\documentclass[11pt]{amsart} %\includeonly{intro,unfolding} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{amssymb} \usepackage{enumerate} \usepackage[dvips]{graphics} \newcommand{\N}{{\mathbf N}} \newcommand{\Z}{{\mathbf Z}} \newcommand{\Q}{{\mathbf Q}} \newcommand{\R}{{\mathbf R}} \newcommand{\C}{{\mathbf C}} \renewcommand{\H}{{\mathbb H}} \newcommand{\coker}{{\mbox{\rm coker}}} \newcommand{\kk}{{\mathbf k}} \newcommand{\cat}{{\rm {cat }}} \newcommand{\Cat}{\rm {Cat}} \newcommand{\id}{{\rm {id }}} \newcommand{\I}{{\mathcal I}} \newcommand{\K}{{\mathcal K}} \newcommand{\cw}{{\rm cwgt}} \newcommand{\E}{{\mathcal E}} \newcommand{\F}{{\mathcal F}} \newcommand{\Id}{\rm Id} \newcommand{\Hom}{{\rm Hom}} \newcommand{\im}{\rm im} \newcommand{\tc}{{\rm\bf {TC}}} %\newcommand{\qed}{\begin{flushright} $\Box$\end{flushright}} \newcommand{\supp}{\rm {supp}} \newcommand{\rk}{\rm {rk}} \newcommand{\comment}[1]{} \def\U{\mathcal U} \def\V{\mathcal V} \def\A{\mathcal A} \def\ker{{\rm Ker }} \def\im{{\rm Im }} \def\rank{{\rm rank }} \def\lra{\longrightarrow} \def\ra{\rightarrow} \def\bbr{{\mathbf R}} \def\bbz{{\mathbf Z}} \def\bbn{{\mathbf N}} \def\cu{{\mathcal U}} \def\cv{{\mathcal V}} \def\x{\times} \def\diam{{\rm diam }} \def\eps{\varepsilon} \def\del{\partial} \def\pHi{\varphi} \def\wt{\widetilde} \def\wh{\widehat} \def\R{\mathbf R} \def\L{\mathcal L} \newtheorem{theorem}{Theorem}%[section] \newtheorem{maintheorem}[theorem]{Main Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}{Example} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}{Definition} \newtheorem{remark}{Remark} %\numberwithin{equation}{section} %\newenvironment{proof}{\par\noindent{\bf Proof.} }{\par\noindent} %\newenvironment{example}{\par{\bf Example.}}%{\par\noindent} %\newenvironment{definition}{{\bf Definition.} }{\par\noindent} %\parindent0em %\parskip0.5ex \begin{document} \title{Closed 1-forms with at most one zero} \author[M. ~Farber, D. ~Sch\"utz]{M.~Farber, D. Sch\"utz} \address{Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK} \email{Michael.Farber@durham.ac.uk} \address{Fachbereich Mathematik und Informatik, Westf\"alische Wilhelms - Universit\"at M\"unster, Einsteinstr. 62, D-48149 M\"unster, Germany} \email{schuetz@math.uni-muenster.de} % \thanks will become a 1st page footnote. %\thanks{The authors thank the Max-Planck Institute for Mathematics in Bonn for hospitality.} \subjclass{58E05, 57R70} \date{August 31, 2004} \keywords{Closed 1-form, Lusternik-Schnirelman theory, Novikov theory.} \begin{abstract} We prove that in any nonzero cohomology class $\xi\in H^1(M;\R)$ there always exists a closed 1-form having at most one zero. \end{abstract} \maketitle \section{Statement of the result} Let $M$ be a closed connected smooth manifold. By Hopf's theorem, there exists a nowhere zero tangent vector field on $M$ if and only if $\chi(M)=0$. If $\chi(M)\not=0$ one may find a tangent vector field on $M$ vanishing at a single point $p\in M$. A Riemannian metric on $M$ determines a one-to-one correspondence between vectors and covectors; therefore on any closed connected manifold $M$ there exists a smooth 1-form $\omega$ vanishing at most at one point $p\in M$. The question we address in this note is {\it whether the 1-form $\omega$ which is nonzero on $M-\{p\}$ can be chosen to be closed, $d\omega=0$?} The Novikov theory \cite{N2} gives bounds from below on the number of distinct zeros which have closed 1-forms $\omega$ lying in a prescribed cohomology class $\xi\in H^1(M;\R)$. However the Novikov theory imposes an additional requirement that {\it all zeros of $\omega$ are non-degenerate in the sense of Morse}. The number of zeros is then at least the sum $\sum_j b_j(\xi)$ of the Novikov numbers $b_j(\xi)$. If $\omega$ is a closed 1-form representing {\it the zero cohomology class} then $\omega=df$ where $f: M\to \R$ is a smooth function; in this case $\omega$ must have at least $\cat(M)$ geometrically distinct zeros, according to the classical Lusternik-Schnirelman theory \cite{DNF}. Our goal in this paper is to show that in general, with the exception of two situations mentioned above, {\it there are no obstructions for constructing closed 1-forms possessing a single zero.} We prove the following statement: \begin{theorem}\label{thm1} Let $M$ be a closed connected $n$-dimensional smooth manifold, and let $\xi\in H^1(M;\R)$ be a nonzero real cohomology class. Then there exists a smooth closed 1-form $\omega$ in the class $\xi$ having at most one zero. \end{theorem} This result suggests that \lq\lq the Lusternik-Schnirelman theory for closed 1-forms\rq\rq \, (see \cite{Fa1, Fa2} and Chapter 10 of \cite{Fa3}) has a new character which is quite distinct from both the classical Lusternik-Schnirelman theory of functions and the Novikov theory of closed 1-forms. Theorem \ref{thm1} was proven in \cite{Fa1} under an additional assumption that the class $\xi$ is integral, $\xi\in H^1(M;\Z)$. See also \cite{Fa3}, Theorem 10.1. This essentially covers all rank 1 cohomology classes $\xi\in H^1(M;\R)$ since any such class is a multiple of an integral class. Theorem \ref{thm1} has interesting implications in the theory of symplectic intersections, compare \cite{Sik}, \cite{EG}. Y. Eliashberg and M. Gromov mention in \cite{EG} that a statement in the spirit of Theorem \ref{thm1} was made by Yu. Chekanov at a seminar talk in 1996. No written account of his work is available. Let us mention briefly a similar question. We know that if $\chi(M)=0$ then there exists a nowhere zero 1-form $\omega$ on $M$. Given $\chi(M)=0$, one may ask if it is possible to find a nowhere zero 1-form $\omega$ on $M$ which is closed $d\omega=0$? The answer is negative in general. For example, vanishing of the Novikov numbers $b_j(\xi)=0$ is a necessary condition for the class $\xi$ to be representable by a closed 1-form without zeros. The full list of necessary and sufficient conditions (in the case $\dim M>5$) is given by the theorem of Latour \cite{La}. \section{Preliminaries} Here we recall some basic terminology. We refer to \cite{Fa3} for more detail. A smooth 1-form $\omega$ is a smooth section $x\mapsto \omega_x$, $x\in M$ of the cotangent bundle $T^\ast(M)\to M$. A {\it zero} of $\omega$ is a point $p\in M$ such that $\omega_p=0$. If $\omega$ is a closed 1-form on $M$, i.e. $d\omega=0$, then in any simply connected domain $U\subset M$ there exists a smooth function $f: U\to R$ such that $\omega|_U=df$. Zeros of $\omega$ are precisely the critical points of $f$. A zero $p\in M$, $\omega_p=0$ is said to be {\it Morse type} iff $p$ is a Morse type critical point for $f$. The {\it homomorphism of periods} \begin{eqnarray}\label{per} {\rm {Per}}_\xi: H_1(M)\to \R\end{eqnarray} is defined by \begin{eqnarray} {\rm {Per}}_\xi([\gamma]) = \int_\gamma \omega\, \in \, \R. \end{eqnarray} Here $\xi=[\omega]\in H^1(M;\R)$ is the de Rham cohomology class of $\omega$ and $\gamma$ is a closed loop in $M$; the symbol $[\gamma]\in H_1(M)$ denotes the homology class of $\gamma$. The image of the homomorphism of periods (\ref{per}) is a finitely generated free abelian subgroup of $\R$; it is called the {\it group of periods}. Its rank is denoted $\rk(\xi)$ -- the {\it rank of the cohomology class} $\xi\in H^1(M;\R)$. A closed 1-form $\omega$ with Morse zeros determines a {\it singular foliation} $\omega=0$ on $M$. It is a decomposition of $M$ into leaves: two points $p,q \in M$ belong to the same leaf if there exists a path $\gamma:[0,1]\to M$ with $\gamma(0)=p$, $\gamma(1)=q$ and $\omega(\dot \gamma(t)) = 0$ for all $t$. Locally, in a simply connected domain $U\subset M$, we have $\omega|_U =df$, where $f:U\to \R$; each connected component of the level set $f^{-1}(c)$ lies in a single leaf. If $U$ is small enough and does not contain the zeros of $\omega$, one may find coordinates $x_1, \dots, x_n$ in $U$ such that $f\equiv x_1$; hence the leaves in $U$ are the sets $x_1=c$. Near such points the singular foliation $\omega=0$ is a usual foliation. On the contrary, if $U$ is a small neighborhood of a zero $p\in M$ of $\omega$ having Morse index $0\le k\le n$, then there are coordinates $x_1, \dots,x_n$ in $U$ such that $x_i(p)=0$ and the leaves of $\omega=0$ in $U$ are the level sets $-x_1^2-\dots -x_k^2 + x_{k+1}^2+\dots +x_n^2 =c.$ The leaf $L$ with $c=0$ contains the zero $p$. It has a {\it singularity} at $p$: a neighborhood of $p$ in $L$ is homeomorphic to a cone over the product $S^{k-1}\times S^{n-k-1}$. There are finitely many {\it singular leaves}, i.e. the leaves containing the zeros of $\omega$. \begin{figure}[h] \begin{center} \resizebox{2.5cm}{3.4cm}{\includegraphics[176, 400][370,622]{singleaf1.eps}} \end{center} \end{figure} We are particularly interested in the singular leaves containing the zeros of $\omega$ having Morse indices 1 and $n-1$. Removing such a zero $p$ {\it locally} disconnects the leaf $L$. However globally the complement $L-p$ may or may not be connected. The singular foliation $\omega=0$ is {\it co-oriented}: the normal bundle to any leaf at any nonsingular point has a specified orientation. We shall use the notion of a weakly complete closed 1-form introduced by G. Levitt \cite{Levitt}. A closed 1-form $\omega$ is called {\it weakly complete} if it has Morse type zeros and for any smooth path $\sigma:[0,1]\to M^\ast$ with $\int_{\sigma}\omega=0$ the endpoints $\sigma(0)$ and $\sigma(1)$ lie in the same leaf of the foliation $\omega=0$ on $M^\ast$. Here $M^\ast$ denotes $M-\{p_1, \dots, p_m\}$ where $p_j$ are the zeros of $\omega$. A weakly complete closed 1-form with $\xi=[\omega]\not=0$ has no zeros with Morse indices $0$ and $n$. According to Levitt \cite{Levitt}, {\it any nonzero real cohomology class $\xi\in H^1(M;\R)$ can be represented by a weakly complete closed 1-form.} The plan of our proof of Theorem \ref{thm1} is as follows. We start with a weakly complete closed 1-form $\omega$ lying in the prescribed cohomology class $\xi\in H^1(M;\R)$, $\xi\not=0$. We show that assuming $\rk(\xi)>1$ all leaves of the singular foliation $\omega=0$ are dense (see \S \ref{density}). We perturb $\omega$ such that the resulting closed 1-form $\omega'$ has a single singular leaf (see \S \ref{modification}). After that we apply the technique of Takens \cite{Ta} allowing us to collide the zeros in a single (highly degenerate) zero. We first prove Theorem \ref{thm1} assuming that $n=\dim M >2$; the special case $n=2$ is treated separately later. \section{Density of the leaves}\label{density} In this section we show that {\it if $\omega$ is weakly complete and $\rk(\xi)>1$ then the leaves of $\omega=0$ are dense}. Note that in general the assumption $\rk(\xi)>1$ alone does not imply that the leaves are dense, see the examples in \S 9.3 of \cite{Fa3}. Let $\omega$ be a weakly complete closed 1-form in class $\xi$. Consider the covering $\pi:\tilde M\to M$ corresponding to the kernel of the homomorphism of periods ${\rm {Per}}_\xi : H_1(M) \to \R$, where $\xi=[\omega]\in H^1(M;\R)$. Let $H\subset \R$ be the group of periods. The rank of $H$ equals $\rk(\xi)$; since we assume that $\rk(\xi)>1$, the group $H$ is dense in $\R$. The group of periods $H$ acts on the covering space $\tilde M$ as the group of covering transformations. We have $\pi^\ast\omega =dF$ where $F:\tilde M\to \R$ is a smooth function. The leaves of the singular foliation $\omega=0$ are the images under the projection $\pi$ of the level sets $F^{-1}(c)$; this property follows from the weak completeness of $\omega$, see \cite{Levitt}, Proposition II.1. For any $g\in H$ and $x\in \tilde M$ one has \begin{eqnarray} F(g x)-F(x)\, =\, g\, \in\, \R. \end{eqnarray} Let $L=\pi(F^{-1}(c))$ be a leaf and let $x\in M$ be an arbitrary point. Our goal is to show that $x$ lies in the closure $\bar L$ of $L$. Let $U\subset M$ be a neighborhood of $x$. We want to show that $U$ intersects $L$. We shall assume that $U$ is \lq\lq small\rq\rq\, in the following sense: $\xi|_U =0$. Consider a lift $\tilde x\in \tilde M$, $\pi(\tilde x)=x$. Let $\tilde U$ be a neighborhood of $\tilde x$ which is mapped by $\pi$ homeomorphically onto $U$. We claim that {\it the set of values $F(\tilde U)\subset \R$ contains an interval $(a-\epsilon, a+\epsilon)$ where $a=F(\tilde x)$ and $\epsilon>0$.} This claim is obvious if $\tilde x$ is not a critical point of $F$ since in this case one may choose the coordinates $x_1, \dots, x_n$ around $\tilde x$ such that $F(x)=a+x_1$. In the case when $\tilde x$ is a critical point of $F$, one may choose the coordinates $x_1, \dots, x_n$ near the point $\tilde x\in \tilde M$ such that $F(x)$ is given by $a\pm x_1^2\pm x_2^2+\dots+\pm x_n^2$ and our claim follows since we know that the Morse index is distinct from $0$ and $n$. \begin{figure}[h] \begin{center} \resizebox{7cm}{4.4cm}{\includegraphics[48,473][497,768]{approx1.eps}} \end{center} \end{figure} Because of the density of the group of translations $H\subset \R$ one may find $g\in H$ such that the real number $F(g\tilde x)=F(\tilde x)+g=a+g$ lies in the interval $(c-\epsilon,c+\epsilon)$. Then we obtain \begin{eqnarray} c\, \in \, (a+g-\epsilon,\, a+g+\epsilon)\, \subset\, g+F(\tilde U)\, =\, F(g\tilde U). \end{eqnarray} Hence we see that the sets $F^{-1}(c)$ and $g\tilde U$ have a nonempty intersection. Therefore the neighborhood $U=\pi(g\tilde U)$ intersects the leaf $L=\pi(F^{-1}(c))$ as claimed. An obvious modification of the above argument proves a slightly more precise statement: {\it Given a point $x\in M$ and a leaf $L\subset M$ of the singular foliation $\omega=0$, there exist two sequences of points $x_k\in L$ and $y_k\in L$ such that \begin{eqnarray} x_k\to x\quad \mbox{and}\quad y_k\to x, \end{eqnarray} and, moreover, \begin{eqnarray}\label{last} \int_x^{x_k} \omega\, >0,\quad \mbox{while}\quad \int_x^{y_k}\omega \, <0. \end{eqnarray} The integrals in (\ref{last}) are calculated along an arbitrary path lying in a small neighborhood of $x$.} This can also be expressed by saying that the leaf $L$ approaches $x$ from both the positive and the negative sides. \section{Modification}\label{modification} Our next goal is to replace $\omega$ by a Morse closed 1-form $\omega'$ which has the property that all its zeros lie on the same singular leaf of the singular foliation $\omega'=0$. In this section we assume that $n=\dim M>2$. Let $\omega$ be a weakly complete Morse closed 1-form in class $\xi$ where $\rk(\xi)>1$. Let $p_1, \dots, p_m\in M$ be the zeros of $\omega$. For each $p_j$ choose a small neighborhood $ U_j\ni p_j$ and local coordinates $x_1, \dots, x_n$ in $U_j$ such that $x_i(p_j)=0$ for $i=1, \dots, n$ and \begin{eqnarray} \label{morse}\quad\omega|_{U_j} =df_j, \quad \mbox{where}\quad f_j= -x_1^2-\dots - x_{m_j}^2+x_{m_j+1}^2+\dots+x_n^2. \end{eqnarray} Here $m_j$ denotes the Morse index of $p_j$. We assume that the ball $\sum_{i=1}^n x_i^2\leq 1$ is contained in $U_j$ and that $U_j\cap U_{j'}=\emptyset$ for $j\not=j'$. Denote by $W_j$ the open ball $\sum_{i=1}^n x_i^2< 1$. Let $\phi:[0,1]\to [0,1]$ be a smooth function with the following properties: %\begin{enumerate} %\item[(a)] $\phi\equiv 0$ on $[3/4, 1]$; \item[(b)] $\phi\equiv \epsilon>0$ on %$[0,1/2]$; \item[(c)] $-1<\phi'\leq 0$. %\end{enumerate} (a) $\phi\equiv 0$ on $[3/4, 1]$; (b) $\phi\equiv \epsilon>0$ on $[0,1/2]$; (c) $-1<\phi'\leq 0$. \begin{figure}[h] \begin{center} \resizebox{5.9cm}{2.7cm}{\includegraphics[92,456][454,662]{phi.eps}} \end{center} \end{figure} Such a function exists assuming that $\epsilon>0$ is small enough. (a), (b), (c) imply that \begin{eqnarray}\label{stam} \phi'(r)>-2r, \quad\mbox{for}\quad r>0.\end{eqnarray} We replace the closed 1-form $\omega$ by \begin{eqnarray}\label{form} \omega'=\omega - \sum_{j=1}^m \mu_j \cdot dg_j\end{eqnarray} where $g_j: M\to \R$ is a smooth function with support in $U_j$. In the coordinates $x_1, \dots, x_n$ of $U_j$ (see above) the function $g_j$ is given by $g_j(x) = \phi(||x||).$ The parameters $\mu_j\in [-1,1]$ appearing in (\ref{form}) are specified later. One has $\omega\equiv \omega'$ on $M-\cup_j U_j$ and near the zeros of $\omega$. Let us show that $\omega'$ has no additional zeros. We have $\omega'|_{U_j}=d(f_j-\mu_j g_j)$ (where $f_j$ is defined in (\ref{morse})) and \begin{eqnarray} \frac{\partial}{\partial x_i}(f_j-\mu_jg_j) = \pm 2x_i -\mu_j \phi'(||x||)\frac{x_i}{||x||} \end{eqnarray} If this partial derivative vanishes and $x_i\not=0$ then $\phi'(r) =\pm 2r\mu_j^{-1}$ which may happen only for $r=||x||=0$ according to (\ref{stam}). We now show how to choose the parameters $\mu_j$ so that the closed 1-form $\omega'$ given by (\ref{form}) has a unique singular leaf. Let $L$ be a fixed nonsingular leaf of $\omega=0$. Since $L$ is dense in $M$ (see \S \ref{density}) for any $j=1, \dots, m$ the intersection $L\cap U_j$ contains infinitely many connected components approaching $p_j$ from below and from above and the function $f_j$ is constant on each of them. \begin{figure}[h] \begin{center} \resizebox{4.0cm}{3.7cm}{\includegraphics[104,347][489,697]{leaves.eps}} \end{center} \end{figure} We say that a subset $T_c\subset L\cap W_j$ is a \em level set\em\, if $T_c=f_j^{-1}(c)\cap W_j$ for some $c\in \R$. Note that $f_j(p_j)=0$. The level set $c=0$ contains the zero $p_j$; it is homeomorphic to the cone over the product $S^{m_j-1}\times S^{n-m_j-1}$. Each level set $T_c$ with $c<0$ is diffeomorphic to $S^{m_j-1}\times D^{n-m_j}$ and each level set $T_c$ with $c>0$ is diffeomorphic to $D^{m_j}\times S^{n-m_j-1}$. Recall that $m_j$ denotes the Morse index of $p_j$. Let $\V_j=f_j(L\cap W_j)\, \subset \R$ denote the set of values of $f_j$ on different level sets belonging to the leaf $L$. The zero $0$ does not lie in $\V_j$ since we assume that the leaf $L$ is nonsingular. However, according to the result proven in \S \ref{density}, the zero $0\in \R$ is a limit point of $\V_j$ and, moreover, the closure of either of the sets $\V_j\cap (0,\infty)$ and $\V_j\cap (-\infty, 0)$ contains $0\in \R$. For the modification $\omega'$ (given by (\ref{form})) one has $\omega'|_{U_j}=dh_j$ where $h_j=f_j-\mu_jg_j$. The level sets $T'_c$ for $h_j$ are defined as $h_j^{-1}(c)\cap W_j$. Clearly $T'_c$ is given by the equation $$f_j(x)=\mu_j\phi(||x||) +c, \quad x\in W_j.$$ Hence for $||x||\geq 3/4$ this is the same as $T_c$; for $||x||\leq 1/2$ the level set $T'_c$ coincides with $T_{c+\mu_j\epsilon}$. In the ring $1/2\leq ||x||\leq 3/4$ the level set $T'_c$ is homeomorphic to a cylinder. The following figure illustrates the distinction between the level sets $T_c$ and $T'_c$ in the case $\mu_j>0$. \begin{figure}[h] \begin{center} \resizebox{5.8cm}{5.5cm}{\includegraphics[94,326][461,698]{leaves3.eps}} \end{center} \end{figure} Examine the changes which undergoes the leaf $L$ when we replace $\omega$ by $\omega'$. Here we view $L$ with the {\it leaf topology}; it is the topology induced on $L$ from the covering $\tilde M$ using an arbitrary lift $L\to \tilde M$. First, let us assume that: (1) the Morse index $m_j$ satisfies $m_j0$ is positive; (3) the number $-\epsilon\mu_j$ lies in the set $\V_j$. Then the complement $$L-\underset{-\epsilon\mu_j2$ we have in this case $n-m_j-11$. The case $\rk(\xi)=1$ is covered by Theorem 2.1 from \cite{Fa1}. The results of the preceding sections allow to complete the proof of Theorem \ref{thm1} in the case $n=\dim M>2$. Indeed, we showed in \S \ref{modification} how to construct a Morse closed 1-form $\omega'$ lying in the prescribed cohomology class $\xi$ such that all zeros of $\omega'$ are Morse and belong to the same singular leaf $L'$ of the singular foliation $\omega'=0$. Now we may apply the colliding technique of F. Takens \cite{Ta}, pages 203--206. Namely, we may find a piecewise smooth tree $\Gamma\subset L'$ containing all the zeros of $\omega'$. Let $U\subset M$ be a small neighborhood of $\Gamma$ which is diffeomorphic to $\R^n$. We may find a continuous map $\Psi: M\to M$ with the following properties: {\it $\Psi(\Gamma)$ is a single point $p\in \Gamma$; $\Psi|_{M-\Gamma}$ is a diffeomorphism onto $M-p$; $\Psi(U)=U;$ $\Psi$ is the identity map on the complement of a small neighborhood $V\subset M$ of $\Gamma$ where the closure $\overline V$ is contained in $U$.} Consider a smooth function $f: U\to \R$ such that $df=\omega'|_U$; it exists and is unique up to a constant. The function $g=f\circ \Psi^{-1}:U\to \R$ is well-defined (since $f|_\Gamma$ is constant). $g$ is continuous by the universal property of the quotient topology. Moreover, $g$ is smooth on $M-p$. Applying Theorem 2.7 from \cite{Ta}, we see that we can replace $g$ by a smooth function $h:U\to \R$ having a single critical point at $p$ and such that $h=f$ on $U-\overline V$. Let $\omega^{''}$ be a closed 1-form on $M$ given by \begin{eqnarray} \omega^{''}|_{M-\overline V}=\omega'|_{M-\overline V}\quad \mbox{and}\quad \omega^{''}|_U=dh. \end{eqnarray} Clearly $\omega^{''}$ is a smooth closed 1-form on $M$ having no zeros in $M-\{p\}$. Moreover, $\omega^{''}$ lies in the cohomology class $\xi=[\omega']$ (since any loop in $M$ is homologous to a loop in $M-\overline{V}$). Now we prove Theorem \ref{thm1} the case $n=2$. We shall replace the construction of \S \ref{modification} (which requires $n>2$) by a direct construction. The final argument using the Takens' technique \cite{Ta} remains the same. Let $M$ be a closed surface and let $\xi\in H^1(M;\R)$ be a nonzero cohomology class. We can split $M$ into a connected sum $$M=M_1\sharp M_2\sharp \dots \sharp M_k$$ where each $M_j$ is a torus or a Klein bottle and such that the cohomology class $\xi_j=\xi|_{M_j}\in H^1(M_j;\R)$ is nonzero. Let $\omega_j$ be a closed 1-form on $M_j$ lying in the class $\xi_j$ and having no zeros; obviously such a form exists. \S 9.3.2 of \cite{Fa3} describes the construction of connected sum of closed 1-forms on surfaces. Each connecting tube contributes two zeros. In fact there are three different ways of forming the connected sum, they are denoted by A, B, C on Figure 9.8 in \cite{Fa3}. In the type C connected sum the zeros lie on the same singular leaf. Hence by using the type C connected sum operation we get a closed 1-form $\omega$ on $M$ having $2k-2$ zeros which all lie on the same singular leaf of the singular foliation $\omega=0$. The colliding argument based on the technique of Takens \cite{Ta} applies as in the case $n>2$ and produces a closed 1-form with at most one zero lying in class $\xi$. \bibliographystyle{amsalpha} \begin{thebibliography}{99} \bibitem{DNF} B. Dubrovin, S. Novikov and A. Fomenko, \textit{Modern Geometry; Methods of the homology theory}, 1984 \bibitem {EG} Y. Eliashberg, M. Gromov, \textit{ Lagrangian Intersection Theory, finite-dimensional approach}, In: \lq\lq Geometry of Differential Equations\rq\rq\, , A. Khovanskii, A. Varchenko, V. Vassiliev, editors, AMS, volume 186, 1998. \bibitem{Fa1} M. Farber, {\em Zeros of closed 1-forms, homoclinic orbits and Lusternik--Schnirelman theory}, "Topological Methods in Nonlinear Analysis", \textbf{19}(2002), 123 - 152. \bibitem{Fa2} M. Farber, \textit{Lusternik--Schnirelman Theory and Dynamics}, in: ``Lusternik - Schnirelmann Category and related topics'', Contemporary Mathematics, \textbf{316}(2002), 95 - 111. \bibitem{Fa3} M. Farber, {\em Topology of closed one-forms}, Mathematical Surveys and Monographs, Vol 108, American Mathematical Society, 2004. \bibitem{La} F. Latour, \textit{Existence de 1-formes ferm\'ees non singuli\`eres dans une classe de cohomologie de De Rham}, Publ. IHES, \textbf{80}, 1994. \bibitem{Levitt} G. Levitt, \textit{1-formes ferm\'ees singuli\`eres et groupe fondamental}, Invent. Math., \textbf{88}(1987), 635 - 667. \bibitem {N2} S. P. Novikov, \textit{The Hamiltonian formalism and a multi-valued analogue of Morse theory}, Russian Math. Surveys, \textbf{37:5}(1982), pp. 1--56. \bibitem{Sik} J.-Cl. Sikorav, \textit{Un problem de disjonction par isotopie symplectique dans un fibr\'e cotangent}, Ann. Sci. \'Ecole Norm. Sup. (4)\textbf{19}(1986), 543-552. \bibitem {Ta} F. Takens, \textit{ The minimal number of critical points of a function on a compact manifold and the Lusternik - Schnirelman category}, Invent. Math., \textbf{6}(1968), pp. 197 - 244. \end{thebibliography} \end{document}