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\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
$m_j<n-1$; (2) the coefficient $\mu_j>0$ is positive; (3) the number
$-\epsilon\mu_j$ lies in the set $\V_j$. Then the complement
$$L-\underset{-\epsilon\mu_j<c<0}{\bigcup_{c\in \V_j}} T_c$$
is connected and it lies in a single leaf $L'$ of the singular foliation
$\omega'=0$. We see that the new leaf $L'$ is obtained from $L$ by infinitely
many surgeries. Namely, each level set $T_c\subset L$, where $c\in \V_j$
satisfies $-\epsilon\mu_j<c<0$, is removed and replaced by a copy of
$D^{m_j}\times S^{n-m_j-1}$; besides, the set $T_c\subset L$ where
$c=-\epsilon\mu_j$, is removed and gets replaces by a cone over the product
$S^{m_j-1}\times S^{n-m_j-1}$. Hence the new leaf $L'$ contains the zero $p_j$.

Let us now show how one may modify the above construction in the case
$m_j=n-1$. Since $n>2$ we have in this case $n-m_j-1<n-2$; hence removing the
sphere $S^{n-m_j-1}$ from the leaf $L$ does not disconnect $L$. We shall assume
that the coefficient $\mu_j$ is {\it negative} and that the number
$-\epsilon\mu_j$ lies in $\V_j\subset \R$. The complement
$$L-\underset{0<c<-\epsilon\mu_j}{\bigcup_{c\in \V_j}} T_c$$
is connected and it lies in a single leaf $L'$ of the singular foliation
$\omega'=0$. Clearly, $L'$ is obtained from $L$ by removing the level sets
$T_c$ where $c\in \V_j$ satisfies $0<c<-\epsilon \mu_j$ (each such $T_c$ is
diffeomorphic to $D^{m_j}\times S^{n-m_j-1}$) and by replacing them by copies
of $S^{m_j-1}\times D^{n-m_j}$. In addition, the set $T_c\subset L$ where
$c=-\epsilon\mu_j$, is removed and is replaces by a cone over the product
$S^{m_j-1}\times S^{n-m_j-1}$.

We see that $L'$ is a leaf of the singular foliation $\omega'=0$ containing all
the zeros $p_1, \dots, p_m$.

\section{Proof of Theorem \ref{thm1}}\label{proof}

Below we assume that $\rk(\xi)>1$. 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
$\xi$.


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