Simple Whitney towers, half-gropes and the Arf invariant of a knot.
Rob Schneiderman
A geometric characterization of the Arf invariant of a knot in the 3--sphere is given in terms of two kinds of 4--dimensional bordisms, {\em half-gropes} and {\em Whitney towers}. These types of bordisms have associated complexities {\em class} and {\em order} which filter the condition of bordism by an embedded annulus, i.e. knot concordance, and it is shown constructively that the Arf invariant is exactly the obstruction to cobording pairs of knots by half-gropes and Whitney towers of arbitrarily high class and order. This illustrates geometrically how, in the setting of knot concordance, the Vassiliev (isotopy) invariants ``collapse'' to the Arf invariant.
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\begin{document}

\title{Simple Whitney towers, half-gropes and the Arf invariant of a knot.}
\author{Rob Schneiderman}
\address{Courant Institute of Mathematical Sciences\\
New York University\\
251 Mercer Street\\New York NY 10012-1185 USA}
\email{schneiderman@courant.nyu.edu}



\begin{abstract}

A geometric characterization of the Arf invariant of a knot in the 3--sphere is given in terms of
two kinds of 4--dimensional bordisms, {\em half-gropes} and {\em Whitney towers}. These types of
bordisms have associated complexities {\em class} and {\em order} which filter the condition of
bordism by an embedded annulus, i.e. knot concordance, and it is shown constructively that the Arf
invariant is exactly the obstruction to cobording pairs of knots by half-gropes and Whitney towers
of arbitrarily high class and order. This illustrates geometrically how, in the setting of knot
concordance, the Vassiliev (isotopy) invariants ``collapse'' to the Arf invariant.
\end{abstract}

\maketitle


\section{Introduction}



This paper gives a geometric characterization of the Arf invariant of a knotted circle in the
3--sphere that is related to recent developments in knot theory. Conant and Teichner have shown
(\cite{CT1},\cite{CT2}) that the Vassiliev filtration (\cite{B}) on isotopy classes of knots
corresponds to a geometric equivalence relation called {\em 3--dimensional capped grope-cobordism}
and that this equivalence relation is generated by certain simplified {\em half-gropes} in $S^3$.
Gropes are built by gluing together embedded surfaces and in this setting the Vassiliev degree
corresponds to a measure of grope complexity called {\em class}, which counts the layers of
attached surfaces. The Arf invariant $\Arf(k)\in\Z_2$ of a knot $k\subset S^3$ is the mod 2
reduction of the lowest degree nontrivial Vassiliev invariant (the degree two coefficient of the
Conway polynomial of $k$) and a result of Ng (\cite{Ng}) says that $\Arf(k)$ is the only finite
type invariant of $k$ up to {\em concordance}, that is, up to bordism of $k$ by an embedded annulus
in the product $S^3\times I$ of the 3--sphere with an interval. Gropes have been extensively
studied in 4--dimensional topology (e.g. \cite{FQ}, \cite{FT}, \cite{Kr}, \cite{KQ}, \cite{KrT})
and are closely related to {\em Whitney towers} which measure the failure of the Whitney move in
terms of intersections among higher order Whitney disks (\cite{COT}, \cite{S1}, \cite{ST1},
\cite{ST2}).

The following theorem illustrates geometrically how, in the setting of knot concordance, the
Vassiliev isotopy invariants ``collapse'' to the Arf invariant and the failure of the Whitney move
can be ``pushed out'' to arbitrarily high order Whitney disks:
\begin{thm}\label{arf-thm}
For knots $k_0$ and $k_1$ in $S^3$ the following are equivalent:
\begin{enumerate}


\item $\mathrm{Arf}(k_0)=\mathrm{Arf}(k_1)$,

\item $k_0$ and $k_1$ cobound a properly embedded class $n$ half-grope in $S^3\times I$ for all $n\in\N$,

\item  $k_0$ and $k_1$ cobound a properly immersed annulus in $S^3\times I$ admitting
an order $n$ Whitney tower for all $n\in\N$.

\end{enumerate}

\end{thm}
Definitions of Whitney towers, $\Arf(k)$ and half-gropes will be given in sections
\ref{sec:w-tower}, \ref{sec:Arf} and \ref{sec:half-grope-simple-tower}, respectively.
Theorem~\ref{arf-thm} will be proved constructively by exploiting the flexibility of the Whitney towers
in (iii). The infinite cyclic Vassiliev (isotopy) invariant which lifts the Arf invariant can be
interpreted as the obstruction to ``pushing down'' this construction into the 3--sphere.

\subsubsection*{Remark} The equivalence of (i) and (ii) also follows from results in \cite{CT1},
\cite{CT2} and \cite{COT} (Proposition 3.8 in \cite{CT2}).
\vspace{.07 in}

It should be mentioned that slight variations of the bordism equivalence relations of {\em class
$n$ Grope concordance} and {\em order $n$ Whitney concordance} suggested by (ii) and (iii) are
highly nontrivial, for instance when using ``height'' instead of class and order to measure
complexity (\cite{COT}) or when working with links rather than knots (\cite{ST2}).



\subsection*{Simple Whitney towers} The essential arguments in the proof of Theorem~\ref{arf-thm} 
are contained in two lemmas. The
first describes a close relationship between half-gropes and certain {\em simple} Whitney towers
(\ref{simple-w-tower-subsec}) both of which are geometric analogues of simple (right- or
left-normed) commutators in a group (\cite{MKS}).


\begin{lem}\label{Grope-tower-lem}

Let $l$ be a link in the boundary of a simply connected 4--manifold. Then $l$ is the boundary of
the bottom stage of a properly embedded half-grope of class $n$ if and only if $l$ is the boundary
of a properly immersed planar surface which admits a simple Whitney tower of order $n-1$.

\end{lem}

\subsubsection*{Remark} A much more general relation between class $n$ gropes and order $n-1$ Whitney towers in (not
necessarily simply connected) 4--manifolds is described in \cite{S1}.
\vspace{.07 in}

Whitney towers are of interest in their own right, in part because an order $(n-1)$ Whitney tower
comes equipped with an $n$th order geometric intersection obstruction to the existence of an $n$th
order tower which is related to Milnor's ($\overline{\mu}$) link invariants and the reduced
Kontsevich integral (\cite{ST2}). The next lemma illustrates how in the present setting (of knots
in a simply connected manifold) the obstruction theory collapses after order 2.

\begin{lem}\label{w-tower-lem}

A properly immersed connected surface in a simply connected 4--manifold admitting an order 2
Whitney tower admits an order $n$ simple Whitney tower for all $n$.

\end{lem}

The connectivity condition on the properly immersed surface in Lemma~\ref{w-tower-lem} is crucial.
For instance, in the setting of {\em link} concordance, the first non-vanishing Milnor invariants
of length $n+1$ give obstructions to building a Whitney tower of order $n$ on a collection of
immersed disks in the 4--ball bounded by the link components in $S^3$ (\cite{ST2}).



\subsection*{Outline} Whitney towers are defined in Section~\ref{sec:w-tower}, which also describes
the basic geometric manipulations of immersed surfaces in 4--manifolds that will be used
throughout. The Arf invariant is defined in Section~\ref{sec:Arf}. Lemma~\ref{Grope-tower-lem} is
proved in Section~\ref{sec:half-grope-simple-tower}, which also contains definitions of half-gropes
and simple Whitney towers. A proof of Lemma~\ref{w-tower-lem} is given in
Section~\ref{sec:w-tower-lem-proof} and the proof of Theorem~\ref{arf-thm} is assembled in
Section~\ref{sec:arf-thm-proof}. All manifolds are assumed smooth and oriented.


%\tableofcontents


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\section{Whitney towers}\label{sec:w-tower}
Whitney towers are introduced in this section, along with some fundamental techniques from the theory
of immersed surfaces in 4--manifolds. More information about Whitney towers can be found in
\cite{COT}, \cite{CST}, \cite{ST1} and \cite{ST2}. For more details on surfaces in 4--manifolds
the reader is referred to \cite{FQ}.

It will be convenient to illustrate local sheets of surfaces in 4--space by picturing 3--dimensional
slices in which a surface may appear either as a 2-disk or as an arc which extends into neighboring
slices; surfaces may also appear as a ``movie of arcs'' in a sequence of 3--dimensional slices
(Figure~\ref{int-point-fig}).

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.75]{int-point-fig.eps}}
        \caption{A generic transverse intersection point $p$
         between surfaces $A$ and $B$ in a 4--manifold.}
        \label{int-point-fig}

\end{figure}





\subsection{Whitney disks}\label{w-disk-subsec}
In a simply connected 4--manifold $X$, a pair of intersection points between oriented connected
surfaces $A$ and $B$ are a called a {\em cancelling pair} if they have opposite signs (via the
usual sign convention that compares the orientations of the surfaces at an intersection point with
the orientation of the ambient manifold). Such a cancelling pair $p$ and $q$ in $A\cap B$ can be
{\em paired by a Whitney disk} as follows: The union of an arc $\alpha$ from $p$ to $q$ in $A$ and
an arc $\beta$ from $q$ to $p$ in $B$ forms a loop in $X$ which bounds an immersed 2--disk $W$
meeting $A$ and $B$ along $\partial W$ in the standard way.  Such a $W$ is a {\em Whitney disk}
pairing $p$ and $q$. (An embedded Whitney disk is shown in Figure~\ref{W-disk-fig}.)

The normal disk bundle $\nu W$ of $W$ in $X$ is a trivial $D^2$--bundle since $W$ is contractible.
The restriction of $\nu W$ to $\partial W$ has a canonical 1--dimensional sub-bundle
$\nu_{\partial}$ which restricts along $\alpha$ to the normal bundle of $\alpha$ in $A$ and
restricts along $\beta$ to the orthogonal complement (in $\nu W$) of the normal bundle of $\beta$
in $B$. Since $p$ and $q$ have opposite signs, $\nu_{\partial}$ is a trivial $I$-bundle over
$\partial W$. The obstruction to extending a non-vanishing section of $\nu_{\partial}$ to a
non-vanishing section of $\nu W$ is an element of $\pi_1 SO(2)\cong \Z$. If this obstruction
vanishes then $W$ is said to be (correctly) {\em framed}.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.50]{W-disk-fig.eps}}
        \caption{A Whitney disk.}
         \label{W-disk-fig}

\end{figure}


\subsection{Definition of a Whitney tower}\label{subsec:w-tower}
A Whitney disk can be used to eliminate its cancelling pair of intersection points via a {\em
Whitney move} (a motion of one of the sheets guided by the Whitney disk), as introduced by Whitney
for higher dimensional manifolds immersed in Euclidean space (\cite{Wh}). In the present
4--dimensional setting, a Whitney move will create new intersections if the interior of the Whitney
disk has any ``higher order'' intersections with sheets of surfaces or Whitney disks. (In
dimensions greater than 4, such higher order intersection points can eliminated by general
position.) The following notion of a {\em Whitney tower} filters the condition that a properly
immersed surface in a 4--manifold (boundary embedded in boundary and interior immersed in interior)
is homotopic (rel boundary) to an embedding.
\begin{defi}\label{w-tower-defi}

A properly immersed surface in a 4--manifold is by definition a {\em surface of order 0}. The {\em
order of a (transverse) intersection point} between a surface of order $n$ and a surface of order
$m$ is $n+m+1$.

The {\em order of a Whitney disk} pairing intersection points of
order $n$ is $n$.

A {\em Whitney tower of order 0} is an order 0 surface in a 4--manifold. A {\em Whitney tower of
order n} for $n\geq 1$ is a Whitney tower $T$ of order $n-1$ together with Whitney disks pairing
all order $n$ intersection points of $T$. Thus, in an order $n$ Whitney tower all intersection
points of order less than or equal to $n$ occur in cancelling pairs with respect to (arbitrarily)
chosen orientations of all Whitney disks (see Figure~\ref{W-tower-part-fig}).

The Whitney disks in $T$ are required to be framed and have
disjointly embedded boundaries.


If $A$ is a properly immersed surface in a 4--manifold and there exists an order $n$ Whitney tower
containing $A$ as its order 0 surface, then $A$ is said to {\em admit} an order $n$ Whitney tower.

\end{defi}
If a Whitney tower of order $n$ has no intersection points of order greater than $n$, then the
Whitney disks can be used to guide a regular homotopy (rel boundary) of the order $0$ surface to an
embedding.

\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.75]{W-tower-part-fig.eps}}
         \caption{Part of an order $n$ Whitney tower.
         The Whitney disk $V$ has an unpaired intersection
         point which must be of order greater than $n$.}
         \label{W-tower-part-fig}

\end{figure}


\subsection{Modifying Whitney
disks}\label{modifying-w-disk-subsec} There are several moves that allow for controlled
modification of Whitney towers. Since the moves are supported in a neighborhood of an arc or a
point they commute with each other and can be iterated disjointly arbitrarily many times.


\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.70]{cusp-fig.eps}}
         \caption{A local cusp homotopy.}
         \label{cusp-fig}

\end{figure}


The first two {\em twisting moves} change the framing of a Whitney disk.
\subsubsection{Interior twisting.}\label{interior-twist-subsec}
Introducing an self-intersection in the interior $\int W$ of $W$ by a {\em cusp homotopy} (see
Figure~\ref{cusp-fig}) changes the framing by $\pm2$ as can be seen by counting the intersections
between a local kink and its parallel push off.
\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.65]{boundary-twist-fig.eps}}
        \caption{Boundary twisting a Whitney disk.}
         \label{boundary-twist-fig}

\end{figure}

\subsubsection{Boundary twisting.}\label{boundary-twist-subsec}
Introducing a {\em boundary twist} by changing a collar of $W$ near a point in $\partial W$ (as in
Figure~\ref{boundary-twist-fig}) changes the framing by $\pm 1$ and creates an intersection between
$\int W$ and the sheet containing $\partial W$.


The next two moves do not affect framing but can be used to make
Whitney disks disjointly embedded.
\subsubsection{Boundary push-off.}\label{boundary-push}

Intersections or self-intersections between boundaries of Whitney disks can always be eliminated by
a regular homotopy in a collar at the cost of creating an intersection between the interior of a
Whitney disk and a surface sheet (Figure~\ref{boundary-push-fig}).
\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.55]{boundary-push-fig.eps}}
        \caption{Boundary push-off.}
         \label{boundary-push-fig}

\end{figure}

\subsubsection{Pushing down an intersection point.}\label{push-down}
\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.55]{push-down-fig.eps}}
         \caption{Pushing down an intersection between the interior of
         a Whitney disk and a sheet of a surface.}
         \label{push-down-fig}

\end{figure}
An intersection point between $\int W$ and any surface $S$ can be removed by a {\em finger move} on
$S$, a homotopy of $S$ supported in a neighborhood of an arc, in this case an arc in $W$ from the
intersection to a point in $\partial W$ (Figure~\ref{push-down-fig}). Such a finger move is called
``pushing $S$ down'' into either sheet and creates two new cancelling pairs of intersection points
between $S$ and the sheet. This move can also be used to remove interior self-intersections of $W$.
\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.55]{push-down-w-disk-fig.eps}}
         \caption{A Whitney disk $V$ for the cancelling pair created by pushing down.}
         \label{push-down-w-disk-fig}

\end{figure}
Note that the newly created cancelling pair can be paired by an embedded Whitney disk $V$ whose
boundary is disjoint from $W$ by applying the boundary push-off move to the obvious small embedded
Whitney disk near the cancelling pair. The interior of $V$ has a single intersection point with the
sheet that was not ``pushed into'' as illustrated in Figure~\ref{push-down-w-disk-fig}.


\subsection{Order 1 towers for knots}
Applying the moves of \ref{modifying-w-disk-subsec} yields the following lemma, which will be used
in the definition of the Arf invariant in the next section.
\begin{lem}\label{1-tower-lem}
Any knot $k$ in $S^3$ bounds a properly immersed 2-disk in $S^3\times I$ admitting a Whitney tower
of order 1.
\end{lem}
\begin{proof}
A finite number of crossing changes, leading from $k$ to the unknot, describes a properly immersed
2-disk $D$ in $S^3\times I$ (with the unknot capped off by an embedded disk). Fixing orientations,
the signs of the self-intersections of $D$ correspond to the signs of the crossing changes and,
after introducing trivial crossing changes (if necessary), the self-intersections of $D$ can be
made to occur in cancelling pairs which are paired by Whitney disks as in
Section~\ref{w-disk-subsec}. By applying boundary twists (\ref{boundary-twist-subsec}) and boundary
push-off (\ref{boundary-push}) as needed, it can be arranged that the Whitney disks are framed with
disjointly embedded boundaries.
\end{proof}




\section{The Arf invariant}\label{sec:Arf}
The Arf invariant (or Kervaire invariant) of a knot in $S^3$ was defined by Robertello in
\cite{Ro}, based on Kervaire and Milnor's generalization (\cite{KM}) of Rochlin's Theorem. The
following geometric definition, which we have translated into the language of Whitney towers, is
due to Matsumoto (\cite{Ma}) using Freedman and Kirby's geometric proof of Rochlin's Theorem
(\cite{FK}).


\begin{defi}\label{arf-defi}
For a knot $k$ in $S^3$, let $D$ be any properly immersed 2-disk immersed in $S^3\times I$
admitting a Whitney tower $T$ of order 1, with $k=\partial D\subset S^3\times \{0\}$. Define the
{\em Arf invariant} of $k$, $\Arf(k)\in\Z_2$, to be the number (modulo 2) of second order
intersection points in $T$.
\end{defi}
By Lemma~\ref{1-tower-lem}, such a $T$ always exists. A direct combinatorial proof that this
definition of $\Arf(k)$ is well-defined can be found in $10.8$ of \cite{FQ}.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{trefoil-arf-fig.eps}}
         \caption{The trefoil has nontrivial Arf invariant.}
         \label{trefoil-arf-fig}

\end{figure}
\subsection{The trefoil.}
That the trefoil knot has nontrivial Arf invariant can be seen in Figure~\ref{trefoil-arf-fig},
where changing the crossings labelled $p^+$ and $p^-$ creates a cancelling pair of first order
intersections in a null-homotopy $D$. This cancelling pair has a framed embedded Whitney disk $W$
which intersects $D$ in a single second order intersection point.



\section{Half-gropes and simple Whitney towers}\label{sec:half-grope-simple-tower}
In this section a proof of Lemma~\ref{Grope-tower-lem} is given after first defining half-gropes
and simple Whitney towers, two geometric analogues of a simple (right- or left-normed) commutator
of elements in a group (\cite{MKS}).

\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.45]{half-grope-fig.eps}}
         \caption{A half-grope of class 4.}
         \label{half-grope-fig}

\end{figure}



\subsection{Half-gropes}\label{half-grope-subsec}
In general, gropes are 2-complexes consisting of surfaces joined along certain essential curves
(see e.g. \cite{CT1}, \cite{FQ}). Requiring that the curves form a ``half-basis'' yields the
``half-gropes'':
\begin{defi}\label{grope-defi}

A {\em half-grope of class $2$} is a compact connected orientable surface $A$ with a single
boundary circle. To form a {\em half-grope of class $n>2$}, start with an orientable surface $A$
with a single boundary circle and choose a symplectic basis $\{a_i,b_i\}$, that is, the $a_i$ and
$b_i$ are embedded curves which represent a basis for $H_1(A)$ and the only intersections among
them occur when $a_i$ meets $b_i$ in a single point. Now attach half-gropes of class $n-1$ along
their boundary circles to a $1/2$-symplectic basis for $A$, i.e., a maximal pairwise disjoint
subset of $\{a_i,b_i\}$, for instance $\{a_i\}$.

The surface $A$ is called the {\em bottom stage} of the half-grope and the boundary circle of $A$
is the {\em boundary} of the half-grope. Half-gropes with more than one boundary component are
formed by removing disks from the bottom stage of a half-grope. The attached punctured surfaces are
also referred to as {\em (higher) stages}. The basis curves that do not have higher stages attached
to them are the {\em tips} of the half-grope.


A half-grope $H$ is {\em properly embedded} in a 4--manifold $X$ if the boundary of $H$ is embedded
in $\partial X$ and the rest of $H$ is embedded in $\int X$. It is also required that $H$ satisfy
the following {\em normal framing condition}: A regular neighborhood of $H$ in $X$ must factor as a
standard embedding of $H$ into 3--space followed by taking the product with an interval.

\end{defi}


\subsection{Simple Whitney towers}\label{simple-w-tower-subsec}
A Whitney tower is {\em simple} if all of its Whitney disks have disjointly embedded interiors.
Thus, every intersection point of order $m$ in a simple Whitney tower is an intersection between a
surface of order $0$ and a surface of order $(m-1)$.

\subsection{Proof of Lemma~\ref{Grope-tower-lem}}\label{Grope-tower-lem-proof}

\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.5]{surgered-grope-fig.eps}}
         \caption{Intersections created by surgering caps can be paired by Whitney disks
         constructed from surgered higher stages of the half-grope. Surgering a cap bounded by
         the dotted circle indicating a tip of $S$ creates a Whitney disk.}
         \label{surgered-grope-fig}
\end{figure}


Let $H$ be a half-grope of class $n$ properly embedded in $X$ and bounded by $l$. Since $X$ is
simply connected, the tips of $H$ bound immersed 2-disks called {\em caps} and the plan is to
create the desired Whitney tower by surgering the caps. Each cap has a normal framing determined by
pushing its boundary along $H$ and after boundary twisting the cap (just as in
\ref{boundary-twist-subsec}) this framing can be made to vanish. We may also arrange, by repeatedly
pushing down intersections as for Whitney disks (\ref{push-down}), that the caps of $H$ are
disjointly embedded (except for the single boundary point intersections between dual caps in the
top stages) with interiors disjoint from all stages of $H$ except for perhaps the bottom stage.

Assume first that the bottom stage intersects the interior of each cap in at most a single point.
Let $A$ be the result of surgering those caps attached to the bottom stage surface of $H$. Then $A$
is a properly immersed planar surface with self-intersection points coming in cancelling pairs
which were created by surgering a cap whose interior intersected the bottom stage surface. Such a
cancelling pair has an embedded (first order) Whitney disk $W$ which is the union of a small band
and the result of surgering the caps on the next stage surface $S$, which was attached along the
dual curve to the boundary of the cap (Figure~\ref{surgered-grope-fig}). The framing condition (in
Definition~\ref{grope-defi}) on the normal bundle of $H$ in $X$ ensures that $W$ is correctly
framed. The only possible intersections between the interior of $W$ and anything else are
intersections with $A$ coming from intersections between $A$ and the surgered caps on $S$, hence
occur in cancelling pairs with an embedded second order Whitney disk gotten by similarly surgering
the next surface stage. This construction terminates at the top $(n-1)$th stage surfaces, where
only a half-basis of caps are surgered to make the order $(n-2)$ Whitney disks and the dual caps
(together with bands) form the order $(n-1)$ Whitney disks yielding the desired simple Whitney
tower.

The above assumption that each cap has at most a single interior intersection with the bottom
surface can always be arranged by Krushkal's grope splitting technique (\cite{Kr}); alternatively,
the above construction can still be carried out for ``near-by'' cancelling pairs created by
surgering caps containing multiple interior intersections by using parallel disjoint copies
(guaranteed by the normal framing condition) of the higher surfaces stages of $H$ to build the
higher order Whitney disks. This completes one direction of the proof of
Lemma~\ref{Grope-tower-lem}.

\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.55]{w-grope-fig.eps}}
         \caption{A simple Whitney tower of order
         $n-1$ yields a half-grope of class $n$ by ``tubing along $A$''.}
         \label{w-grope-fig}

\end{figure}

For the other direction, let $T$ be a simple order $n-1$ Whitney tower on a properly immersed
planar surface $A$ in $X$ bounded by $l$. Since $T$ is simple, each of its Whitney disks has (at
least one) boundary arc lying in $A$ (the order 1 Whitney disks have both boundary arcs on $A$).
The desired half-grope $H$ of class $n$ is constructed by ``tubing the Whitney disks of $T$ along
$A$'' as illustrated in Figure~\ref{w-grope-fig}: More specifically, let $W$ be any order $m$
Whitney disk ($1\leq m\leq n-1$) for a pair of cancelling order $m$ intersections between $A$ and
an order $m-1$ Whitney disk $V$ (if $m=1$, then $V$ is just the order zero surface $A$). Denote by
$\partial_AW$ the part of the boundary of $W$ that lies in $A$ (if $m=1$, choose a boundary arc of
$\partial W$). Using the boundary annulus of the normal disk bundle to $A$ in $X$ restricted to
$\partial_AW$ to perform 0-surgery on $V$ eliminates the cancelling pair of intersections between
$A$ and $V$. If $m=n-1$, then $W$ is discarded; if $1\leq m<n-1$, then $W$ (minus a small collar
near $\partial_AW$) becomes an ($m+1$)th stage surface of $H$ by 0-surgering the interior of $W$ to
eliminate any intersections with $A$.  Applying this construction to all the Whitney disks of $T$
yields $H$, with the bottom stage surface of $H$ consisting of 0-surgery (one for each first order
Whitney disk) on $A$ and each $m$th stage surface 0-surgery on an $(m-1)$th order Whitney disk. The
normal framing condition on $H$ is satisfied since all the Whitney disks of $T$ were framed.
\hfill$\square$





\section{Proof of Lemma~\ref{w-tower-lem}}\label{sec:w-tower-lem-proof}
The idea of the proof of Lemma~\ref{w-tower-lem} is that, for an order $(n-1)$ simple Whitney tower
whose order $0$ surface is {\em connected}, any order $n$ intersection point can be cancelled by
boundary twisting its order $(n-1)$ Whitney disk into the order 0 surface (in a simply connected
4--manifold). The framing on the Whitney disk can then be recovered by boundary twisting along the
other boundary arc of the Whitney disk, which only creates an intersection point of order greater
than $n$, provided $n\geq 2$.


\begin{proof}
Let $A$ be a properly immersed connected surface in a simply connected 4--manifold $X$ admitting an
order 2 Whitney tower $T$. A {\em simple} order 2 Whitney tower can be constructed from $T$ by
pushing down any intersections among its Whitney disks into $A$: Pushing down (third order)
intersections among the first order Whitney disks creates cancelling pairs of second order
intersections which can be equipped with disjointly embedded second order Whitney disks having a
single interior intersection with $A$ (Figure~\ref{push-down-w-disk-fig}). Pushing down
intersections among second order disks and between second and first order disks creates cancelling
pairs of third order intersections between the second order disks and $A$.


Now assume inductively that $A$ admits a simple Whitney tower
$T_n$ of order $n\geq 2$. Since the interiors of all Whitney disks
in $T_n$ are disjointly embedded, the only possible unpaired
intersection points are $(n+1)$th order intersections between
$n$th order Whitney disks and $A$.

Let $p$ be such an intersection point between an $n$th order Whitney disk $W$ and $A$. Since $T_n$
is simple, $W$ pairs intersections between $A$ and an $(n-1)$th order Whitney disk (recall
$n\geq2$). By performing a boundary twist around the arc of $\partial W$ that lies on $A$, we can
create an intersection point $q\in \int W\cap A $ of opposite sign as $p$. Since $A$ is connected
(and the 4--manifold is simply-connected), $p$ and $q$ can be paired by an $(n+1)$th order Whitney
disk.

To restore the framing of $W$, perform a boundary twist along the (other) arc of $\partial W$ that
lies on the $(n-1)$th order Whitney disk, creating a $2n$th order intersection point between $W$
and the $(n-1)$th order Whitney disk. This $2n$th order intersection point can be eliminated by
repeatedly pushing $\int W$ down into Whitney disks of lower order as in
Figure~\ref{multi-pushdown-fig} until eventually reaching $A$, where $2^{(n-2)}$ cancelling pairs
of order $(n+1)$ intersections between $\int W$ and $A$ will be created.

Since this modification of $T_n$ takes place in a neighborhood of a 1-complex, it may be repeated
(in disjoint neighborhoods) until all order $(n+1)$ intersections are paired by Whitney disks. The
boundaries of the $(n+1)$th order Whitney disks can be made disjointly embedded (and disjoint from
all other Whitney disk boundaries) by applying boundary push-off moves (\ref{boundary-push}).
Finally, intersections between any Whitney disk and the $(n+1)$th order Whitney disks can be
eliminated by repeatedly pushing the $(n+1)$th order Whitney disks down until they only intersect
$A$, yielding a simple order $(n+1)$ Whitney tower.
\end{proof}


\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.65]{multi-pushdown-fig.eps}}
         \caption{}
         \label{multi-pushdown-fig}

\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








%--------------------------------------------------------------------------------------------



\section{Proof of Theorem~\ref{arf-thm}}\label{sec:arf-thm-proof}




\subsection*{{\boldmath $\mathrm{(iii)}\Rightarrow\mathrm{(i)}$}}
Let $A$ be an annulus admitting an order $n\geq 2$ Whitney tower $T$ as in (iii). Then any 2-disk $D_0$
admitting an order 1 Whitney tower $T^0$ in $S^3\times I$ and bounded by $k_0$ can be extended by
$A$ to a 2-disk $D_1=A\cup D_0$ in $S^3\times I$ admitting an order 1 Whitney tower $T^1=T\cup T^0$
and bounded by $k_1$. Since $n\geq 2$, all second order intersection points in $T$ occur in
cancelling pairs, so $T^1$ has the same number (modulo 2) of order 2 intersection points as $T^0$
and $\Arf(k_1)=\Arf(k_0)$.

\subsection*{{\boldmath $\mathrm{(i)}\Rightarrow\mathrm{(iii)}$}}
Let $D_i$, $i=0,1$, be disjoint properly immersed 2-disks in $S^3\times I$, where $I=[0,1]$,
bounded by the knots $k_i \subset S^3\times\{i\}$ and admitting order 1 Whitney towers $T^i$. We
may assume that the $T^i$ are disjoint, so the $D_i$ can be tubed together by a thin embedded
annulus to get an annulus $A$ co-bounded by the $k_i$ such that $A$ admits an order 1 Whitney tower
$T$ whose Whitney disks are just the union of the Whitney disks in $T^i$. The assumption that
$\Arf(k_0)=\Arf(k_1)$ means that $T$ has an even number of second order intersection points. By
using the move illustrated in Figure~\ref{Arf-move-fig} (details in \cite{Y}, also \cite{ST1}),
which does not affect framings, it can be arranged that each Whitney disk contains an even number
of second order intersection points. After introducing an even number of (like-signed) boundary
twists (\ref{boundary-twist-subsec}) on each Whitney disk, the second order intersection points on
each Whitney disk occur in cancelling pairs (have opposite signs) admitting second order Whitney
disks. If $2m$ boundary twists were done on a first order Whitney disk $W$, then the framing of $W$
can be recovered by performing $m$ interior twists (\ref{interior-twist-subsec}), which create only
third order intersection points (self-intersections of $W$).

Having thus far constructed a second order Whitney tower, the proof of the implication
$\mathrm{(i)}\Rightarrow\mathrm{(iii)}$ is completed by Lemma~\ref{w-tower-lem}.



\subsection*{{\boldmath $\mathrm{(ii)}\Leftrightarrow \mathrm{(iii)}$}}

By Lemma~\ref{w-tower-lem}, we may assume that the Whitney tower in (iii) is simple, thus the proof
of $\mathrm{(ii)}\Leftrightarrow \mathrm{(iii)}$ follows from Lemma~\ref{Grope-tower-lem}.\hfill$\square$


\begin{figure}[ht!]
         \centerline{\includegraphics[scale=.65]{Arf-move-fig.eps}}
         \caption{}
         \label{Arf-move-fig}

\end{figure}


%-------------------------------------------------------------------------------






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\\ Geometry and Topology Vol. 4 (2000) 407--430.

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\bibitem{MKS} W. Magnus,  A. Karass and  D. Solitar. {\em Combinatorial group theory.}\\
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\bibitem{S1} R. Schneiderman.
{ \em Whitney towers and gropes in 4--manifolds.}\\
Preprint.

\bibitem{ST1} R. Schneiderman and P. Teichner. { \em Higher order
intersection numbers of 2-spheres in 4--manifolds. }\\ Algebraic and Geometric Topology Vol. 1
(2001) 1--29.
\bibitem{ST2} R. Schneiderman and P. Teichner.
{ \em Whitney towers, Milnor invariants and the Kontsevich integral.}\\
In preparation.





% \bibitem{T} P. Teichner. { \em Pulling apart 2-spheres in 4--space. }\\
%to appear in Topology.

%\bibitem{W}  C. T. C. Wall. { \em Surgery on Compact Manifolds.}
% \\ London Math.Soc.Monographs~1, Academic Press~1970 or Second Edition,
%edited by A. Ranicki, Math. Surveys and Monographs 69, A.M.S.

\bibitem{Wh} H. Whitney. {\em The self intersections of a smooth
$n$-manifold in $2n$-space.}\\ Annals of Math. 45 (1944) 220-246.

\bibitem{Y} M. Yamasaki. {\em Whitney's trick for three 2-dimensional
homology classes of 4--manifolds.}\\ Proc. Amer. Math. Soc. 75 (1979) 365-371.


\end{thebibliography}








\end{document}

math0310304.extracted_bib
\begin{thebibliography}{HS1}

\bibitem{B} D. Bar-Natan. { \em On the Vassiliev knot invariants.
}\\ Topology 34 (1995) 423--472.

\bibitem{COT} T. Cochran, K. Orr and P. Teichner. { \em Knot
concordance, Whitney towers and $L^2$-signatures. }\\
Annals of Math. Vol. 157 No.2 (2003) 433-519.

\bibitem{CST} J. Conant, R. Schneiderman and P. Teichner. { \em A geometric Jacobi relation in 3 and 4 dimensions.}\\
(In preparation.)

\bibitem{CT1} J. Conant and P. Teichner. { \em Grope coborbism of classical knots.}\\
(To appear in Topology) Posted at arXiv:math.GT/0012118 14 Dec 2000.


\bibitem{CT2} J. Conant and P. Teichner. { \em Grope coborbism and Feynman diagrams.}\\
(To appear in Math. Annalen.) Posted at arXiv:math.GT/0209075 6 Sep 2002.

\bibitem{FK} M. H. Freedman and R. Kirby. { \em A geometric proof of Rochlin's Theorem. }\\
Proceedings of Symposia in Pure mathematics Vol. 32 Part 2 (1978) 85--98.

\bibitem{FQ}  M. H. Freedman and F. Quinn. { \em The topology of
4--manifolds.}\\ Princeton Math. Series 39 Princeton NJ (1990).

\bibitem{FT}  M. H. Freedman and P. Teichner. { \em 4--manifold topology II:
Dwyer's filtration and surgery kernels.}\\ Invent.  Math. 122 (1995) 531--557.




\bibitem{KM} M. Kervaire and J. Milnor. { \em On 2-spheres in 4--manifolds. }\\
Proc. of Nat. Acad. Sci. U.S.A. 47 (1961) 1651--1657.



\bibitem{Kr} V. Krushkal. { \em Exponential separation in 4--manifolds. }\\
Geometry and Topology Vol. 4 (2000) 397--405.

\bibitem{KQ}  V. Krushkal and F. Quinn. { \em Subexponential groups in 4--manifold topology.}
\\ Geometry and Topology Vol. 4 (2000) 407--430.

\bibitem{KrT} V. Krushkal and P. Teichner. { \em Alexander duality, gropes and link homotopy. }\\
Geometry and Topology Vol. 1 (1997) 51--69.



\bibitem{MKS} W. Magnus,  A. Karass and  D. Solitar. {\em Combinatorial group theory.}\\
Dover Publications Inc. (1976).

\bibitem{Ma} Y. Matsumoto. { \em Secondary intersectional properties of 4--manifolds and
Whitney's trick. }\\ Proceedings of Symposia in Pure mathematics Vol. 32 Part 2 (1978) 99--107.



\bibitem{Ng} K.Y. Ng. {\em Groups of ribbon knots.}\\ Topology 37 (1996) 441-458.

\bibitem{Ro} R. A. Robertello. { \em An invariant of knot cobordism. }\\
Comm. Pure Appl. Math. Vol. 18 (1965) 543--555.

\bibitem{S1} R. Schneiderman.
{ \em Whitney towers and gropes in 4--manifolds.}\\
Preprint.

\bibitem{ST1} R. Schneiderman and P. Teichner. { \em Higher order
intersection numbers of 2-spheres in 4--manifolds. }\\ Algebraic and Geometric Topology Vol. 1
(2001) 1--29.
\bibitem{ST2} R. Schneiderman and P. Teichner.
{ \em Whitney towers, Milnor invariants and the Kontsevich integral.}\\
In preparation.





% \bibitem{T} P. Teichner. { \em Pulling apart 2-spheres in 4--space. }\\
%to appear in Topology.

%\bibitem{W}  C. T. C. Wall. { \em Surgery on Compact Manifolds.}
% \\ London Math.Soc.Monographs~1, Academic Press~1970 or Second Edition,
%edited by A. Ranicki, Math. Surveys and Monographs 69, A.M.S.

\bibitem{Wh} H. Whitney. {\em The self intersections of a smooth
$n$-manifold in $2n$-space.}\\ Annals of Math. 45 (1944) 220-246.

\bibitem{Y} M. Yamasaki. {\em Whitney's trick for three 2-dimensional
homology classes of 4--manifolds.}\\ Proc. Amer. Math. Soc. 75 (1979) 365-371.


\end{thebibliography}