(x)=g(x).[f(x)]^{-1}.$$

We also define $\overline{f},\overline{g}:M\to M$ by $\overline{f}(x)=f(x,1)$ and $\overline{g}(x)=g(x,1),$ where $1$ denotes the point $(1,0)\in S^{1}.$ If $\overline{O_{M}}$ is the dual of the fundamental class $O_{M}$ of $M$ then by [7, Theorem 7.1] we obtain;

$$\Lambda_{fg}(z)=\langle\overline{O_{M}},{\psi_{fg}}_{\#}(z)\rangle$$

for $z\in H_{m}(X).$ Let $j:H_{m}(M)\to H_{m}(M\times S^{1})$ be given by $j(w)=w\otimes 1,$ where we are using the K$\ddot{u}$nneth isomorphism to identify

$$H_{m}(M\times S^{1})\approx\bigoplus_{k+l=m}H_{k}(M)\otimes H_{l}(S^{1}),$$

and define $\overline{\psi_{fg}}={\psi_{fg}}_{\#}\circ j.$

From [11, Theorem 3] we have $\deg(\overline{\psi_{fg}})=L(\overline{f},\overline{g}).$ Thus, considering $z=w\otimes 1,$ where $w\in H_{m}(M)$ we have

$$\begin{array}[]{lll}\Lambda_{fg}(z)&=&\langle\overline{O_{M}},{\psi_{fg}}_{\#}(w\otimes 1)\rangle\\ &=&\langle\overline{O_{M}},\overline{\psi_{fg}}(w)\rangle\\ &=&\langle\overline{O_{M}},w\rangle L(\overline{f},\overline{g}).\end{array}$$

Therefore, if $w=O_{M}$ we obtain;

$$\Lambda_{fg}(O_{M}\otimes 1)=L(\overline{f},\overline{g}).$$

Equation (8) gives a calculation for two maps. For $p$-maps $f_{1},\cdots,f_{p}:M\times S^{1}\to M$ with $p\geq 3,$ the element $O_{M}\otimes 1$ has degree $m<m(p-1),$ and therefore

$$\Lambda_{f_{1}\cdots f_{p}}(O_{M}\otimes 1)=0.$$

Thus the non triviality of the pairwise numbers $L(\overline{f_{1}},\overline{f_{j}})$ does not by itself imply the non triviality of $\Lambda_{f_{1}\cdots f_{p}}$ on the same element. The coproduct in Theorem 4.6 has to be taken into account.