\documentstyle[12pt]{article} \renewcommand{\baselinestretch}{1.4} \setlength{\textheight}{54pc} \setlength{\textwidth}{38pc} \setlength{\parindent}{1.4\parindent} \setlength{\topmargin}{-16mm} \addtolength{\oddsidemargin}{-1cm} \setlength{\footheight}{36pt} \setlength{\footskip}{48pt} %\setlength{\textheight}{200mm} %\setlength{\textwidth}{160mm} \newtheorem{prop}{Proposition}[section] \newtheorem{cor}[prop]{Corollary} \newtheorem{defi}[prop]{Definition} \newtheorem{lema}[prop]{Lemma} \newtheorem{fundlema}[prop]{Fundamental Lemma} \newtheorem{teo}[prop]{Theorem} \newcommand{\bea} {\begin{eqnarray*}} \newcommand{\beq} {\begin{equation}} \newcommand{\bey} {\begin{eqnarray}} \newcommand{\eea} {\end{eqnarray*}} \newcommand{\eeq} {\end{equation}} \newcommand{\eey} {\end{eqnarray}} \newcommand{\af}{\alpha} \newcommand{\bt}{\beta} \newcommand{\dt}{\delta} \newcommand{\Dt}{\Delta} \newcommand{\eps}{\epsilon} \newcommand{\gm}{\gamma} \newcommand{\Gm}{\Gamma} \newcommand{\lb}{\lambda} \newcommand{\Lb}{\Lambda} \newcommand{\na}{\nabla} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\sg}{\sigma} \newcommand{\ta}{\theta} \newcommand{\vf}{\varphi} \newcommand{\vps}{\varepsilon} \newcommand{\nb}{{\bf n}} \newcommand{\calc}{{\cal C}} \newcommand{\calf}{{\cal F}} \newcommand{\calr}{{\cal R}} \newcommand{\bs}{\backslash} \newcommand{\C} {\begin{picture}(8,8) \put(0,0){\normalsize C} \put(3.7,0.1){\line(0,1){7}} \end{picture}} \newcommand{\coin}{{\rm coin}} \newcommand{\coker}{{\rm coker}} \newcommand{\degr}{{\rm deg}} \newcommand{\dif}{{\rm d}} \newcommand{\dps}{\displaystyle} \newcommand{\hbx}{\hfill$\Box$} \newcommand{\hseis}{\hspace{6mm}} \newcommand{\intr}{{\rm int}} \newcommand{\lge}{\langle} \newcommand{\lra}{\longrightarrow} \newcommand{\Lt}{\Leftarrow\ } \newcommand{\nd}{\noindent} \newcommand{\N}{{\rm I\! N}} \newcommand{\tr}{{\rm tr}} \newcommand{\Z}{{\rm Z\!\! Z}} \newcommand{\ovl}{\overline} \newcommand{\proof}{\noindent {\bf Proof:\ }} \newcommand{\vv}{\vspace{4mm}} \newcommand{\wh}{\widehat} \newcommand{\wt}{\widetilde} \title{{\bf Coincidence of maps between surfaces}} \author{Daciberg L. Gon\c calves\thanks{This work has been supported by the international cooperation program GMD/Germany-CNPq/Brasil and Universidade de S\~ao Paulo during the author visit at the Mathematisches Institut der Universit\"at Heidelberg.}} \date{} \begin{document} \maketitle \vv \begin{abstract} We will consider $f, g : S_1 \to S_2$ a pair of maps between two orientable compact surfaces. The purpose of this paper is to study when the pair $(f,g)$ has the property that $(f,g)$ can be deformed to a pair $(f',g')$ such that $\#\coin(f'g') = N(f,g)$ the Nielsen coincidence number of $(f,g)$. We derive an equivalent algebraic condition and show that if we compose $(f,g)$ with certain maps $h : S \to S_1$ then the question has a positive answer. Finally we analize the case of roots i.e. $g$ is the constant map. When $S_2$ is the torus we give a new proof of the converse of the Lefschetz theorem for coincidence. \vv AMS Subject Classification: Primary: 55M20 \qquad Secondary: 55M25 \quad 57N75 \vv {\bf Running title:} Coincidence on surfaces. \end{abstract} \normalsize \baselineskip=8.5mm \nd {\bf Introduction:} The purpose of this work is to study the Wecken problem for coincidence. Namely, given $f,g : S_1 \to S_2$, two continuous maps, when does it exist $f'$ homotopic to $f$, $g'$ homotopic to $g$ such that $\#\coin(f',g') = N(f,g)$ (where $\#$ stands for cardinality and $coin(f,g)=\{x \in S_1 | f(x)=g(x)\}$. This question even for fixed points is known to have a positive answer for some pairs ($id, f$) and negative answer for others. See for example [J$1$] and [J$2$]. Let us say that $(f,g)$ has the Wecken property if the question above has a positive answer. In section 1 we derive algebraic conditions, in terms of the Braid groups, for a pair $(f,g)$ to have the Wecken property. This is the Fundamental Lemma 1.2. We also prove Theorem 1.3 which says: {\it Given any pair of maps $(f,g) : S_l \to S_h$ between two surfaces, there is an integer $n$ such that $(f \circ p_{l,n}, g\circ p_{l,n})$ has the Wecken property. Also $N(f\circ p_{l,n}, g\circ p_{l,n}) = N(f,g)$, for any integer $n$.} Here $p_{l,n}$ looks like a projection. In section 2 we specialize to the case where the second space is the torus T. We reduce the problem where one of the maps is the constant map. Then we write the algebraic equations which are equivalent to the geometric problem. These are propositions 2.1 and 2.3 respectively. Finally we give a proof of the converse to the Lefschetz theorem for coincidence of maps on the torus. This is Theorem 2.8 which says: {\it Given $f, g : S \to T$ such that $N(f,g) = 0$ then $(f,g)$ can be deformed to be a pair $(f',g')$ which is coincidence free.} This theorem relies on the fact that a map $h : S \to T$, which has degree zero, can be deformed to a map which is not surjective. This result is known, but we give a new proof, that we expect to have its own interest. We would like to thank Prof. E. Fadell for many helpful conversations, for his interesting questions about the subject and for his encouragement. Finally, we would like to thank Prof. R.F. Brown for pointing out the overlapping of this work with others and his suggestions, which improved the exposition of this work. \section{Algebraic version of the Wecken property and further results} We will start with a Fundamental Lemma, which is a natural generalization of Theorem 1.1 of [J$3$] for coincidence; the proof basically follows the same kind of arguments therein. Let $S_l,S_h$ be the orientable compact surfaces of genus $l,h$ respectively, and ${e_1,...e_{2l}}$ a fixed basis of $\pi_1(S_l,x_0)$. Let us assume that the chosen basis satisfies the relation $\prod_{i=1}^l [e_{2i - 1}, e_{2i}]=[T]$, where $[T]$ is the element defined in [J$3$], page 125. Suppose we are given two maps $f, g : S_l \to S_h$, where we have $f_{\#}(e_i) = w_i$, $g_{\#}(e_i) = v_i$, $i = 1, 2, \ldots, 2l$, and $w_i,v_i$ belong to $\pi_1(S_h,y_1),\pi_1(S_h,y_2)$ respectively and $f(x_0)=y_1 \ne y_2=g(x_0)$. Following [FH$1$] section 4, we have two subgroups of $\pi_1(S_h \times S_h - \Dt,(y_1,y_2))$ generated by $(\rho_{1,1}$, $\rho_{2,1}$, $\ldots$, $\rho_{2h-1, 1}$, $\rho_{2h, 1})$, $(\rho_{1,2}$, $\rho_{2,2}$, $\ldots$, $\rho_{2h-1, 2}$, $\rho_{2h,2})$, which we denote by $F_1,F_2$ respectively. Let $W_i$, $V_i$ be elements of $F_1,F_2$ which project over $w_i$, $v_i$ respectively under the map $\pi_1(S_h \times S_h - \Dt,(y_1,y_2)) \to \pi_1(S_h\times S_h,(y_1,y_2))$ induced by the inclusion. Let $N(f,g) = r$ and $k_1, \ldots, k_r$ be the indices of the essential Nielsen classes. Also, let $\{1, \af_2, \ldots, \af_r\}$ be a set of representatives of the Reidemeister classes of $(f,g)$, which corresponds to the essential Nielsen classes. Let the class defined by $\af_i$ have index $k_i$ and the class defined by 1 have index $k_1$. (For the definition of the local coincidence index, see e.g. [V]). We choose a base point in the class defined by $1$. \begin{defi} A pair $(f,g)$ has the Wecken property if we can find $f'$ homotopic to $f$, $g'$ homotopic to $g$ such that $\#\coin(f',g') = N(f,g)$. \end{defi} Now let $F = \pi_1(S_h - y_1,y_2)$, $N = {\rm Ker}(F \to \pi_1(S_h,y_2))$, $B = \prod_{i=1}^h [\rho_{2i - 1,2}, \rho_{2i, 2}^-1]$ and $B_\af = \af B\af^{-1}$ as in [FH$1$]. \begin{fundlema} The pair $(f,g)$ satisfies the Wecken property if and only if \begin{itemize} \item[(a)] If $N(f,g) = 0$ then we can find a solution $\ta_i \in N$, $i = 1, 2, \ldots, 2g$ of the equation \[\prod_{i=1}^g [\ta_{2i- 1}W_{2i-1}V_{2i-1}, \ta_{2i}W_{2i}V_{2i}] = 1\ .\] \item[(b)] If $N(f,g) = r \ne 0$ and $k_1, \ldots, k_r$ are the indices of the Nielsen classes, then we can find a solution $\ta_i \in N$, $i = 1, \ldots, 2g$ of the equation \[\prod_{i=1}^g [\ta_{2i- 1}W_{2i-1}V_{2i-1}, \ta_{2i}W_{2i}V_{2i}] = B^{k_1}B_{\af_2}^{k_2} \cdots B_{\af_r}^{k_r}\] for some set $\{1, \af_2, \ldots, \af_r\}$ of representatives of the correspondents essential Nielsen classes. \end{itemize} \end{fundlema} \proof The case (a) can be proved in the same way as case (b) and it is simpler. So we will show only case (b). Suppose $(f,g)$ satisfies the Wecken property. Let $f' \sim f$, $g' \sim g$, where $\sim$ means homotopic, such that $\#\coin(f',g') = N(f,g)$. Denote by $x_1, \ldots, x_r$ the points of $\coin(f',g')$. Around each point we draw a small circle and connect them by a path. See picture below. \vspace{11cm} Using the notation of the picture above, the small circle around the point $x_j$ is $\lb_j *\bt_j^{-1}$. The base point is $x_0$ which belongs to the circle around the first point. Two consecutive circles are connected by a path $\gm_j$. Call $D_j$ the closed disk which boundary the circle around $x_j$ and $\stackrel{\circ}{D}_j$ its interior. Since $(f',g')$ have no coincidence in $S_g - \bigcup_{j=1}^r\stackrel{\circ}{D}_j$, this means that $(f',g')$ is in fact a map $(f',g') : S_g - \bigcup_{j=1}^r\stackrel{\circ}{D}_j \to S_n \times S_n - \Dt$. But the loop $\xi = \prod_{i=1}^g[\rho_{2i - 1}, \rho_{2i}]\prod_{i=1}^r\xi_i$ is trivial in $S_g - \bigcup_{j=1}^r \stackrel{\circ}{D}_j$ where $\xi_i$ is the class of the loop \[\lb_1 * \gm_1 * \cdots * \lb_{i-1}*\gm_{i-1}*\bt_i*\lb_i^{-1}*\gm_{i-1}^{-1} *\lb_{i-1}^{-1}*\cdots *\gm_1^{-1}*\lb_1^{-1}\ .\] So $(f \times g)_{\#}(\xi)$ is 1 in $\pi_1(S_n \times S_n - \Dt)$. Let $\vf_i$ be the path $\lb_1*\gm_1 * \cdots * \lb_{i-1}*\gm_{i-1}$ which goes from the base point $x_0$ to the base point of the circle around $x_i$. Since the coincidence index of $x_i$ is $k_i$ we have that $(f'(\vf_i * \lb_i * \bt_i^{-1}* \vf_i^{-1})$, $g'(\vf_i * \lb_i * \bt_i^{-1} * \vf_i^{-1}))$ as a Braid represents the class of $\af_iB^{k_i}\af_i^{-1}\approx B_{\af_i}^{k_i}$ where $\af_i \approx g(\wt\vf_i)f(\wt\vf_i)^{-1}$ and $\wt\vf_i$ is just $\vf_i$ followed by the radius from the end of $\vf_i$ to $x_i$. Also $(f'(\rho_i), g'(\rho_i))$ as a braid is $\ta_iW_iV_i$ for some $\ta_i \in N$. So it follows from $(f\times g)_{\#}(\xi) = 1$ the equation given in part (b). Now suppose that we have a solution for the equation given in (b). Let us define two functions $f, g : S_g \to S_h$. Using the same picture and notation as above define $f(x) = y_0$ for $x \in D_1 \cup \gm_1 \cup D_2 \cup \cdots \cup \gm_{r-1} \cup D_r$, where $y_0$ is a fixed point of $S_h$. Define $g$ on the boundary of $D_i$ s.t. the image is a small circle around $y_0$ and $g$ has degree $k_i$. Extend $g$ to $D_i$ radially and define $g$ on the path $\gm_i$ in such way that $g(\wt\vf)$ represents the element $\af_i$ given by the equation. Finally on the edges $\gm$, $\rho_i$ define $f$ and $g$ such that $f(e_i)$, $g(e_i)$ represent the words $w_i$, $v_i$ respectively. There is no problem to define $f, g$ in these edges without having coincidence. But the given equation is precisely the algebraic condition to extend the map $\vf = (f,g)$ over the 2-skeleton. See theorem 4.3.1 of [B]. If $\wt\vf$ is the extension and $p_i : S_h \times S_h - \Dt \to S_h$ is the projection on the $i$-th coordinate, then the maps $f_1 = p_1 \circ\wt\vf$, $g_1 = p_2\circ\wt\vf$ are certainly homotopic to $f$, $g$ respectively and we get the result. \hbx \vv This type of equation is basically the same one which appears in [J3], theorem 1.1 where they treat the fixed point case. \vv Let $S_l$ be the orientable surface of genus $l$. Define $p_{l,n} : S_{l+n} \to S_l$ as the map which takes the last $n$-handles to a point $p$, while the complement is mapped almost like the identity (see picture below)\\ \vspace*{14cm} Now we will state the main result of this section. For its proof we need Proposition 1.4. \begin{teo} Given any pair of maps $(f,g) : S_l \to S_h$ among two surfaces, there is an integer $n$ such that $(f \circ p_{l,n}, g\circ p_{l,n})$ has the Wecken property. Also $N(f\circ p_{l,n}, g\circ p_{l,n}) = N(f,g)$ for any integer $n$. \end{teo} Now let us consider the abelianized obstruction, (see [FH$1$]), to lift a map to the 2-skeleton. If we consider the diagram \[\begin{array}{ccccc} &&&& S_h \times S_h - \Dt \\ &&&& \\ &&&& \downarrow \\ &&&& \\ (f,g): S_l && \lra && S_h\times S_h \end{array}\] by [FH$1$] the abelianized obstruction to lift $(f,g)$ from the 1-skeleton to the 2-skeleton is an element of $H^2(S_l, Z[\pi])$, where $\pi = \pi_1(S_h)$. This cohomology class can be represented by a 2-cocycle of the form $\sum n_i[\af_i]$ where $n_i$ is the index of a Nielsen class and $[\af_i]$ is the corresponding Reidemeister class. \begin{prop} We can find $\ta_1, \ldots, \ta_{2l} \in N$ which are a solution for the equation given in part (b) of the fundamental lemma 1.2, on $N_{ab}$ i.e. the abelianized of $N$. \end{prop} \proof By classical obstruction theory, if a 2-cocycle represents an obstruction class which is a two dimensional cohomology class, then we can deform the function over the 1-skeleton such that the cocycle defined from this new function is precisely the one given. This new function together with the original one provide us with $\ta_1, \ldots, \ta_{2l}$ which is a solution for the equation in part (b) on $N_{ab}$. \hbx \vv \nd {\bf Remark:} This can be done directly working with $N_{ab} \cong Z[\pi]$. \vv \nd {\bf Proof of theorem 1.3:} By propostion 1.4 if we are given a set of representatives $\{1, \af_2$, $\ldots$, $\af_r\}$ of the Reidemeister classes which correspond to the essential Nielsen classes, we can find $\ta_1, \ldots, \ta_{2l}$ such that \[W = B_{\af_k}^{-n_k} \cdots B_{\af_2}^{-n_2}B^{-n_1}[\ta_1W_1V_1, \ta_2W_2 V_2]\cdots [\ta_{2l-1}W_{2l-1}V_{2l-1}, \ta_{2l}W_{2l}V_{2l}] \in [N, N]\] or this element is zero in $N_{ab}$. So we have $\ta_{2l+1}, \ldots, \ta_{2l+2n} \in N$ such that \[W = \prod_{j=1}^n[\ta_{2l + 2n - 2j + 2}, \ta_{2l + 2n - 2j + 1}]\qquad \hbox{or}\] \beq \prod_{i=1}^l[\ta_{2i-1}W_{2i-1}V_{2i-1}, \ta_{2i}W_{2i}V_{2i}] \prod_{j=l+1}^{l+n} [\ta_{2j-1}, \ta_{2j}] = B^{n_1}B_{\af_2}^{n_2} \cdots B_{\af_k}^{n_k}\ . \eeq Now we have \[f\circ p_{l,n}(e_i) = \cases{f_{\#}(e_i) , \ 1 \le i \le 2l \cr 1 , \ 2l + 1 \le i \le 2(l+n) \cr} \quad\hbox{and}\quad g\circ p_{l,n}(e_i) = \cases{g_{\#}(e_i), \ 1 \le i \le 2l \cr 1 , \ 2l + 1 \le i \le 2(l+n) \cr}\] So by the Fundamental Lemma 1.2, equation (1) above implies that $(f\circ p_{l,n}, g \circ p_{l,n})$ has the Wecken property. Finally, we can assume that the point $p$ which appears in the definition of the function $p_{l,n}$, does not belong to $\coin (f,g)$. So $\coin(f\circ p_{l,n}, g\circ p_{l,n}) \stackrel{p_{l,n}}{\longrightarrow} \coin (f,g)$ is an homeomorphism, which induces a map among the Nielsen classes. This induced map is certainly surjective and is also injective because $p_{l,n\#} (\pi_1(S_{l+n}) \to \pi_1(S_l))$ is surjective. So we have a bijection among the Nielsen classes and the corresponding classes certainly have the same index. So the result follows. \hbx \vv \nd {\bf Remark:} The result above suggests the natural question which is to find the minimum integer $n$ such that $(f\circ p_{n,l}, g\circ p_{n,l})$ has the Wecken property. In a reasonable number of cases, the answer is known mainly when one of the maps is the identity. See e.g. [J$1$]. \vv To finish this section, we prove a proposition which tells us that, in order to minimize coin, it suffices to change one function, at least when the target space is a manifold. This geometric fact is useful. Let $f, g : M \to N$ be two continuous maps where $M$ is a topological space, and $N$ is a manifold. Let $\mu(f, g) = \min\limits_{f'\in [f], g' \in [g]} \#\coin(f',g')$ and $\mu_1(f,g) = \min\limits_{g'\in [g]} \# \coin (f,g')$. \begin{prop} For any pair $(f,g) : M \to N$ we have $\mu(f,g) = \mu_1(f,g)$. \end{prop} \proof Consider the fibered pair $(N \times N, N\times N - \Dt) \stackrel{p_1}{\lra} N$ (see [F]), where $p_1$ is the projection on the first coordinate. Certainly we have $\mu(f,g) \le \mu_1(f,g)$. So it suffices to show that $\mu_1(f,g) \le \mu(f,g)$. For this let $(f',g')$ such that $\#\coin(f',g') = k$. We have that $\ovl f, \ovl f' : M \times M \to N$ are homotopic where $\ovl f(x,y) = f(x)$ and $\ovl f'(x,y) = f'(x)$. Call $H$ such homotopy. The map $f'$ has a lift, namely $(f',g')$ such that $(f',g')(M - \coin(f',g')) \subset N \times N - \Dt$. By the lifting property of fibered pairs it follows that there is a lift $\wt H$ of $H$ such that $\wt H(M - \coin(f',g')) \subset N \times N - \Dt$. So $\wt H(\ , 1) = (f,g'')$ and $\coin(f,g'') \subset \coin(f',g')$. Therefore $\#\coin(f,g'') \le \#\coin(f',g')$ and the result follows. \hbx \vv Although known in a more general form, we decided to include the above proof since it is simpler than the one appearing in [Br]. \vv \section{Root case and coincidence of maps into the torus} Let $S$ be an orientable surface, $T$ the torus and $f,g : S \to T$ two maps. Now, if we identify $T$ with $S^1 \times S^1$, then $T$ has a group structure given by the complex multiplication in each coordinate. So we consider the maps $h(x) = g(x)/f(x)$ and $c(x) = e$, where $e \in T$ is the identity element with respect to the above multiplication. \begin{prop} We have $\coin(f,g) = \coin(h,c)$ and $N(f,g) = N(h,c)$. Further $(f,g)$ has the Wecken property if and only if we can find a map $h_1$ homotopic to $h$ such that $\# h_1^{-1}(1) = N(f,g)$. \end{prop} \proof It is clear that $\#\coin(f,g) = \#\coin(h,c)$. Also, the Nielsen classes of $(f,g)$ and $(h,c)$ are the same. For let $x, y \in \coin(f,g)$, $\lb : I \to S$, $\lb(0) = x$, $\lb(1) = y$ with $f(\lb) \sim g(\lb)$ relative to the end points and let $H$ be a homotopy. Then $H_1(s, t) = H(s,t)/f(\lb(s))$ gives a homotopy relative to the end points between $h(\lb)$ and $c(\lb)$. To see that a class has the same index with respect to both pair of maps, let us consider the map $\psi : (T \times T, T\times T - \Dt) \to (T, T - \{1\})$ given by $\psi(x,y) = y/x$. This is a well defined map of the pairs and $\psi^* : H^n(T, T - \{1\}) \to H^n(T \times T, T\times T - \Dt)$ takes the fundamental cohomology class to the Thom class. But $\psi \circ (f,g) = \psi \circ (h,c)$. So a Nielsen class has the same index either with respect to $(f,g)$ or $(h,c)$. Finally, by Proposition 1.5, in order to minimize coincidence, it suffices to deform one of the maps, let us say $h$, so the result follows. \hbx \vv From now on let $h : S \to T$ be a map and let $h_{\#} : H_1(S) \to H_1(T)$ be given by $h_{\#} = \left(\begin{array}{ccccc} a_1 & b_1 & \cdots & a_g & b_g \\ c_1 & d_1 & \cdots & c_g & d_g \end{array}\right)$, where $S$ is a compact surface of genus $g$. Let $\Lb(h,c)$ be the Lefschetz coincidence number and $\deg(h)$ the degree of $h$. \begin{prop} We have $\Lb(h,c) = \degr(h)$. If $\degr(h) = 0$ then $N(h,c) = 0$. If $\degr(h) \ne 0$ then $N(h,c) = \#\coker(h_{\#})$ and each Nielsen class has index equal to $\degr(h)$ divided by $N(h,c)$. \end{prop} \proof The above result is true in general, i.e. whenever $S$ and $T$ are orientable manifolds of the same dimension. This follows from Proposition 5 of [L] and Corollary 7.3 of [K], or [O1]. \hbx \vv Now we will derive the algebraic condition for $(h,c)$ to have the Wecken property. When $\degr(h)$ is not zero, then we certainly have that $h_{\#}(\pi_1(S)) \lhd \pi_1(T)$ has a finite index. Denote by $\{\af_1, \ldots, \af_n\}$ a set of representatives of the elements of the Reidemeister classes $\pi_1(T)/h_{\#}(\pi_1(S))$. Let us pick a base point $x_0$ for $S$ and $y_0$ for $T$ where $y_0$ is close to $1 \in T$. Denote by $j : T - \{1\} \hookrightarrow T$ the inclusion, by $F(x,y) = \pi_1(T - \{1\})$ the free group on two generators and $B_\af = \af B\af^{- 1}$, $w_{2i - 1} = x^{a_i}y^{b_i}$, $w_{2i} = x^{c_i}y^{d_i}$. As before let $B = [x, y]$, $N = [F, F]$ and $h_{\#} = \left(\begin{array}{ccccc} a_1 & b_1 & \cdots & a_g & b_g \\ c_1 & d_1 & \cdots & c_g & d_g \end{array}\right)$. \begin{prop} Let $h : S \to T$. Then there exists $h_1 \sim h$ such that $\#h_1^{-1}(1) = N(h,c)$ iff \begin{itemize} \item[(a)] If $\degr(h) = 0$ then the equation \[\prod_{i=1}^g[\ta_{2i-1}w_{2i-1},\ta_{2i}w_{2i}] = 1\] has a solution $\ta_i \in N = [F, F]$, $i = 1, \ldots 2g$. \item[(b)] If $\degr(h) = m \ne 0$, let $r = \#\coker(h_{\#})$ and $k = m/r$. Then the equation \[\prod_{i=1}^g[\ta_{2i-1}w_{2i-1}, \ta_{2i}w_{2i}] = B^k.B_{\af_2}^k \cdots B_{\af_r}^k\] has a solution for some $\ta_j \in N$ and for some set of representatives $\{1, \af_2, \ldots, \af_r\}$ of the quotient ${\pi_1(T) \over h_{\#} (\pi_1(S))}$. \end{itemize} \end{prop} \proof The proof follows from the Fundamental Lemma 1.2. It is enough to see that because the second map is the constant map, then the equation given in part b) of the Fundamental Lemma 1.2 is in fact an equation in the subgroup $F_1$, where $F_1$ is defined before definition 1.1. So the result follows. \hbx \vv In the special case where $N(f,g) = 0$, we will show, in a geometric way, that $(f,g)$ satisfies the Wecken property. \begin{lema} In order to show that a pair with $N(h,c) = 0$, has the Wecken property, it suffices to consider the case where $h_{\#}(\pi_1(S)) = \pi_1(T)$. \end{lema} \proof If the ${\rm rank}(h_{\#}(\pi_1(S))) < 2$ then we can deform $h$ to $h_1$ such that $h_1(S)$ lies inside a curve which does not contain $1 \in T$. If $h_{\#}(\pi_1(S))$ has rank two, take $\wt T \stackrel{p}{\longrightarrow} T$ the finite cover which corresponds to the subgroup $h_{\#}(\pi_1(S))$ and let $\wt h : S \to \wt T$ be the lifting of $h$. So $\wt h : S \to \wt T$ has the property that $\wt h_{\#} : \pi_1(S) \+to \pi_1(\wt T)$ is surjective. If $\wt h$ can be deformed to a map which is not surjective, then it can be deformed to the one skeleton of $\wt T$ which we can assume that does not intersect $p^{-1}(1)$, where $p : \wt T \to T$ is the cover map. So the result follows. \hbx \vv Let $\gm \subset S_g$ be an embedded curve. We let $h_\gm : S_g \to S^1$ be a map defined as follows: take a tubular neighbourhood of $\gm$ and let $\vf : \gm \times [-\vps, \vps] \to S_g$ be an homeomorphism onto that tubular neighborhood of $\gm$. Now we define \[h_\gm(x) = \cases{e^{\pi ti \over \vps} &if $x = \vf(x_1, t)$ \cr -1 &otherwise. \cr}\] Roughly speaking, the homology class represented by $\gm$ is the Poincar\'e dual of the cohomology class $h_{\gm}^{\#}(w_1) \in H^1(S_g)$ with proper orientations, where $w_1$ is a choosen generator of $H^1(S^1)$. See [G], Part II and [T] for more details. Let $h : S_g \to T$ and \[h_{\#} = \left(\begin{array}{ccccccc} a_1 & b_1 & a_2 & b_2 & \cdots & a_g & b_g \\ c_1 & d_1 & c_2 & d_2 & \cdots & c_g & d_g \end{array}\right)\ .\] We will assume from now on that ${\rm deg}\,h = \Lb(h,c) = 0$ and $h_\#(\pi_1 (S)) = \pi_1(T)$. \begin{prop} The elements $(a_1, b_1, \ldots, a_g, b_g)$ and $(c_1, d_1, \ldots, c_g, d_g)$ in $\underbrace{\Z \oplus \cdots \oplus \Z}_{2g}$ are indivisible. \end{prop} \proof This follows from the fact that $h_\#(\pi_1(S)) = \pi_1(T)$. \hbx \vv Let $\gm_1$ resp. $\gm_2$ be two connected closed simple curves which represent the homology classes $(a_1, b_1, \ldots, a_g, b_g)$ resp. $(c_1, d_1, \ldots, c_g, d_g)$. Such curves do exist. See e.g. [G] or [ZVC] section 3.6. We can assume that these two curves intersect transversally. \begin{prop} The intersection number of these two curves is zero. \end{prop} \proof See [D], chapter VIII, 13 or [ZVC] section 3.6. \hbx \vv Now we will modify one of the curves, $\gm_1$ for example, such that the new curve $\gm_1'$ has the following properties: a) $\gm_1'$ is a simple curve, not necessarily connected; b) $\gm_1'$ represents the same homology class as $\gm_1$; c) $\gm_1' \cap \gm_2 = \emptyset$. \begin{prop} Given $\gm_1$ and $\gm_2$ we can construct $\gm_1'$ as above. \end{prop} \proof The argument how to construct $\gm_1'$ is contained in [G], appendix. The picture below gives the idea of the construction. Namely, for each two consecutive points with intersection number $+1$ and $-1$ we get a new curve as shown below. \newpage \vspace*{10cm} Since the number of points in the intersection is finite the process ends after a finite number of steps and we get the curve $\gm_1'$ with the required properties. \hbx \begin{teo} Given $f, g : S \to T$ such that $N(f,g) = 0$ then $(f,g)$ can be deformed to a pair $(f',g')$ which is coincidence free. \end{teo} \proof It suffices to consider the map $h(x) = g(x)/f(x)$ and to show that $h$ can be deformed to $h'$ such that $1 \not\in h'(S)$. We can also assume, by lemma 2.4, that $h_\#(\pi_1(S)) = \pi_1(T)$. Let $\gm_1'$ and $\gm_2$ be the two curves given by proposition 2.7. So we have two maps $h_{\gm_1'}$, $h_{\gm_2} : S \to S^1$. Therefore we have $h_1 = (h_{\gm_1'}, h_{\gm_2}) : S \to T$. Certainly \[h_1^{-1}(1) = h_{\gm_1'}^{-1}(1) \cap h_{\gm_2}^{-1}(1) = \gm_1' \cap \gm_2 = \emptyset \] So $h_1^{-1}(1) = \emptyset$ and $h_1$ is certainly homotopic to $h$ because they induce the same homomorphism on $\pi_1$ or $H_1$. Therefore the result follows. \hbx \vv \nd {\bf Comments on the general case of degree zero 1:} Given a map $h : S_1 \to S_2$ between any two compact orientable surfaces, it is known that if $\degr(h) = 0$, then $h$ can be deformed to a map which is not surjective. This is a consequence of deep results of Kneser, as it was pointed out in [E]. For this purpose you can also see [K$1$], [K$2$] and [ZVC] section 3.3. This, in particular, shows by means of the Fundamental Lemma 2.1, that certain quadratic equations on free group have solutions. It is not clear how to explictly provide a solution or even how to show algebraically, that such equations have a solution. \vv \nd {\bf Comment on our result 2:} Theorem 2.8 certainly is not new. In fact it is weaker than the results pointed out above. Nevertheless, we believe that the proof given of Theorem 2.8 may have its own interest. \vv \nd {\bf Comment on the case general case 3:} In general, one can not expect to have $(h,c)$ to satisfying the Wecken property. The first example of a map $h : S_2 \to T$ $S_2$=the surface of genus $2$ where $(h,c)$ does not satisfy the Wecken property, was given in [L]. This naturally brings up the question of trying to classify the maps $h$ such that $(h,c)$ satisfies the Wecken property. More has been done recently about this question in a joint work in preparation. \vv {\bf General comment 4:} The connection between geometry and algebra given by the Fundamental Lemma, sounds promissing for both areas. \section*{References} \parindent=0pt \baselineskip=5mm \begin{description} \item{[B] Baues, H.J.}\ \ Obstruction theory. \ \ {\it Lectures Notes in Mathematics} {\bf 628}, Springer-Verlag, Berlin-Heidelberg-New York, 1977. \item{[Br] Brook, R.B.S.}\ \ On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy. \ \ {\it Pacific Journal of Mathematics}, {\bf 40}, n.1, 45--52. \item{[D] Dold, A.}\ \ Lectures on algebraic topology.\ \ {\it Grundlehren der Mathematischen Wissenschaften} {\bf 200}. 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Fixed point theory, 163--170 (1981). \item{[J2] Jiang, B.}\ \ Fixed points and braids II. \ \ {\it Math. Ann.}, 272, 249--256 (1985). \item{[J3] Jiang, B.}\ \ Surface maps and braid equations, I.\ \ {\it Lecture Notes in Math.}, {\bf 1369}, 125--141. \item{[K] Kiang, T.H.}\ \ The theory of fixed point classes.\ \ Springer-Verlag 1989. \item{[K1] Kneser, H.}\ \ Gl\"attung von Fl\"achenabbildungen.\ \ {\it Math. Ann.} {\bf 100}, 609--617 (1928). \item{[K2] Kneser, H.}\ \ Die Kleinste Bedeckungszahlinnerhalb einer Klasse von Fl\"a\-chen\-abbil\-dun\-gen.\ \ {\it Math. Ann.} {\bf 103}, 347--358 (1930). \item{[L] Lin, X.}\ \ On the root classes of mapping.\ \ {\it Acta Math. Sinica}, {\bf 2}, 199--206 (1986). \item{[O1] Oliveira, E.} \ \ { Teoria de Nielsen para coincid\^encia e algumas aplica\c c\~oes}.\ \ Ph.D. Thesis. \item{[T] Thom, R.} \ \ Quelques propri\'et\'es globales des vari\'et\'es differentiables.\ \ {\it Comm. Math. Helv.} {\bf 28}, 17--86 (1954). \item{[V] Vick, J.}\ \ Homology theory.\ \ {\it Academic Press}, New York, 1976. \item{[ZVC] Zieschang, H.}, Vogt, E., Coldewey, H.D.\ \ Surfaces and planes discontinuous groups. \ \ {\it Lectures Notes in Mathematics} {\bf 835}. Springer-Verlag, Berlin-Heidelberg-New York, 1980. \end{description} \vv\vv \begin{center} Daciberg Lima Gon\c calves Departamento de Matem\'atica - IME-USP Caixa Postal 66.281 - Ag. Cidade de S\~ao Paulo 05315-970 - S\~ao Paulo - SP - Brasil E-mail: DLGONCAL@IME.USP.BR \end{center} \end{document} % PARTE QUE FOI ELIMINADA NA PENULTIMA VERSAO: PROPOSICAO 2.9 E TEOREMA 2.10 \begin{prop} Given $\gm_1$ and $\gm_2$ we can construct $\gm_1$ as above. \end{prop} \proof The argument for construct $Y_1'$ is contained in [G], appendix. The picture below gives the idea of the construction. Namely for each two consecutives points with intersection number $+1$ and $-1$ we get \vspace{5cm} \hbx \begin{teo} Given $f, g : S \to T$ such that $N(f,g) = 0$ then $(f,g)$ can be deformed to be a pair $(f',g')$ which is coincidence free. \end{teo} \proof It suffices to consider the map $h(x) = g(x)/f(x)$ and to show that $h$ can be deformed to $h'$ such that $1 \not\in h'(S)$. We can also assume that $h_\#(\pi_1(S)) = \pi_1(T)$. Let $\gm_1'$ and $\gm_2$ be the two curves given by proposition 2.7. So we have two maps $h_{\gm_1'}$, $h_{\gm_2} : S \to S^1$. Therefore we have $h' = (h_{\gm_1'}, h_{\gm_2}) : S \to T$. Certainly \[h^{'-1}(1) = h_{\gm_1'}^{-1}(1) \cap h_{\gm_2}^{-1}(1) = \gm_1' \cap \gm_2 = \emptyset\ .\]