\documentstyle[a4,11pt]{article} \begin{document} %%%%%%%%%%%%% Definicao de novos comandos %%%%%%%%%%%%%% %%% DECLARACAO DE EQUACOES NUMERADAS COM COLCHETES %%%%% \newcounter{eqccounter}[section]% contador por secao % Faz com que as equacoes tenham o numero da secao \renewcommand{\theeqccounter}{\thesection.\arabic{eqccounter}} \newenvironment{eqc}{% % \refstepcounter{eqccounter}% seta valor do label \vspace{\abovedisplayskip}% espaco estra igual a do "displaymath" \par\noindent\begin{minipage}{0.5in}% $\rm[\theeqccounter]$% numero da equacao \end{minipage}% \begin{minipage}{3.7in}\vspace{-\abovedisplayskip}% retira espaco extra \[}{\]% % equacao \end{minipage}\hfill\vspace{\belowdisplayskip}\\}% finaliza \newcommand{\Interior}{\mbox{int}} \newcommand{\Int}[1]{{\stackrel{{\scriptscriptstyle\circ}}{#1}}} %%% DECLARACAO DE EQUACOES COM ROTULOS NAO NUMERICOS %%%%% \newenvironment{eqr}[1]{% % \begin{flushleft}% //\begin{minipage}{0.5in}% #1% rotulo da equacao \end{minipage}% \begin{minipage}{4in}% \vspace{-\abovedisplayskip}% retira espaco extra \[}{\]% equacao \end{minipage}\end{flushleft}% }% finaliza \newcommand{\Z} {\mbox{$\sf Z\!\!Z$\,}} \newcommand{\dps} {\displaystyle} \newcommand{\pn} {\par\noindent} \newcommand{\vs} {\vskip} \newcommand{\N} {{\rm I}\!{\rm N}} \newcommand{\R} {{\rm I}\!{\rm R}} \newcommand{\calc} {{\cal C}} \newcommand{\bigdownarrow}{\downarrow\hspace{-1.43ex}^|\,} \title {Equations in free groups and \\ coincidence of mappings on surfaces} \author{Daciberg L. Gon\c calves\\ Departamento de Matem\'atica - IME-USP\\ Caixa Postal 66281 - Agencia Cidade de S\~ao Paulo\\ 05315-970 - S\~ao Paulo - SP - Brasil\\ e-mail: dlgoncal@ime.usp.br\\ \\ Heiner Zieschang\\ Fakult\"at f\"ur Mathematik\\ Ruhr - Universit\"at Bochum\\ 44780 Bochum - Deutschland \\ e-mail: marlene.schwarz@rz.ruhr-uni-bochum.de} \date{} \maketitle \begin{center} \begin{minipage}{3.25in} \begin{center} {\large\it Abstract} \end{center} When has a pair of maps among orientable closed surfaces, $(f,c) \colon S_h \to S_g$, the Wecken property for the case of roots, that is when $c$ is a constant map, and primitive $f$, i.e. $f$ induces a surjective homomorphism on the fundamental group? It turns out that the pair $(f,c)$ has the Wecken property if and only if $degree(f) \leq {{2h-1} \over {2g-1}}$. Certain quadractic equations in free groups closely related to the coincidence problem are solved. \end{minipage} \end{center} \vskip 2.5cm \pn ................................... \pn This work was done during the visit of the second author to the Departamento de Matem\'atica - USP - Brasil, August 1996 and during the Visit of the first Author to the Fakult\"at f\"ur Mathematik, Ruhr-Universit\"at Bochum-Deutschland, June 1997. \pn The first visit has been supported by the international cooperation pro\-gram GMD/Germany - CNPq/Brasil. The second visit has been supported by FAPESP-S\~ao Paulo-Brasil. \vfill\break \large \par\noindent {\bf Introduction} \normalsize \baselineskip=6.5mm Let $f_1, f_2\colon M\to N$ be a pair of maps. The number of points $x \in M$ with $f_1(x) = f_2(x)$ is called the {\it coincidence number} and denoted by $\# coin(f_1,f_2)$. Two coincidence points $x,y \in M$ are of the same {\it Nielsen coincidence class} if there is a curve $\gamma \colon [0,1] \to M$ such that $\gamma (0) = x, \gamma (1) = y$ and a homotopy $\gamma_t \colon ([0,1], \{0,1\}) \to (M, \{ f_1(x),f_2(y)\})$ with $\gamma_0 = f_1 \circ \gamma$ and $\gamma_1 = f_2 \circ \gamma$. This defines an equivalence relation and the number of essential equivalence classes is called the {\it Nielsen coincidence number } and is denoted by $N(f_1,f_2)$. By essential, we mean a class which has index different from zero where the index is defined in a similar form as for fixed points. In general, the coincidence Nielsen number $N(f_1,f_2)$ cannot be realized as $\# coin (f'_1,f'_2)$ for some mappings $f'_1, f'_2$ homotopic to $f_1, f_2$, respectively, and the problem is to find necessary and/or sufficient conditions for the existence of such homotopic maps. \pn {\bf Definition.} $(f_1,f_2)$ has the {\it Wecken property} if $N(f_1, f_2)$ = $\# coin (f'_1,f'_2)$ for some $f'_1, f'_2$ homotopic to $f_1,f_2$, respectively. In this work we study this question for mappings between surfaces for the case of roots, namely, when $f_2$ is a constant map. This problem is closely related to the decision whether certain quadratic equations in free groups have solutions. For the connections between the two problems, see [G], for example. Let $S_n$ denote the closed orientable surface of genus $n$ and, in particular, $T= S_1$ the torus. By $deg(f)$ we denote the degree of a mapping $f$. Our main result is \pn {\bf Theorem 3.2}. {\it Let $f\colon S_h \to S_g$ be a primitive map, that is $f_\# (\pi_1 (S_h)) = \pi_1 (S_g)$. Then: \pn (a) $(f,c)$ does not have the Wecken property if $\dps |deg (f)| > {{2h-1}\over{2g-1}}$. \pn (b) $(f,c)$ satisfies the Wecken condition if $\dps |deg (f)| \leq {{2h-1}\over {2g-1}}$.} %{\it Let $f \colon S_h \to S_g$ $g,h \geq 1$ % be a map with $|deg (f)| = k$ and $f_\# (\pi_1(S_h))=\pi_1(S_g)$. Then: % %\pn a) If $k> {{2h-1} \over {2g-1}}$ \ then \ $(f,c)$ does %not satisfies the Wecken condition. % %\pn b) If $k\leq {{2h-1} \over {2g-1}}$ \ then \ $(f,c)$ has the %Wecken property.} Now let the target be the torus $T$. Since the torus $T$ is a topological group, the coincidence problem for maps $(f_1,f_2) \colon S \to T$ is equivalent to that of the pair $(\varphi,c)$ where $\varphi$ is the function defined by $\varphi(x)=f_1(x)\cdot f_2(x)^{-1}$ and c is a constant map. Let $\Lambda (f_1,f_2)$ be the Lefschetz coincidence number defined for a pair of maps among two compact orientable manifolds of the same dimension. See [V], pg. 195. Therefore we also get from Theorem 3.2: \pn {\bf Corollary 3.4.} {\it Let $(f_1,f_2) \colon S_h \to T$ be a primitive pair, that is $im(f_{1\#}-f_{2\#})=\pi_1(T)$. \ Then \ $(f_1,f_2)$ has the Wecken property if and only if $\left |\Lambda (f_1,f_2)\right | \leq 2h-1$. } %{\it Given $(f_1,f_2) \colon S_h \to T$ %where %$im(f_{1\#}-f_{2\#})=\pi_1(T)$, \ then \ %$(f_1,f_2)$ has the Wecken property if %and only if $\left |\Lambda (f_1,f_2)\right | \leq 2h-1$.} The first example of a map $f$ among surfaces such that $(f,c)$ does not satisfy the Wecken condition was given in [L]. This paper is divided into three sections. In Section 1 we get some results about branched covers and construct some special ones. These are Propositions 1.1, 1.3 and 1.4. In Section 2 we show that some quadractic equations in the free group on $2g$ generators have solutions, see Proposition 2.2. In particular, when the free group has two generators, $F(a,b)$, we do not only prove the existence of a solution, but we give an explicit solution for the equation $W(z_1,z_2,...,z_{2h-1}, z_{2h}) = B^{2h-1}$ where $W(z_1,z_2,...,z_{2h-1}, z_{2h})$ is an alternating quadractic word and B=[a,b], the commutator of a and b (Proposition 2.3). Such results are important for our applications. In section 3 we give some applications to the coincidence theory, but mainly for roots. We consider the case where the map $f$ is primitive and decide when $f$ has the Wecken property (Theorem 3.2). Finally when the target is the torus we consider a primitive pair $(f_1,f_2)$ and get similar results (Corollary 3.4). \vs 1cm \large \par\noindent {\bf 1. On special branched covers and the degree of maps} \normalsize \baselineskip=6.5mm In this section we study the existence of branched covers $f\colon S_h\to S_g$ with respect to the number of branch points, the order of the pre-image of such points and the degree of the maps. Let $F$ and $F'$ be orientable compact surfaces with boundary components $\rho _1,..,\rho_l$ and $\rho'_1,...,\rho'_{l'}$, resp.; let $F'$ be connected. We choose orientations on $F$ and $F'$ and take the induced orientations on the boundary, following the rule that the positive boundary curves $\rho_j$ \ or \ $\rho'_j$, resp., appear in the positive boundary of the disks obtained by cutting along a canonical, for instance, system of curves of the surfaces. Let $\Phi \colon (F, \partial F) \to (F', \partial F')$ be a continuous mapping with $\Phi^{-1} (\partial F') = \partial F$. Take the notation such that $\rho_1,..,\rho_{r_1}$ are mapped to $\rho'_1$ and so on: for suitable $0= r_0 < r_1 < ... 0$ for all boundary components of $F$ and let $\chi(F)$ denote the Euler characteristic of $F$. } \vs 0.2cm \pn (a) $\chi (F) \leq |d| \cdot \chi (F')$ \vs 0.2cm \pn (b) {\it If the restriction of $\Phi$ to the boundary is a covering $\partial F \to \partial F'$ and if $\chi (F) = |d| \cdot \chi (F')$ then $\Phi$ is homotopic to a $($unbranched$)$ covering $\Psi$ such that $\Psi (x) = \Phi (x)$ for all $x\in \partial F$; during the homotopy the images of the boundary points do not change. } \vs 0.2cm \pn {\it Proof:} We follow the arguments of Seifert (see [S], [ZVC, 3.3.4]). Assume that $F$ and $F'$ are connected and take the orientation on every boundary component induced by the orientation of $F$ or $F'$, resp. After a deformation we may assume that $\Phi|\partial F \colon \partial F \to \partial F'$ defines a covering on each component. By assumption $\Phi$ induces on every boundary component of $\partial F$ a mapping of positive degree. Next we take triangular cell complexes on both surfaces and deforme $\Phi$ such that every simplex of $F$ is mapped onto a simplex of $F'$ of the same dimension and that $\Phi$ defines coverings between regular neighborhoods $N(\partial F)$ and $N(\partial F')$ of the boundary components. Moreover we may assume that $\Phi^{-1}(N(\partial F')) = N(\partial F) $. The $2$-simplexes of $F$ and $F'$ obtain orientations induced by the orientations of $F$ and $F'$, resp. Now we repeat the arguments of Seifert, see [ZVC, 3.3.4], without changing the mapping on $N(\partial F)$ and $N(\partial F')$: If every $2$-simplex of $F$ is mapped with degree $1$ then $\Phi$ is a -- perhaps -- branched cover. If we glue into every boundary component a disk we obtain a covering between two closed surfaces $\overline{\Phi }\colon \overline{F} \to \overline{F' }$ of degree $c$. By Kneser's formula we have $$ \chi(F) + l = \chi(\overline{F }) \leq c\cdot \chi(\overline{F' }) = c \cdot (\chi(F') +l') ;$$ here equality holds only if we have an unbranched covering and if $l = c \cdot l'$, that is $d_i = 1$ for $1 \leq i \leq l$, otherwise $l < cl'$. Hence $$ \chi(F) \leq c\cdot \chi(F') $$ and equality only if $\Phi$ is a covering and $l = c \cdot l'$. If $\Phi$ maps some $2$-simplexes with degree $1$, others with degree $-1$ then we find a pair of closed $2$-simplexes $\sigma_1^2, \sigma_2^2$ in $F- N(\partial F)$ with at least one common $1$-simplex in their boundaries which are mapped - with inverse degrees - to the same simplex in $F'$. If $\sigma_1^2 \cap \sigma_2^2$ consists of one or two closed $1$-simplexes, hence is contractible, then we remove the interior of $\sigma_1^2 \cup \sigma_2^2$ and identify the remaining boundary $1$-simplexes of them pairwise. The new surface $F^*$ has the same Euler characteristic and number of boundary components. $\Phi$ induces a map $F^* \to F'$ with the same degree and behavior at the boundary. Thus, if we prove the degree-Euler-characteristic formula for $F^* \to F'$ it is also valid for the original case. The remaining case is that $\sigma_1^2 \cap \sigma_2^2$ consists of a common edge and that the remaining vertices of $\sigma_1^2$ and $\sigma_2^2$ coincide. The other four $1$-simplexes are mapped in pairs to two $1$-simplexes of $F'$ and each pair defines a simple closed curve on $F$. We cut $F$ along such a curve, add two discs and obtain a not necessarily connected surface $F^*$ with Euler characteristic increased by $2$. If $F^*$ contains a $S^2$-component we drop it. $\Phi$ induces a mapping $\Phi^*\colon F^* \to F'$ which has the same properties as $\Phi$, but maps less $2$-simplexes of $F^*$ with degree $-1$. By induction we may assume that the assertion is true for $\Phi^*$ and, thus, it also holds for $\Phi$. \hfill $\Box$ %{\bf << {{2h-1}\over{2g-1}}$ in $F_{2g}$. } \vs 0.2cm \pn {\it Proof:} Suppose that the equation has a solution. Consider $S_h - \Int{D}_h$ and $S_g - \Int{D}_g$ where $D_h$ and $D_g$ are closed disks and let $S^1_h$ and $S^1_g$ be their boundaries. Let $\pi_1(S_h - \Int{D}_h) = \langle e_1, \ldots, e_{2h}\vert \rangle$ and $S^1_h = \prod_{j=1}^h [e_{2j-1}, e_{2j}]$. Define a function $f'$ on the 1-skeleton of $S_h-\Int{D}_h$ such that $f_{\#}'(e_i) = z_i , \ i=1,...,2h$ \ and \ $f'^{-1}(S^1_g) = S^1_h$ \ where $f'$ restricted to $S^1_h$ is a covering between the boundaries of degree $d$ (see also [G], Fundamental Lemma 1.2 for similar techniques, and more details). Then $f'$ extends to a map of pairs $f\colon (S_h - \Int{D}_h , S_h^1) \to (S_g-\Int{D}_g, S_g^1)$. By Proposition 1.1, we have that $\dps |d|\leq {{2h-1}\over{2g-1}}$ and the result follows. \hfill $\Box$ \vs 0.2cm \pn {\bf Proposition 2.2.} {\it The equation $W(z_1,z_2,...,z_{2h-1}, z_{2h}) = B_g^c,\ c={{2h-1} \over {2g-1}}$ has a so\-lu\-tion in the free group on $2g$ generators $F(a_1,b_1,...,a_g,b_g)$; here $B_g=[a_1,b_1]...[a_g,b_g]$. } \vs 0.2cm \pn {\it Proof:} Let us consider the $c$-fold cover given by Proposition 1.4. Since the cover is homeomorphic to $S_h-\Int{D}$, \ let \ $\{e_1, e_2, ... ,e_{2h-1}, e_{2h}\}$ be a set of generators of the fundamental group of $S_h$, such that $W(e_1,e_2,...,e_{2h-1}, e_{2h})$ is homotopic to $\partial (S_h-\Int{D})$. Therefore $z_i = \gamma (e_i) , \ i=1,...,2h$ give a solution of the above equation. \hfill $\Box$ Sometimes one might be interested in an explicit solution of the above equation. The cover given by Proposition 1.3 has given us the idea how to find an explicit solution of the above equation, at least when F is the free group on two generators. \vs 0.2cm \pn {\bf Proposition 2.3.} {\it The equation $W(z_1,...,z_{2h}) = B_1^{2h-1}$ has the following so\-lu\-tion: } \begin{center} $\begin{array}{ccc} w_1 = aba^{-1} && w_2 = [b^{-1},a]^{h-1} a^{h-1} a^{-h} \\ \\ \vdots && \vdots \\ \\ w_{2l-1} = a^{2-l}b^{-1}aba^{l-2} && w_{2l} = a^{1-l} [b^{-1},a]^{h-l} a^{h-1} a^{l-h} b^2 a^{l-2} \\ \\ l=2,...,h && 1 < l < h \\ \\ && w_{2h} = b^2a^{h-2} . \end{array}$ \end{center} \vs 0.2cm \pn {\it Proof:} The cases $h= 1,2$ can be checked easily. Let $h>2$. Then we have \vs 0.5cm \pn $\dps [w_1,w_2] \left (\prod\limits^{h-1}_{l=2} \left [w_{2l-1}, w_{2l}\right ]\right ) \left [w_{2h-1}, w_{2h}\right ] =$ \vskip 0.2cm \pn $= \left [aba^{-1}, [b^{-1},a]^{h-1} a^{h-1} a^{-h}\right ] \prod\limits^{h-1}_{l=2} \left (\left [ a^{2-l} b^{-1} aba^{l-2},\ a^{1-l} [b^{-1},a]^{h-l} a^{h-1} a^{l-h} b^2 a^{l-2}\right ]\right ) $ \vskip 0.2cm $\left [a^{2-h}b^{-1}aba^{h-2},\ b^2a^{h-2}\right ]=$ \vskip 0.2cm \pn $= aba^{-1}b^{-1}b \left [b^{-1}, a\right ]^{h-1} a^{h-1} a^{-h} ab^{-1} a^{-1} a^h a^{1-h} \left [a,b^{-1}\right ]^{h-1} \prod\limits^{h-1}_{l=2} \bigg (a^{2-h} b^{-1} aba^{l-2}a^{1-l}$ \vskip 0.2cm $\left [b^{-1}, a\right ]^{h-l} a^{h-1} a^{l-h} b^2 a^{l-2} a^{2-l} b^{-1} a^{-1} ba^{l-2}a^{2-l} b^{-2} a^{h-l} a^{1-h} \left [a,b^{-1}\right ]^{h-l} a^{l-1} \bigg )\times $ \vskip 0.2cm $\times a^{2-h}b^{-1}aba^{h-2} ba^{-1}b^{-1}=$ \vskip 0.2cm \pn $= [a,b]^h . [a,b^{-1}]^{h-1} \prod\limits^{h-1}_{l=2} \bigg (a^{2-l} \left [b^{-1},a\right ]^{h-l+1} a^{l-1} ba^{-1}b^{-1}a^{1-l} \left[a,b^{-1}\right ]^{h-l} a^{l-1}\bigg )\times $ %\pn %$= [a,b]^h . [a,b^{-1}]^{h-1} \prod\limits^{h-1}_{l=2} \bigg (a^{2-l} %b^{-1} %aba^{-1}\left [b^{-1},a\right ]^{h-l} a^{l-1} ba^{-1}b^{-1}a^{1-l}$ \vskip 0.2cm $\times a^{2-h}b^{-1}aba^{h-3} [a,b]$. \vskip 0.2cm \noindent Since \vskip 0.2cm %\pn %$\left (a^{2-l}b^{-1}aba^{-1}\left [b^{-1},a\right ]^{h-l} %a^{l-1}ba^{-1}b^{-1}a^{1-l} %\left [a,b^{-1}\right ]^{h-l} a^{l-1}\right ) .$ %\vskip 0.2cm %$\left (a^{1-l} b^{-1} aba^{-1} \left [b^{-1},a\right ]^{h-l-1} a^l %ba^{-1}b^{-1}a^{-l} %\left [a,b^{-1}\right ]^{h-l-1} a^l\right ) =$ %\vskip 0.2cm %\pn %$= a^{2-l}b^{-1}aba^{-1}\left [b^{-1},a\right ]^{h-l} a^{l-1} ba^{-1} b^{-1} %aba^{-1} b^{-1}a^{-l} \left [a,b^{-1}\right ]^{h-l-1}a^l$ %\vskip 0.2cm %\pn we have %\vskip 0.2cm %\pn %$\prod\limits^{h-1}_{l=2} a^{2-l}b^{-1}aba^{-1}\left [b^{-1},a\right ]^{h-l} %a^{l-1}ba^{-1}b^{-1}a^{1-l}\left [a,b^{-1}\right ]^{h-l} a^{l-1} =$ %\vskip 0.2cm %\pn $ %= b^{-1} aba^{-1} \left [b^{-1},a\right ]^{h-2}aba^{-1}b^{-1}a^{-1} %\left (\prod\limits^{h-2}_{l=3} a^{l-1}ba^{-1}b^{-1}a^{1-l}\right )$ %\vskip 0.2cm %$a^{h-2} ba^{-1}b^{-1}a^{2-h}\left [a,b^{-1}\right ] a^{h-2}$ = %\vskip 0.2cm %\pn %$= \left [b^{-1},a\right ]^{h-1} [a,b]^{h-2} a^{2-h} \left [a,b^{-1}\right ] %a^{h-2}$ %vskip 0.2cm \pn $\prod\limits^{h-1}_{l=2} a^{2-l}\left [b^{-1},a\right ]^{h-l+1} a^{l-1}ba^{-1}b^{-1}a^{1-l}\left [a,b^{-1}\right ]^{h-l} a^{l-1} =$ \vskip 0.2cm \pn $\left [b^{-1}, a \right ]^{h-1} aba^{-1}b^{-1} \prod\limits^{h-1}_{l=3} \left (a^{2-l} \left [a, b^{-1} \right ]^{h+1-l} a^{l-2} a^{2-l} \left [b^{-1}, a \right ]^{h-l+1} a^{l-1}ba^{-1}b^{-1} \right ) \times$ \vskip 0.2cm $\times a^{2-h}\left [a,b^{-1} \right ] a^{h-2} =$ \vskip 0.2cm \pn $= \left [b^{-1},a \right ]^{h-1} aba^{-1}b^{-1} \prod\limits^{h-1}_{l=3} \left ( aba^{-1}b^{-1} \right ) \cdot a^{2-h} \left [a, b^{-1} \right ] a^{h-2} $ \vskip 0.2cm \pn $= \left [b^{-1},a\right ]^{h-1} [a,b]^{h-2} a^{2-h} \left [a,b^{-1}\right ] a^{h-2}$ . \noindent Thus \vskip 0.2cm \pn $\left [w_1,w_2\right ] \left (\prod\limits^{h-1}_{l=2} \left [w_{2l-1}, w_{2l}\right ]\right ) \left [w_{2h-1}, w_{2h}\right ] =$ \vskip 0.2cm \pn $= [a,b]^h \left [a,b^{-1}\right ]^{h-1} \left [b^{-1},a\right ]^{h-1} \left [a,b\right ]^{h-2} a^{2-h} \left [a,b^{-1}\right ] a^{h-2} a^{2-h} b^{-1} aba^{h-3} [a,b] =$ \vskip 0.2cm \pn $= [a,b]^h [a,b]^{h-2} [a,b] = [a,b]^{2h-1} . $ \hfill $\Box$ \vs 1cm \large \par\noindent {\bf 3. Applications to coincidence theory} \normalsize \baselineskip=6.5mm \vs 0.5cm We briefly review parts of the Nielsen Coincidence theory relevant to our applications. Given $f_1, f_2 : M\to N$ a pair of maps among compact orientable manifolds of the same dimension, let $C= \{x\in M \mid f_1(x) = f_2(x)\}$. \ If \ $F\subset C$ is an isolated subset of $C$ then one can define an integer $I(f_1,f_2; F) \in \Z$ called {\it local index}. (See [V], pg. 195 for details.) The {\it Nielsen relation } is defined by $x\sim y$ \ for \ $x, y\in C$ if there exists $\lambda : I\to M$ such that $\lambda (0) = x, \ \lambda (1) =y$ \ and \ $f_1(\lambda )$ is homotopic to $f_2(\lambda )$ relative end points. Under this Nielsen relation $C$ is divided into a finite number of equivalence classes where each class is an isolated subset of $C$. Now the {\it Nielsen coincidence number} $N(f_1,f_2)$ is defined as the number of Nielsen classes which have index different from zero. Let $\mu (f_1,f_2) = min\ \#\ coin (f'_1,f'_2)$ where $f'_1$, $f'_2$ runs over $[f_1]$, $[f_2]$, respectively; here [ ] denotes the homotopy class. Finally, with respect to the pair $(f_1,f_2)$, one can define the {\it Lefschetz coincidence number } $\Lambda (f_1,f_2)$. If the map $f_2=id$ then this number is the usual Lefschetz number and, if $f_2$ is the constant map, then this number is the degree of $f_1$. The precise definition of $\Lambda (f_1,f_2)$ one can find in [V], pg. 195. \pn Now we give two definitions in order to study the relation between the Nielsen coincidence number $N(f_1,f_2)$ and $\mu (f_1,f_2)$. \vskip 0.2cm \pn {\bf Definitions} \pn (a) A pair $(f_1,f_2)$ {\it has the Wecken property} if $\mu (f_1,f_2) = N(f_1,f_2)$. \pn (b) A pair $(M,N)$ of spaces has the {\it Wecken property} if every pair of maps $(f_1,f_2) \colon M\to N$ satisfies the Wecken condition. \bigskip Let us consider the obvious action of the group $Homeo(M)$ of homeomorphisms of $M$ on the set of pairs of maps $f_1, f_2 \colon M\to N$ given by $\phi(f_1,f_2) = (f_1\circ \phi,\ f_2\circ \phi)$ for $\phi \in $Homeo$(M)$. \vs 0.2cm \pn {\bf Proposition 3.1.} {\it If $(f_1, f_2)$ \ and \ $(f_3,f_4)$ belong to the same orbit, with respect to the above action, then either both pairs have the Wecken property or both do not have it. } \vs 0.2cm \pn {\it Proof:} The proof is straightforward. We certainly have that $N(f_1,f_2)=N(f_3,f_4)$. Since $\phi$ is a homeomorphism of $M$, \ $coin (f_1,f_2)\ {\stackrel{\phi^{-1}}{\longrightarrow}}\ coin (f_1\circ \phi,\ f_2\circ \phi)$\ is also a homeomorphism and we certainly have that $\mu (f_1,f_2) = \mu (f_1\circ \phi,\ f_2\circ \phi)$. \ If \ $(f'_1, f'_2)$ is homotopic to $(f_1,f_2)$, where $\# coin (f'_1,f'_2) = \mu (f_1,f_2)$ \ then \ $\# coin (f'_1\circ \phi,\ f'_2 \circ \phi) = N(f_3,f_4)$. So if $(f_1,f_2)$ satisfies the Wecken condition then $(f_3, f_4)$ also. \hfill $\Box$ \bigskip Now let us consider the particular case where the second function $f_2$ is the constant map denoted by c. This is so called the {\it root case}. \vs 0.2cm \pn {\bf Theorem 3.2.} {\it Let $f\colon S_h \to S_g$ be a primitive map, that is $f_\# (\pi_1 (S_h)) = \pi_1 (S_g)$. Then: \pn (a) $(f,c)$ does not have the Wecken property if $\dps |deg (f)| > {{2h-1}\over{2g-1}}$. \pn (b) $(f,c)$ satisfies the Wecken condition if $\dps |deg (f)| \leq {{2h-1}\over {2g-1}}$.} \vs 0.2cm \pn {\it Proof:} Suppose that $(f,c)$ satisfies the Wecken condition. Then, by the Fundamental Lemma 1.2 of [G], an equation of the form $W(z_1,...,z_{2h})= B_g^k$ has solutions when $k= |deg(f)|$. By Proposition 2.1, this implies that $\dps |deg (f)| = k \leq {{2h-1}\over{2g-1}}$. So part (a) follows. For part (b) let $n_0$ be the greatest integer less or equal to ${{2h-1} \over {2g-1}}$. By [GK] Theorem 9.2, any two primitive maps $f_1$ and $f_2$ with $deg (f_1) = deg (f_2)$ are strictly equivalent, i.e. there is a homeomorphism $\phi$ such that $f_1\circ \phi$ is homotopic to $f_2$. By this result and Proposition 3.1, it suffices to show for {\it one } primitive $f$ with $ \dps |deg(f)|=n_0$ that $(f,c)$ satisfies the Wecken condition. If $n_0={{2h-1} \over {2g-1}}$ then, by Proposition 2.2, the problem has a solution where $f$ plays the role of the branched covering map. So let $n_0 < {{2h-1} \over {2g-1}}$. Let $h_0$ be the greatest integer less than $h$ such that ${{2h_0-1} \over {2g-1}}$ is an integer, denoted by $n_1$. Certainly $n_0$ is equal to $n_1$ or $n_1+1$. Now we are going to use the fact that $(f,c)$ has the Wecken property if and only if the equation $W(z_1,...,z_{2h}) = B^k_g$ \ for \ $\dps k=|deg(f)|$ has a solution. In the first case, by Proposition 1.4, an equation $W(z_1,...,z_{2h_0}) = B^{n_0}_g$ has a solution. In the second case, $n_0=n_1+1$ and, hence, $h-h_0 > {{2g-1} \over 2}$. But we know that any quadratic equation $W'(z_1,...,z_{2h_0})= B^{n_1}_g$ has a solution. Then we multiply both sides with $B_g$, which is a product of $g$ commutators, and obtain a solution of the equation $W(z_1, \ldots, z_{2h}) = B_g^{n_0}$ where some of the new generators may get trivial values. Consequently, by the Fundamental Lemma 1.2 of [G], our pair $(f,c)$ satisfies the Wecken condition. The remaining cases where $\dps |deg(f)| < n_0$ are similar to the last case and simpler. So we leave it to the reader. \hfill $\Box$ \bigskip For the case where the target is the torus we can get the following corollaries: \vs 0.2cm \pn {\bf Corollary 3.3.} {\it Let $f \colon S_h \to T$ be a primitive map, that ist $f_\# (\pi_1(S_h))=\pi_1(T)$. Then: \pn $(a)$ $(f,c)$ does not have the Wecken property if $| \deg (f)|> 2h-1$. \pn $(b)$ $(f,c)$ satisfies the Wecken condition if $| def (f)| \le 2h-1$. } \vs 0.2cm \pn {\it Proof:} Since the genus of T is one, the result follows by Theorem 3.2. \hfill $\Box$ \vs 0.2cm \pn {\bf Corollary 3.4.} {\it Let $(f_1,f_2) \colon S_h \to T$ be a primitive pair, that is $im(f_{1\#}-f_{2\#})=\pi_1(T)$. \ Then \ $(f_1,f_2)$ has the Wecken property if and only if $\left |\Lambda (f_1,f_2)\right | \leq 2h-1$. } \vs 0.2cm \pn {\it Proof:} Since $T$ is a group, $\varphi (x) = f_1(x) \cdot f_2(x)^{-1}$ is defined and $coin (f_1,f_2) = coin (f_1 \cdot f_2^{-1},c)$. This implies that $\Lambda (f_1,f_2) = \Lambda (f_1 \cdot f_2^{-1}, c)$. >From the definition of the Lefschetz coincidence number $\Lambda (\ \ ,\ \ )$, we easily obtain that $\Lambda (f_1 \cdot f_2^{-1}, c) = deg(f_1 \cdot f_2^{-1})$. Now the result follows from Corollary 3.3. \hfill $\Box$ \vs 1cm \large \par\noindent {\bf Appendix} \normalsize \baselineskip=6.5mm \vs 0.5cm The picture shows the obvious $(2h-1)$-fold cover of the disjoint union of $(2h-1)$ polygons over one polygon. Further, after performing the identifications as labelled, it shows a $(2h-1)$-fold cover of the surface $S_h$ of genus $h$ minus one open disk over the torus minus one open disk. We hope that the picture is self-explanatory. \vs 1cm \begin{thebibliography}{9} \bibitem[E] { } Edmonds, A.: {\it Deformations of maps to branched coverings in dimension two.} Ann. Math. {\bf 110}, 113-125 (1979). \bibitem[Ep] { } Epstein, D.B.A.: {\it The degree of a map.} Proc. London Math. Soc. {\bf (3) 16}, 369-383 (1966). \bibitem[G] { } Gon\c calves, D.L.: {\it Coincidence of maps between surfaces.} Preprint Mathematisches Institut, Universit\"at Heidelberg, {\bf Heft Nr. 120}, Juni 1995. \bibitem[GK] { } Gabai, D., Kazez, W.H.: {\it The classification of maps of surfaces.} Invent. Math. {\bf 90}, 219-242 (1987). \bibitem[L] { } Lin, X.: {\it On the root classes of mapping.} Acta Math. Sinica, New Series {\bf 2(3)}, 199-206 (1996). \bibitem[S] { } Seifert, H.: {\it Bemerkungen \"uber stetige Abbildungen von Fl\"achen.} Abh. math. Sem. Univ. Hamburg {\bf 12}, 23-37 (1937). \bibitem[V] { } Vick, J.: {\it Homology theory.} Grad. Text Math. 145. New York-Berlin-Heidelberg: Springer 1994. \bibitem[ZVC] { } Zieschang, H., Vogt, E., Collewey, H.D.: {\it Surfaces and planes discon\-tinuous groups.} Lecture Notes Math. {\bf 835}, Springer-Verlag, Berlin-Heidelberg-New York, 1980. \end{thebibliography} \end{document}