Whitney towers and Gropes in 4--manifolds.
Rob Schneiderman
Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called {\em gropes}. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these {\em Whitney towers} is used to demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability, $k$-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves.
W-tower-grope.tex

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\title{Whitney towers and Gropes in 4--manifolds.}
\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{document}

\begin{abstract}
Many open problems and important theorems in low-dimensional topology have been formulated as
statements about certain 2--complexes called {\em gropes}. This paper describes a precise
correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms
of iterated `towers' of Whitney disks. The `flexibility' of these {\em Whitney towers} is used to
demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability,
$k$-cobordism and grope concordance. The key observation is that the essential structure of gropes
and Whitney towers can be described by embedded unitrivalent trees which can be controlled during
surgeries and Whitney moves.

\end{abstract}

\maketitle
%-------------------------------------------------------------------------------

\section{Introduction}
Many open problems and important theorems in low-dimensional topology, including the classification
theory of topological 4--dimensional manifolds and the study of knots and links in 3--manifolds,
have been formulated in terms of statements about smooth maps of certain 2--complexes called {\em
gropes}. As suggested by its name, a grope is built inductively from layers of surfaces which
``reach into'' a 4--manifold in an attempt to approximate an embedded 2--disk. The existence of
generically embedded 2--disks in dimensions greater than 4 allows for use of the Whitney move, a
vital part of the surgery programs which yield classification theorems for higher dimensional
manifolds and knotted submanifolds. The failure in general of the Whitney move in dimension 4 is a
defining characteristic of low-dimensional topology and this paper describes a precise
correspondence between gropes and an approximation to the Whitney move via certain ``towers'' of
iterated Whitney disks.

A Whitney move eliminates a pair of singularities between immersed surfaces in a 4--manifold but
will also create new intersections if the guiding Whitney disk contains singularities in its
interior. A {\em Whitney tower} is constructed by pairing such singularities in the interiors of
Whitney disks with ``higher order'' Whitney disks in an attempt to find embedded Whitney disks via
``higher order'' Whitney moves (Figure~\ref{W-tower5-fig} and Section~\ref{w-tower-sec}).



The {\em order} of a Whitney tower (Definition~\ref{w-tower-defi}) is determined by the number of
layers of Whitney disks added to the immersed surfaces and the {\em class} of a grope
(Figure~\ref{class4gropeA-fig} and Definition~\ref{grope-defi}) measures its complexity in terms of
the number of its layers or {\em stages} of surfaces.



In the case of gropes, all singularities are usually contained in {\em caps} which are 2--disks
that are mapped in after all stages of embedded surfaces have been attached. The caps are attached
along essential curves called {\em tips} (details in Section~\ref{grope-sec}).


The correspondence between class and order is described by the following basic version of our main
result:


\begin{thm}\label{class-order-thm}
For any collection of embedded closed curves $\gamma_i$ in the boundary of a 4--manifold $X$, the
following are equivalent:
\begin{enumerate}


\item The $\gamma_i$ bound disjoint properly embedded class $n$
gropes $g_i$ with null-homotopic tips in $X$.

\item The $\gamma_i$ bound properly immersed 2--disks $D_i$ admitting an order $(n-1)$
Whitney tower $T$ in $X$.

\end{enumerate}


\end{thm}
The proof of Theorem~\ref{class-order-thm} is given by a local construction meaning that, given
caps for the $g_i$, the $D_i$ and $T$ are constructed in a neighborhood of the capped gropes
$g^c_i$ and, given the $D_i$ contained in $T$, the gropes $g_i$ (and caps) are constructed in a
neighborhood of $T$.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{W-tower5-fig.eps}}
        \caption{Part of a Whitney tower.}
        \label{W-tower5-fig}

\end{figure}

\subsection*{Unitrivalent trees}
In fact, a much more precise correspondence between gropes and Whitney towers can
be described in terms of unitrivalent trees (\ref{grope-trees}, \ref{int-point-trees}) which
capture the essential structure of both gropes and Whitney towers. This correspondence is given in
Theorem~\ref{tower-to-grope-thm} and Theorem~\ref{grope-to-tower-thm}, which are stated and proved
in Section~\ref{sec:thms} (and directly imply Theorem~\ref{class-order-thm}). The key element used
in the proofs of these theorems and their corollaries is that the associated trees, embedded as
subsets of the gropes and Whitney towers, can be preserved and controlled during modifications such
as ambient surgery or Whitney moves.




\begin{remn}\label{rem:tau}
%\subsection{Remark}\label{rem:tau}
Besides providing an interpretation of gropes in terms of Whitney disks, the notion of Whitney
towers comes equipped with an obstruction theory that generalizes the usual (first order)
intersection theory for surfaces in 4--manifolds. The associated invariants $\tau_n$ take values in
graded groups of trees which generalize the well-known target groups of finite type concordance
invariants of links. These higher order intersection trees are enhancements of the trees discussed
in this paper and are used in \cite{ST2} to give a geometric description of Milnor's
$\mu$--invariants and the (reduced) Kontsevich integral for links. (See also \cite{ST3}
and \cite{ST4}.)
\end{remn}

\subsection*{Symmetric gropes and $\mathbf{n}$--solvability}
It should be noted that the Whitney towers described here differ from the towers that appear in the
theory of Casson handles (\cite{F}, \cite{Q}) in that the latter always involve killing the
so-called ``accessory circles'' that run through the singularities. The gropes that arise in the
context of Casson handles are certain capped {\em symmetric} gropes whose singularities are
restricted to intersections among the caps (\cite{FQ}).


The notion of Whitney tower described here {\em is} related to general symmetric gropes (without
restrictions on caps) and the recently discovered filtration of the classical knot concordance
group in terms of {\em n-solvability} by Cochran, Orr and Teichner \cite{COT}: A Whitney tower of
{\em height} $n$ (resp. $n.5$) as defined in \cite{COT} is a Whitney tower of order $2^n-2$ (resp.
$2^n+2^{(n-1)}-1$) with certain additional restrictions on the type of allowable intersections (see
Section~\ref{sec:height-cor}). This definition of height is in rough correspondence with the usual
definition of height for a symmetric (uncapped) grope and in \cite{COT} it is shown that a knot in
the 3--sphere $S^3=\partial B^4$ is $n$-solvable if it bounds an embedded symmetric grope of height
$n+2$ or a Whitney tower of height $n+2$ in $B^4$. It is not known if these geometric conditions
are equivalent to each other (or to being $n$-solvable) but we do have:
\begin{cor}\label{cor:height}
If a knot in $S^3$ bounds a properly embedded grope of height $n$ (resp. $n.5$) in $B^4$, then the
knot bounds a Whitney tower of height $n$ (resp. $n.5$) in $B^4$.
\end{cor}
The proof of Corollary~\ref{cor:height} suggests that bounding a grope of height $n$ is probably a
stronger condition than bounding a Whitney tower of height $n$ (see Section~\ref{sec:height-cor}).


%===========================================================================



\subsection*{Half-gropes, $\mathbf{k}$--cobordism and geometric IHX}
In the general study of knot and link concordance (cobordism) the translation between gropes and
Whitney towers can be helpful in understanding connections between algebraically and geometrically
defined filtrations. This is the case in the next two corollaries which are proved by applying a
geometric ``IHX'' Jacobi relation to manipulate the trees associated to a Whitney tower
(Lemma~\ref{IHX-lemma}).


In analogy with the fact that the $n$th term of the lower central series of a group is generated by
{\em simple} (right- or left-normed) commutators (\cite{MKS}) we have the following geometric
result which implies in particular that class $n$ {\em grope concordance} is generated by class $n$
(annulus-like) {\em half-gropes} (details in Section~\ref{sec:half-grope-cor}):
\begin{cor}\label{cor:half-grope}
If $\gamma_i$ bound disjoint properly embedded $A_i$--like class $n$ gropes $g_i$ with
null-homotopic tips in a 4--manifold $X$, then the $\gamma_i$ bound disjoint properly embedded
$A_i$--like class $n$ {\em half-gropes} $h_i$ with null-homotopic tips in $X$, with the $h_i^c$
contained in a neighborhood of $g_i^c$ for any choice of caps on the $g_i$.
\end{cor}
Since a symmetric grope of height $n$ has class $2^n$, Corollary~\ref{cor:half-grope} also gives a
geometric (embedded) analogue of the fact that the $n$th derived subgroup is contained in the
$2^n$th lower central subgroup of a group.


In the case of knots in $S^3$, it was shown constructively in \cite{S3} (see also \cite{CT2}) that
the Arf invariant is the only obstruction to the existence of a (half)-grope concordance of
arbitrarily high class. Thus, the Von Neumann signatures of \cite{COT} which obstruct
$n$-solvability are obstructions to ``inverting'' the geometric manipulations of gropes and Whitney
towers via geometric IHX constructions (as used, for instance, in the proof of
Corollary~\ref{cor:half-grope} above) to convert class into height.


The filtration of link concordance classes by the notion of $k$-{\em cobordism} was introduced and
studied in \cite{C1}, \cite{C2}, \cite{C3} and \cite{O1}. In particular, a link $L$ in $S^3$ is
$k$-{\em slice} ($k$-null-cobordant) if the link components bound disjoint surfaces in $B^4$ which
``look (algebraically) like'' slice disks modulo the $k$th term of the lower central series of the
link group $\pi_1(S^3- L)$. This was shown in \cite{IO} to be equivalent to $L$ having vanishing
Milnor $\overline{\mu}$-invariants up through length $2k$. The precise relation between grope
concordance and Milnor's invariants is not known, however class $n$ grope concordance implies
length $n$ Milnor--equivalence (\cite{KrT}). Evidence that class $n$ grope concordance is {\em
stronger} than length $n$ Milnor--equivalence is provided by the easy proof of the following
corollary, in contrast to the difficulties encountered in the just mentioned result in \cite{IO}.


\begin{cor}\label{cor:k-slice}
If the components of a link $L$ in $S^3$ bound disjoint properly embedded class $2k$ gropes in the
$B^4$, then $L$ is $k$-slice.
\end{cor}
The proof of Corollary~\ref{cor:k-slice} uses the flexibility of Whitney towers to ``evenly
distribute'' the gropes' higher surface stages over symplectic sets of circles on the bottom
surfaces. In fact, the proof yields a stronger conclusion as described in
Section~\ref{sec:k-slice}. (The same proof also shows more generally that class $2k$ grope
concordance implies $k$-cobordism.)

\subsection*{Further applications}
Although closely related, Whitney towers are slightly more general objects than gropes as is
suggested by the association of {\em rooted} trees to gropes and {\em un}rooted trees to Whitney
towers. This generality is conducive to defining geometric invariants associated to an obstruction
theory (as mentioned in Remark~\ref{rem:tau}) as well as manipulating the shape of gropes (as in
Corollaries~\ref{cor:half-grope} and \ref{cor:k-slice} above). However, in certain constructions it
is useful to convert Whitney towers into gropes, for instance to take advantage of nice properties
of embedded grope complements. This interplay is exploited in \cite{ST4} which investigates the
geometric information that is {\em not} seen by Milnor's $\mu$--invariants and the Kontsevich
integral in the setting of link concordance.
 Recent work of Conant and Teichner (\cite{CT1}, \cite{CT2}) suggests that the study of both
3-- and 4--dimensional grope cobordism of restricted graph type can play an important role in the
general theory of knots and links. The relevant graphs in 4--dimensions are the unitrivalent trees
which occur here naturally in the context of Whitney towers. Perhaps an analogous notion of Whitney
tower projected into 3-dimensions with associated unitrivalent graphs could be useful in
understanding 3--dimensional grope cobordism.


\subsection*{Outline}
Gropes are discussed in Section~\ref{grope-sec}. Section~\ref{w-tower-sec} covers Whitney towers,
including two lemmas (\ref{split-tower-lem} and \ref{subtower-lemma}) which are used in subsequent
constructions. Section~\ref{sec:grope-subtowers} introduces the hybrid ({\em split}) {\em grope
subtowers} which are used to interpolate between gropes and Whitney towers.
Theorem~\ref{class-order-thm} is then proved via the more detailed Theorem~\ref{tower-to-grope-thm}
and Theorem~\ref{grope-to-tower-thm} which are stated and proved in Section~\ref{sec:thms}. 
Corollary~\ref{cor:height} is proved in Section~\ref{sec:height-cor}. 
Corollary~\ref{cor:half-grope} is proved in Section~\ref{sec:half-grope-cor}, which also contains a
geometric (IHX) construction (Lemma~\ref{IHX-lemma}) which is used in \cite{ST2}. 
Corollary~\ref{cor:k-slice} is proved in Section~\ref{sec:k-slice}.
%\tableofcontents

\subsection*{Remark} The figures and their captions contain enough information (on definitions,
notation and proofs) so that most of the content of this paper can be absorbed from the figures,
together with the motivating remarks at the start of each section and the statements of the main
theorems in Section~\ref{sec:thms}.

\subsection*{Conventions}\label{conventions}
All maps and manifolds are assumed smooth and oriented. Surfaces in 4--manifolds are illustrated in
figures showing a 3--dimensional slice of local 4--dimensional coordinates with the understanding
that sheets of surfaces that appear as 1-dimensional arcs in the ``present'' 3--dimensional slice
extend as a product into the ``past and future'' coordinate. Sheets of surfaces which are contained
in the 3--dimensional slice may appear either translucent or opaque.

\subsection*{Acknowledgments} This work was supported in part by an NSF Postdoctoral Fellowship and the
Max-Planck-Institut f\"{u}r Mathematik.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{class4gropeA-fig.eps}}
        \caption{A grope of class 4 (see Definition~\ref{grope-defi}).}
        \label{class4gropeA-fig}

\end{figure}


\section{Gropes}\label{grope-sec}
This section contains mostly standard grope terminology for the purpose of fixing notation since
there are subtle differences in definitions and approaches throughout the literature. See any of
\cite{FQ}, \cite{FT} or \cite{KrT} for detailed discussions of gropes in 4--manifolds.

There will be however be a few mildly non-standard wrinkles which are worth pointing out: It will
be convenient to consider the punctured {\em starting surface} $A^0$ of an $A$--like grope $g$ as a
{\em 0th stage} of $g$,  and it will be helpful think of $g$ as formed from $A^0$ by attaching many
genus one gropes rather than a single higher genus grope. In light of these notational conventions,
it turns out to be convenient to associate to each grope a disjoint union of unitrivalent trees (as
in \cite{CST}, \cite{CT1} and \cite{CT2}) rather than the customary (single) multivalent tree (e.g.
in \cite{KrT}). This point of view is in line with Krushkal's grope-splitting technique \cite{Kr}
which simplifies the combinatorics of gropes. Note also that what we will refer to as a {\em
dyadic} grope is called a ``grope with dyadic branches'' in \cite{KQ}.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{capped-A-like-grope1-and-with-trees-fig.eps}}
        \caption{A dyadic annulus-like capped grope $g^c$ of class 4, also shown
        with the embedding of $t(g^c)$ in $g^c$.
        The 0th stage of $g^c$ is the twice-punctured annulus.}
        \label{capped-A-like-grope1-and-with-trees-fig}

\end{figure}


\subsection{Grope terminology}\label{grope-terminology}



\begin{defi}[\cite{FT}]\label{grope-defi}
A {\em grope} is a special pair (2--complex, circle). A grope has a {\em class}
$n\in\{1,2,\ldots,\infty\}$. A class 1 grope is defined to be the pair (circle, circle). A class 2
grope $(S,\partial S)$ is a compact oriented connected surface $S$ with a single boundary
component. For $n>2$, a grope of class $n$ is defined inductively as follows: Let
$\{\alpha_j,\beta_j, j=1,\ldots,\mbox{genus}\}$ be a chosen standard symplectic basis of circles
for a class~2 grope $S$. For any positive integers $a_j,b_j$, with $a_i+b_j\geq n$ and
$a_{j_0}+b_{j_0}=n$ for at least one index $j_0$, a grope of class $n$ is formed by attaching a
class $a_j$ grope to each $\alpha_j$ and a class $b_j$ grope to each $\beta_j$ (See
Figure~\ref{class4gropeA-fig}).
\end{defi}
Here ``attaching a class 1 grope'' is understood to mean ``not attaching a grope at all''. The
surfaces in a grope $g$ are called {\em stages} and the basis circles in (all stages of) $g$ which
do not have a surface stage attached to them are the {\em tips} of $g$. Attaching 2--disks, called
{\em caps}, to all the tips of $g$ yields a {\em capped grope} $g^c$ and the underlying uncapped
grope $g$ is the {\em body} of $g^c$.

It is customary to omit the boundary of a grope from notation. We adopt the convention that the
tip of a class 1 grope is the grope (a circle) itself, so a class one capped grope is just a
circle bounding a disk.

Note that a class $n$ grope is {\em not} really also a grope of class $m$ for $m<n$, but can always
be made into one by deleting some surface stages (in order to satisfy $a_{j_0}+b_{j_0}=m$ for at
least one index $j_0$).


\subsection{Dyadic {\boldmath $A$}--like gropes with 0th
stages.}\label{dyadic-gropes-defi} The gropes of Definition~\ref{grope-defi} are special cases of
the more general {\em $A$--like (capped) grope} of class $n$ which is gotten from a starting
surface $A$ by replacing disks in $A$ with (capped) gropes of class $n$ as defined above. All the
notions such as stages, tips and caps apply to $A$--like gropes without change with the following
addition: If $g$ is an $A$--like grope then the {\em 0th stage} of $g$ is defined to be the
punctured surface $A^0$ which is $A$ minus the disks that were replaced by gropes to form $g$. The
0th stage is included in the body of an $A$--like capped grope. In this language, the gropes of
Definition~\ref{grope-defi} can be thought of as disk-like gropes.

An $A$--like (capped) grope $g$ is {\em dyadic} if all surface stages of all gropes attached to the
0th stage $A^0$ are genus one surfaces (See Figure~\ref{capped-A-like-grope1-and-with-trees-fig}).
Since we are allowing $A^0$ to have many punctures, any (capped) $A$--like grope can be converted
to a (capped) dyadic $A$--like grope by Krushkal's grope-splitting technique \cite{Kr}.

The notion of dyadic gropes was introduced in \cite{KQ}; our terminology is slightly different in
that instead of allowing higher genus in the first stage we attach many genus 1 gropes to the 0th
stage. Also, for brevity we are using the term ``dyadic grope'' here instead of (the more precise)
``grope with dyadic branches'' as in \cite{KQ}.


\subsection{Rooted trees for gropes}\label{grope-trees}
A {\em tree} is a connected graph without 1--cycles. A {\em rooted} tree has a single preferred
univalent vertex called the {\em root}. To each $A$--like dyadic capped grope $g^c$, we associate a
disjoint union $t(g^c)$ of rooted unitrivalent trees which is essentially the dual one-complex: The
vertices of $t(g^c)$ are (dual to) the stages and caps of $g^c$, with two vertices joined by an
edge if the corresponding stages/caps meet in a circle. The univalent vertices corresponding to the
0th stages are the roots.

It will be helpful to think of $t(g^c)$ as being embedded in $g^c$ in the following way. Choose a
basepoint in each stage (including $A^0$) and a basepoint in each cap of $g^c$. If $g^c$ was formed
by removing just one disk from $A$, then connecting basepoints in adjacent stages and caps by
sheet-changing paths yields an embedded connected unitrivalent tree; here ``adjacent'' means
``intersecting along a circle'' so that dual stages/caps (having a single intersection point in
their boundaries) are not considered to be adjacent. If $g^c$ was formed by replacing $m$ disks of
$A$, then we get $m$ unitrivalent trees (each ``sprouting from'' the basepoint in the $0$th stage
$A^0$) and $t(g^c)$ is the disjoint union of these $m$ trees
(Figure~\ref{capped-A-like-grope1-and-with-trees-fig}). Each trivalent vertex of $t(g^c)$
corresponds to a genus one stage in $g^c$ and each univalent vertex of $t(g^c)$ corresponds to a
cap  of $g^c$, except for one univalent vertex on each connected component of $t(g^c)$ which
corresponds to $A^0$. These univalent vertices on the $A^0$ are the root vertices of $t(g^c)$. To
the underlying uncapped grope $g$ is associated the same disjoint union of rooted trees denoted
$t(g)$. Note that the class of $g^c$ is equal to the minimum of the Vassiliev degrees (half the
number of vertices) of each (connected) tree in $t(g^c)$.


The reader familiar with previous associations of multi-valent trees to arbitrary $A$--like gropes
(e.g. \cite{KrT}) can check that our definition of $t(g^c)$ is essentially what you would get after
applying Krushkal's grope-splitting procedure. In fact, a formal sum of {\em vertex-oriented} trees
can be associated to an {\em oriented} grope (\cite{CST} \cite{CT1} \cite{CT2}); although we will
not work with orientations in this paper, the notation here has been chosen to be compatible with
the just cited works.






\subsection{Proper immersions of gropes}\label{immersions}
A surface is {\em properly} immersed in a 4--manifold if boundary is embedded in boundary and
interior is immersed in interior. Immersions of (capped) gropes into a 4--manifold are required to
factor through an embedding in 3--space followed by standard product thickenings and plumbings, so
that a regular neighborhood of the immersion contains disjoint parallel copies of any embedded
subsets of the (capped) grope. An immersion of (capped) gropes is {\em proper} if the bottom stage
surfaces are properly immersed and all other stages (and caps) are immersed in the interior.

In much of the literature, a proper immersion of a capped grope in a 4--manifold will have
restrictions on allowable grope/cap singularities. We will not make such restrictions, however it
will be convenient to arrange for all intersections to occur among $0$th stages and between caps
and $0$th stages .


%=============================================================================
\section{Whitney towers}\label{w-tower-sec}
After defining {\em Whitney towers}, the main goal of this section is to show how the essential
geometric structure of a Whitney tower $T$ is captured by a disjoint union of unitrivalent
(labelled) trees $t(T)$. The surfaces of $T$ are indexed by brackets which correspond to rooted
trees, and each intersection point $p$ of $T$ is then assigned an unrooted tree $t(p)$ which is a
pairing of the rooted trees that correspond to the intersecting surfaces. After introducing the
notion of ({\em split}) {\em subtowers}, Lemma~\ref{split-tower-lem} then shows how $T$ can be
decomposed into essential parts which are described by the disjoint union $t(T)$ of all the $t(p)$.
Finally, Lemma~\ref{subtower-lemma}, which will play a key role in later proofs, gives further
evidence of the essential nature of $t(T)$ by showing how the trees $t(p)$ are ``stable'' under
certain Whitney moves.

All of the material in this section can be enhanced to take into account orientations as well as
the fundamental group of the ambient manifold by adding vertex-orientations and edge-decorations to
the trees. Such enhancements are used in the obstruction theory of \cite{ST1}, \cite{ST2},
\cite{ST3}, and \cite{ST4}. Notation in this paper has been chosen to be consistent with these
papers where they overlap.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{W-disk-move2-fig.eps}}
        \caption{The embedded Whitney disk $W_{(I,J)}$ guides a Whitney move that
        eliminates a cancelling
        pair of intersections between surface sheets $I$ and $J$ via an isotopy of $I$.
        Since $W_{(I,J)}$ had an interior intersection with a surface sheet $K$,
        this Whitney move creates a new cancelling
        pair of intersections between $I$ and $K$.}
        \label{W-disk-move2-fig}

\end{figure}




\subsection{Whitney disks}\label{W-disk-subsec}
We refer the reader to \cite{FQ} for a detailed description of the {\em Whitney move} in dimension
4. For our purposes it is enough to understand the model Whitney move on an embedded Whitney disk
in 4--space as illustrated in Figure~\ref{W-disk-move2-fig}, which shows the effect of a Whitney
move on three local sheets of surfaces: The Whitney move is guided by an embedded {\em Whitney
disk} labelled $W_{(I,J)}$ which {\em pairs} a (geometrically) {\em cancelling pair} of
intersection points between sheets of surfaces labelled $I$ and $J$. In
Figure~\ref{W-disk-move2-fig}, the $I$-sheet is locally contained in the 3--dimensional ``present''
and the $J$- and $K$-sheets extend into ``past and future''. The Whitney move eliminates the
cancelling pair of intersections between $I$ and $J$ by an isotopy of $I$ across $W_{(I,J)}$ at the
cost of introducing a new cancelling pair of intersection points between $I$ and the sheet labelled
$K$ which intersected the interior of the guiding Whitney disk $W_{(I,J)}$.




\subsection{Whitney towers}\label{W-tower-sub-sec}
In general, a Whitney disk may have multiple interior self-intersections and intersections with
other surface sheets; however we will require that arbitrary Whitney disks resemble the model near
their boundaries. By pairing up interior intersections with higher order Whitney disks we are led
to the notion of a Whitney tower:
\begin{defi}\label{w-tower-defi}

A {\em surface of order 0} is a properly immersed surface in a 4--manifold. 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 a collection of order 0 surfaces. 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 $T$. The Whitney disks in a Whitney tower are required to be {\em
framed} (see \cite{FQ}) and have disjointly embedded boundaries.

\end{defi}
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-tower5-fig}).



It will also be assumed that the order 0 surfaces are {\em 0-framed} (see 1.2 of \cite{FQ}).

Some further terminology: If $T_0$ is an order 0 Whitney tower and there exists an order $n$
Whitney tower $T_n$ containing $T_0$ as its order 0 surfaces, then $T_0$ is said to {\em admit} an
order $n$ Whitney tower and any one of the order 0 surfaces in $T_n$ is said to {\em support} the
$n$th order Whitney tower $T_n$.


\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{bracket-treesA-fig.eps}}
        \caption{Rooted trees and brackets: $t(i)$ and $t(j)$ (upper left) and their rooted product
        $t(i,j)=t(i)\ast t(j)$ (lower left);
        $t(i_1,i_2)$ and $t(j_1,(j_2,j_3))$ (upper right) and their rooted product
        $t((i_1,i_2),(j_1,(j_2,j_3)))=t(i_1,i_2)\ast t(j_1,(j_2,j_3))$
        (lower right).}
        \label{bracket-treesA-fig}

\end{figure}




\subsection{Rooted trees and brackets}\label{trees-brackets}
Non-associative but commutative (unordered) {\em bracketings} of elements from some index set
correspond to rooted labelled (unoriented) unitrivalent trees as follows. A bracketing $(i)$ of a
singleton element $i$ from the index set corresponds to the rooted {\em chord} $t(i)$ having a
single edge with one vertex labelled by $i$ and the other vertex designated as the root. A
bracketing $(I,J)$ of brackets $I$ and $J$ corresponds to the {\em rooted product} $t(I) \ast t(J)$
of the trees $t(I)$ and $t(J)$ which identifies together the roots of $t(I)$ and $t(J)$ to a single
vertex and ``sprouts'' a new rooted edge at this vertex (Figure~\ref{bracket-treesA-fig}). Thus,
the (non-root) univalent vertices of the tree $t(I)$ associated to a bracket $I$ are labelled by
elements from the index set and the trivalent vertices correspond to sub-bracketings of $I$, with
the trivalent vertex adjacent to the root corresponding to $I$. Note that Vassiliev degree is
additive with respect to rooted product.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{W-disk-treeC-fig.eps}}
        \caption{A Whitney disk $W_{((I_1,I_2),I_3)}$ and (part of) its
        associated tree $t(W_{((I_1,I_2),I_3)})$ shown both as a subset of
        the Whitney tower and as an abstract rooted tree. (If any $I_i$ is not
        a singleton, then that label $I_i$ represents a sub-tree.)}
        \label{W-disk-treeC-fig}

\end{figure}

\subsection{Rooted trees for Whitney disks}\label{w-disk-tree-subsec}
The rooted trees and bracketings of the previous subsection are associated to the surfaces in a
Whitney tower $T$ both formally and geometrically as follows: A bracketing $(i)$ of a singleton
element $i$ from the index set is associated to each order zero surface. The bracket $(I,J)$ is
associated to a Whitney disk pairing intersections between surfaces with associated brackets $I$
and $J$. Using brackets as subscripts, we write $A_i$ for an order zero surface (dropping the
brackets around the singleton $i$) and $W_{(i,j)}$ for a first order Whitney disk that pairs
intersections between $A_i$ and $A_j$. In general, we write $W_{(I,J)}$ for a Whitney disk pairing
intersections between $W_I$ and $W_J$, with the understanding that if a bracket $I$ is just a
singleton $(i)$, then the surface $W_I=W_{(i)}$ is just the order zero surface $A_i$. The rooted
labelled tree $t(W_I)$ associated to $W_I$ is defined to be $t(I)$, the tree that corresponds to
the bracket $I$ as before (\ref{trees-brackets}).

In fact, the tree $t(W_I)$ can be mapped into $T$ in a similar fashion to the case of gropes
(\ref{grope-trees}): First fix a basepoint in the interior of each surface (including the Whitney
disks) of $T$. (If an order zero surface is not connected put a basepoint in each component.) Now
map the vertices (other than the root) of $t(W_I)$ to the basepoints of the surfaces corresponding
to the sub-brackets of $I$ and map the edges (other than the edge adjacent to the root) of $t(W_I)$
to sheet-changing paths between these basepoints, as illustrated in Figure~\ref{W-disk-treeC-fig}.
Then embed the root and its edge anywhere in the interior of $W_I$. Note that the order of $W_I$ is
equal to the number of trivalent vertices of $t(W_I)$.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.5]{inner-product-trees-fig.eps}}
        \caption{On the left, a pair of rooted trees $t(I)$ and $t(J)$
        corresponding to first order Whitney disks $W_{I}$ and
        $W_{J}$ with $I=(i_1,i_2)$ and $J=(j_1,j_2)$. On the upper
        right, the inner product $t(p)=t(I)\cdot t(J)$ associated to
        a 3rd order intersection point $p\in W_{I}\cap W_{J}$ and on the
        lower right, the punctured tree $t^{\circ}(p)$ that also
        keeps track of $p$.}
        \label{inner-product-trees-fig}

\end{figure}

\subsection{Trees for intersection points}\label{int-point-trees}


Given a pair $t(I)$ and $t(J)$ of rooted trees, define the {\em unrooted product} $t(I)\cdot t(J)$
to be the labelled unrooted tree gotten by identifying together the root vertices of $t(I)$ and
$t(J)$ to a single (non-vertex) point. The tree $t(p)$ associated to a (transverse) intersection
point $p\in W_I\cap W_J$ between surfaces $W_I$ and $W_J$ in a Whitney tower $T$ is defined to be
the unrooted product $t(W_I)\cdot t(W_J)$ ($=t(I)\cdot t(J)$) of the rooted trees corresponding to
$W_I$ and $W_J$ as illustrated in Figure~\ref{inner-product-trees-fig}. The order of $p$ is equal
to the Vassiliev degree (one half the total number of vertices) of $t(p)$.

The above mentioned mappings of $t(W_I)$ and $t(W_J)$ in $T$ give rise to a mapping of $t(p)$ into
$T$: Just map the root vertices of $W_I$ and $W_J$ to $p$ and the adjacent edges become a path
between the basepoints of $W_I$ and $W_J$ which changes sheets at $p$
(Figure~\ref{W-disk-int-point-treeC-fig}). This mapping can be taken to be an embedding of $t(p)$
into $T$ if all the Whitney disks ``beneath'' $W_I$ and $W_J$ (corresponding to sub-brackets of $I$
and $J$) are distinct.

It is sometimes convenient to keep track of the edge of $t(p)$ that corresponds to $p$ by marking
that edge with a small linking circle as in Figure~\ref{inner-product-trees-fig} and
Figure~\ref{W-disk-int-point-treeC-fig}; such a {\em punctured tree} will be denoted by
$t^{\circ}(p)$.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{W-disk-int-point-treeC-fig.eps}}
        \caption{A local picture of the punctured tree $t^{\circ}(p)$ associated to
        an intersection point $p\in W_I\cap W_J$, shown as a subset
        of the Whitney tower and as an abstract labelled (punctured) tree.}
        \label{W-disk-int-point-treeC-fig}

\end{figure}
\subsection{Trees for Whitney towers}\label{w-tower-trees-def}
For any Whitney tower $T$ define $t(T)$ to be the disjoint union of all the trees $t(p)$ for all
unpaired intersection points $p\in T$.
\begin{remn}
As mentioned at the start of this section, $t(T)$ can be enhanced with vertex-orientations and
edge-decorations and is called the {\em geometric intersection tree} of $T$. The invariants
$\tau_n$ associated to the obstruction theory mentioned in Remark~\ref{rem:tau} are determined by
such trees (e.g. \cite{ST2}).
\end{remn}
\subsection{Split Subtowers}
In order to simplify constructions and combinatorics it will be helpful to ``split'' a Whitney
tower into {\em split subtowers} analogous to Krushkal's grope splitting procedure \cite{Kr} which
splits a grope into genus one ``branches'' (higher stages).

\begin{defi}\label{subtower-defi}
A {\em subtower} is a Whitney tower except that the boundaries of the immersed order zero surfaces
in a subtower are allowed to lie in the interior of the 4--manifold. (The boundaries of the order
zero surfaces in a subtower are still required to be embedded.) The notions of {\em order} for
intersection points and Whitney disks are the same as in Definition~\ref{w-tower-defi}.
\end{defi}



\begin{defi}\label{split-subtower-defi}
A subtower $T_p$ is {\em split} if it satisfies all of the following:
\begin{enumerate}
\item
$T_p$ contains a single unpaired intersection point $p$,
\item
the order zero surfaces of $T_p$ are all embedded 2--disks,
\item
the Whitney disks of $T_p$ are all embedded,
\item
the interior of any surface in $T_p$ either contains $p$ or contains a single Whitney arc of a
Whitney disk in $T_p$,
\item
$T_p$ is connected (as a 2--complex in the 4--manifold).

\end{enumerate}
The tree $t(T_p)$ associated to such a split subtower $T_p$ is just the tree $t(p)$ determined by
its unpaired intersection point $p$ (\ref{int-point-trees}).
\end{defi}
Note that the order of a split subtower $T_p$ is equal to one less than the order of $p$ and that a
normal thickening of $T_p$ in the ambient 4--manifold is just a 4--ball neighborhood of the
embedded tree $t(T_p)=t(p)$.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{splitting-and-split-tower-fig.eps}}
        \caption{Part of a Whitney tower before (top) and after (bottom) applying
        the splitting procedure described in the proof of Lemma~\ref{split-tower-lem}.}
        \label{splitting-and-split-tower-fig}

\end{figure}


\subsection{Split Whitney towers and their trees}\label{split-w-tower-tree-subsec}

\begin{lem}\label{split-tower-lem}
Let $T$ be a Whitney tower on order 0 surfaces $A_i$. Then there exists a Whitney tower
$T_{\mathrm{split}}$ contained in any regular neighborhood of $T$ such that:
\begin{enumerate}
\item The order~0 surfaces $A'_i$ of $T_{\mathrm{split}}$ only differ from the
$A_i$ by finger moves.
\item All unpaired intersection points of $T_{\mathrm{split}}$ are contained
in disjoint split subtowers on sheets of the $A'_i$.
\item $t(T)$ and $t(T_{\mathrm{split}})$ are isomorphic.
\end {enumerate}
\end{lem}

Such a Whitney tower $T_{\mathrm{split}}$ will be called a {\em split} Whitney tower. The disjoint
union of trees $t(T_{\mathrm{split}})$ (\ref{w-tower-trees-def}) associated to $T_{\mathrm{split}}$
is the disjoint union of the trees $t(T_p)$ associated to the split subtowers in
$T_{\mathrm{split}}$. Also, $t(T_{\mathrm{split}})$ sits as an {\em embedded} subset of
$T_{\mathrm{split}}$ via the description in \ref{int-point-trees}.



\begin{proof}
Starting with the highest order Whitney disks of $T$, apply finger moves as indicated in
Figure~\ref{splitting-and-split-tower-fig}. Working down through the lower order Whitney disks
yields the desired $T_{\mathrm{split}}$.
\end{proof}

This decomposition of a Whitney tower into split subtowers corresponds to the idea that the
disjoint union of trees associated to the Whitney tower captures its essential structure. The next
lemma can be interpreted as justifying that this essential structure is indeed captured by the {\em
un}punctured trees rather than the punctured trees, in the sense that a punctured edge
(corresponding to the unpaired intersection point) can be ``moved'' to any other edge of its tree.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{tree-move-trees-fig.eps}}
        \caption{A local picture of the tree associated to the split
        subtower $T$
        before (left) and $T'$ after (right) the Whitney move illustrated in
        Figure~\ref{tree-move-and-W-disk-trees-fig}.}
        \label{tree-move-trees-fig}

\end{figure}
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{tree-move-and-W-disk-trees-fig.eps}}
        \caption{Upper left, the unpaired intersection point $p=W_I\cap W_J$ in the split
        subtower $T$ of Lemma~\ref{subtower-lemma}. Upper right, the unpaired
        intersection point $p'=W_{I'}\cap W_{J'}$ in $T'$ after the Whitney move. The lower part of
        the figure shows that the punctured trees differ as indicated in
        Figure~\ref{tree-move-trees-fig}.}
        \label{tree-move-and-W-disk-trees-fig}

\end{figure}
\begin{lem}\label{subtower-lemma}
Let $T$ be a split subtower on order zero sheets $s_i$, with unpaired intersection point $p=W_I\cap
W_J\subset T$. Denote by $\nu(T)$ a normal thickening of $T$, so that $\partial
s_i\subset\partial\nu(T)\subset\nu(T)\cong B^4$. If $I'$ and $J'$ are any brackets such that
$t(I')\cdot t(J')=t(p)=t(I)\cdot t(J)$, then after a homotopy (rel $\partial$) of the $s_i$ in
$\nu(T)$ the $s_i$ admit a split subtower $T'\subset\nu(T)$ with single unpaired intersection point
$p'=W_{I'}\cap W_{J'}\subset T'$.
\end{lem}
\begin{proof}
It is enough to show that the puncture in $t^{\circ}(p)$ can be ``moved'' to either {\em adjacent}
edge, since by iterating it can be moved to any edge of $t(p)$. Specifically, it is enough to
consider the case where $J=(J_1,J_2)$, $I'=(I,J_1)$ and $J'=J_2$ so that
$I\cdot(J_1,J_2)=(I,J_1)\cdot J_2$ as in Figure~\ref{tree-move-trees-fig}. (Here we are assuming
that $W_J$ is not order zero, since if both $W_I$ and $W_J$ are order zero there is nothing to
prove.) The proof is given by the maneuver illustrated in the upper part of
Figure~\ref{tree-move-and-W-disk-trees-fig}: Use the Whitney disk $W_J$ to guide a Whitney move on
$W_{J_1}$. This eliminates the intersections between $W_{J_1}$ and $W_{J_2}$ (as well as
eliminating $W_J$ and $p$) at the cost of creating a new cancelling pair of intersections between
$W_{J_1}$ and $W_I$. This new cancelling pair can be paired by a Whitney disk $W_{(I,J_1)}$ having
a single intersection point $p'$ with $W_{J_2}$. That this achieves the desired effect on the
punctured tree is clear from the lower part of Figure~\ref{tree-move-and-W-disk-trees-fig}.
\end{proof}





\section{Grope subtowers}\label{sec:grope-subtowers}
In this section a hybrid grope--Whitney tower combination is introduced which will be used to
interpolate between gropes and Whitney towers in the proofs of the next section: A {\em split grope
subtower} is a collection of capped gropes whose caps support certain split subtowers. We assume
from now on that all gropes are dyadic (\ref{dyadic-gropes-defi}).

The content of this section can essentially be grasped by inspecting
Figure~\ref{grope-subtower1-fig} and Figure~\ref{grope-subtower-tree1-with-trees-fig}, which
illustrate split grope subtowers and their trees, and observing that the limiting cases reduce to
gropes (\ref{Order 0 subtowers on caps}) and Whitney towers (\ref{Class 1 grope subtowers})
respectively.




\subsection{Split grope subtowers.}\label{grope-subtowers}





\begin{defi}

Let $g_i^c$ be a collection of (dyadic) $A_i$--like capped gropes properly immersed in a
4--manifold such that:
\begin{enumerate}

\item The higher (greater than zero) stages of the $g_i$ are all
disjointly embedded and disjoint from the interiors of all caps,

\item the interiors of all caps are disjointly embedded,

\item each cap $c$ supports a split subtower $T_c$ whose other order 0 surfaces are sheets of the
0th stage surfaces $A^0_i$,

\item the $T_c$ are disjoint and contain all singularities among the $0$th stages $A^0_i$.

\end{enumerate}
Denote by $g_i^T$ the union of the grope $g_i^c$ and the subtowers on its caps. The union $g^T$ of
all the $g_i^T$ is a {\em split grope subtower} (see Figure~\ref{grope-subtower1-fig}).

The {\em class} of $g_i^T$ is the class of the underlying grope $g_i^c$. The {\em class} of $g^T$
is the minimum of the classes of the $g_i^T$.

The {\em order} of $g_i^T$ is defined inductively as follows: If $g_i^T$ is class 1, then the order
of $g_i^T$ is the minimum of the orders of the split subtowers on the caps of $g_i^T$ (the
immersed disks that fill in the punctures of $A^0_i$). If $g_i^T$ has class 2 and a single genus
one first stage, then the order of $g_i^T$ is the sum of the orders of the split subtowers on the
dual pair of caps of $g_i^T$. If $g_i^T$ has class 2 and more than one first stage, then the order
of $g_i^T$ is the minimum of the orders of the first stages. If the class of $g_i^T$ is greater
than 2, then the order of $g_i^T$ is defined (inductively) to be the minimum of the sums of the
orders of the pairs of dual grope subtowers that are attached to the first stages of $g_i^T$. The
{\em order} of $g^T$ is the minimum of the orders of the $g_i^T$.

\end{defi}
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.70]{grope-subtower1-fig.eps}}
        \caption{A {\em split grope subtower} is a collection of properly immersed
        capped gropes,
        with each cap supporting
        a split subtower whose other order 0 surfaces are sheets of the
        0th stages of the gropes.}
        \label{grope-subtower1-fig}

\end{figure}
\begin{figure}[h!]
        \centerline{\includegraphics[scale=.70]{grope-subtower-tree1-with-trees-fig.eps}}
        \caption{The tree $t(g^T)$ associated to a split grope subtower
        $g^T$ is formed from the trees associated to the caps and gropes
        by gluing univalent vertices associated to common caps (bottom)
        and embeds in $g^T$ (top).}
        \label{grope-subtower-tree1-with-trees-fig}

\end{figure}


\subsection{Trees for dyadic split grope subtowers.}
For each $g_i^T$ in a split grope subtower $g^T$, construct the disjoint union of (rooted)
unitrivalent trees $t(g_i^T)$ from $t(g_i^c)$ (defined in \ref{grope-trees}) by gluing on the trees
$t(T_c)$ (defined in \ref{subtower-defi}) along the univalent vertices that correspond to caps.
Specifically, a univalent vertex of $t(g_i^c)$ which corresponds to a cap $c$ in $g_i^c$ is
identified with the univalent vertex of $t(T_c)$ which corresponds to $c$, where $T_c$ is the
subtower on $c$. This identification is to a single non-vertex point in an edge of $t(g_i^T)$ (see
the lower part of Figure~\ref{grope-subtower-tree1-with-trees-fig}). Doing this for all caps on
$g_i$ and all $i$ yields all the $t(g_i^T)$. The disjoint union of trees $t(g^T)$ associated to the
split grope subtower $g^T$ is defined to be the disjoint union of the $t(g_i^T)$ and sits as a
subset of $g^T$ (upper part of Figure~\ref{grope-subtower-tree1-with-trees-fig}).



\subsection{Order 0 split grope subtowers.}\label{Order 0 subtowers on
caps} If a class $m$ split grope subtower $g^T$ has order 0, then all the split subtowers $T_c$
in $g^T$ are order 0 which just means that each cap of every $g_i^c$ has exactly one interior
intersection point with a sheet of some $A^0_j$. In this case, the trees $t(g_i^T)$ and $t(g_i^c)$
are clearly isomorphic for all $i$ and each univalent vertex corresponds to a sheet of some
$A^0_j$.


\subsection{Class 1 split grope subtowers.}\label{Class 1 grope
subtowers}

If each $g_i^T$ in an order $m$ grope subtower $g^T$ has class 1, then the caps fill in the
punctures in the $0$th stages $A^0_i$ to form the order 0 surfaces in an order $m$ split Whitney
tower $T$ on immersions of the $A_i$ extending the embedded $A^0_i$. The disjoint unions of trees
$t(g^T)$ and $t(T)$ are isomorphic, with the root of each chord in $t(g_i^c)$ corresponding to an
$i$-labelled vertex of a tree in $t(T)$.



\section{Proof of Theorem~\ref{class-order-thm}}\label{sec:thms}
The equivalence of the statements in Theorem~\ref{class-order-thm} in the introduction follows
directly from the more detailed Theorems \ref{tower-to-grope-thm} and \ref{grope-to-tower-thm}
which are stated and proved in this section. A key element of these theorems is that when passing
between gropes and Whitney towers, the associated trees are ``preserved''. In this setting, an {\em
isomorphism} between rooted and unrooted (disjoint unions of) trees will always mean an isomorphism
between the underlying unrooted trees, but will also include a correspondence between the roots and
certain specified univalent vertices, e.g. the roots in $t(g_i)$ will always correspond to
$i$-labelled univalent vertices of $t(T)$ when passing between gropes $g_i$ and a Whitney tower $T$
on order zero surfaces $A_i$. (These isomorphisms also preserve the {\em signed} trees associated
to gropes and Whitney towers as in \cite{CST} and \cite{ST2}.)
\subsection{From Whitney towers to
gropes.}\label{w-tower-to-grope-subsec}
\begin{thm}\label{tower-to-grope-thm}
Let $T$ be an order $(n-1)$ Whitney tower on properly immersed surfaces $A_i$ in a 4--manifold $X$.
Then there exist class $n$ $A_i$--like capped gropes $g_i^c$ in $X$ such that:
\begin{enumerate}
\item The $0$th stage $A_i^0$ of each $g_i^c$ is $A_i$ minus (perhaps) some
sheets containing points or Whitney arcs of $T$ that intersect $A_i$.

\item The union of the $g_i^c$ are contained in any regular
neighborhood of $T$.

\item The $g_i^c$ have disjoint properly embedded bodies $g_i$.

\item Each cap of every $g_i^c$ has a single interior intersection
with some $A^0_j$.

\item $t(T)$ is isomorphic to the disjoint union of the
$t(g_i^c)$, with $j$-labelled univalent vertices in $t(T)$ corresponding to either vertices in the
$t(g_i^c)$ associated to caps which intersect $A_j^0$ or roots in $t(g_j^c)$; furthermore, it may
be arranged that this isomorphism takes any chosen preferred $i$-labelled univalent vertices on the
trees in $t(T)$ to the root vertices of the trees in $t(g_i^c)$.
\end{enumerate}
\end{thm}
The proof of Theorem~\ref{tower-to-grope-thm} is well illustrated by
Figure~\ref{tubing-W-disk1and2with-trees-fig} together with the observation that the pictured case
can always be arranged by Lemma~\ref{subtower-lemma}.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{tubing-W-disk1and2with-trees-fig.eps}}
        \caption{Upper left to right: Reducing the order and increasing the class of
        a split grope subtower $g^T$ by 0-surgering a cap $c$ along the boundary arc of a Whitney disk.
        That $c$ contains such a boundary arc (rather than the unpaired intersection
        point of $T_c$) can be arranged by Lemma~\ref{subtower-lemma}.
        Lower left to right: This procedure preserves $t(g^T)$.}
        \label{tubing-W-disk1and2with-trees-fig}

\end{figure}
\begin{proof}
First split $T$ (Lemma~\ref{split-tower-lem}) so that $t(T)$ is the disjoint union of the split
subtower trees $t(T_p)$ each of Vassiliev degree at least $n$. For each $T_p$, choose a preferred
univalent vertex of $t(T_p)$ and let $A^0_i$ denote the punctured surfaces which are the
complements of the sheets of the $A_i$ that correspond to the chosen preferred vertices. (Each of
these sheets is either a neighborhood of a Whitney disk boundary arc or a neighborhood of an
unpaired intersection point.) These chosen vertices will end up corresponding to root vertices in
the $t(g_i^c)$ (which are associated to the 0th stages of the capped gropes $g_i^c$) as in
statement (v) of the theorem.

Now $T$ is a grope subtower $g^T$ of class 1 and order $(n-1)$: The $0$th stages of $g^T$ are the
$A^0_i$ and the caps of $g^T$ are the sheets of the $A_i$ that correspond to the chosen preferred
vertices. The trees $t(T)$ and $t(g^T)$ are isomorphic. In particular, the theorem is true for
$n=1$ by \ref{Order 0 subtowers on caps}.

Assume now that $n\geq 2$. The proof will be completed by the following construction which shows
how to decrease the order of $g^T$ while increasing the class of $g^T$ in a manner that preserves
the tree $t(g^T)$. When each cap supports an order 0 split subtower the proof is done by
\ref{Order 0 subtowers on caps}.
\subsection{Decreasing the order and increasing the class and of a
split grope subtower.} \label{decreasing-order-subsec} Consider a cap $c$ attached to some stage
$S$ in a grope subtower $g^T$ such that the order of the split subtower $T_c$ supported by $c$ is
greater than or equal to 1. There are two cases to consider: Either $c$ contains a boundary arc of
a Whitney disk in $T_c$, or $c$ contains the unpaired intersection point $p$ of $T_c$.

First assume that $c$ contains a boundary arc of a Whitney disk $W_{(c,I)}$ (see upper left of
Figure~\ref{tubing-W-disk1and2with-trees-fig}). In this case, ``tube'' (0-surger) $c$ along the
(other) boundary arc of $W_{(c,I)}$ that lies in $W_I$ as indicated in the upper right of
Figure~\ref{tubing-W-disk1and2with-trees-fig}. This changes $c$ into a genus one capped surface
stage $S'$. One cap $c'$ is $W_{(c,I)}$ minus a small collar and the other dual cap $d'$ is a
meridional disk to $W_I$. Both of these caps support split subtowers of order strictly less than
$T_c$ since the trees $t(T_{c'})$ and $t(T_{d'})$ are gotten from $t(T_c)$ by removing the edge
adjacent to the vertex associated to $c$ and cutting $t(T_c)$ at the vertex associated to
$W_{(c,I)}$. The tree associated to the new grope subtower is the same as the original tree
$t(g^T)$ since the effect of creating $S'$ from $c$ just isotopes (in $X$) the trivalent vertex
(basepoint) of $t(g^T)$ in $W_{(c,I)}$ down to a trivalent vertex (basepoint) in $S'$ as
illustrated in the lower part of Figure~\ref{tubing-W-disk1and2with-trees-fig}.

Now assume that $c$ contains the unpaired intersection point $p$ of $T_c$. We may assume that $p$
is the intersection between $c$ and a Whitney disk $W_{(I,J)}$ since $T_c$ has order greater than
or equal to 1. Modify $T_c$ by one iteration of the procedure of Lemma~\ref{subtower-lemma}: Do
the $W_{(I,J)}$ Whitney move on $W_I$. This creates a cancelling pair of intersections between $c$
and $W_I$ which are paired by a Whitney disk $W_{(c,I)}$ that has a single intersection with $W_J$
(as in Figure~\ref{tree-move-and-W-disk-trees-fig} but with different labels). The modified split
subtower $T'_c$ on $c$ has the same tree as $T_c$ and we are back to the previous case where $c$
contains a boundary arc of $W_{(c,I)}$.
\end{proof}



\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{surgered-cap-1and2-with-trees-fig.eps}}
        \caption{Upper left to right: Surgering one of a dual pair of
        caps in a split grope subtower $g^T$ reduces the class and increases the order of
        $g^T$. This procedure requires the surgered cap $c$ to contain the unpaired intersection point
        $p$ in its split subtower $T_c$ as can be arranged by
        Lemma~\ref{subtower-lemma}. Lower left to right: This
        surgery preserves $t(g^T)$.}
        \label{surgered-cap-1and2-with-trees-fig}

\end{figure}
\subsection{From gropes to Whitney
towers.}\label{grope-to-w-tower-subsec} The gropes in the statement of
Theorem~\ref{class-order-thm} can be arranged to satisfy the hypotheses of next theorem by using
finger moves to push down cap-intersections (2.5 of \cite{FQ}) into the order 0 surfaces and by
applying Krushkal's grope splitting technique \cite{Kr}.

\begin{thm}\label{grope-to-tower-thm}
Let $g_i^c$ be a collection of class $n$ $A_i$--like dyadic capped gropes in a 4--manifold $X$ with
disjoint properly embedded bodies $g_i$ such that all the caps have disjointly embedded interiors
and each cap contains only a single interior intersection point with some $0$th stage $A_j^0$. Then
there exists an order $(n-1)$ Whitney tower $T$ in $X$ such that:
\begin{enumerate}

\item $T$ is contained in any regular neighborhood of the union of the
$g_i^c$.

\item The order 0 surfaces of $T$ are immersions of the $A_i$
extending the embeddings $A_i^0$ up to regular homotopy (rel $\partial$).

\item $t(T)$ is isomorphic to the disjoint union of the
$t(g_i^c)$ with $j$-labelled univalent vertices in $t(T)$ corresponding to either vertices in the
$t(g_i^c)$ associated to caps which intersect $A_j^0$ or roots in $t(g_j^c)$.

\end{enumerate}
\end{thm}

The proof of Theorem~\ref{grope-to-tower-thm} is well illustrated by
Figure~\ref{surgered-cap-1and2-with-trees-fig} together with the observation that the pictured case
can always be arranged by Lemma~\ref{subtower-lemma}:

\begin{proof}
When $n=1$, the $g_i^c$ form a grope subtower $g^T$ of class 1 and the Theorem is true by
\ref{Class 1 grope subtowers}.

Assuming $n\geq 2$, the proof is completed by the following construction (essentially the inverse
to the construction \ref{decreasing-order-subsec} in the proof of Theorem~\ref{tower-to-grope-thm})
which decreases the class and increases the order of a grope subtower $g^T$ while preserving the
associated trees. When each $g_i^c$ in $g^T$ has class 1 the proof is complete by \ref{Class 1
grope subtowers}.


\subsection{Decreasing the class and increasing the order of a
grope subtower.} Let $c$ and $d$ be a pair of dual caps on a surface stage $S$ in a grope
subtower $g^T$ supporting split subtowers $T_c$ and $T_d$. By applying
Lemma~\ref{subtower-lemma} to $T_c$, we may arrange that $c$ contains the unpaired intersection
point $p=c\cap W_I$ in $T_c$. Using $c$ to ambiently surger $S$ changes $S$ into a cap $c'$ on the
stage $S'$ below $S$. This new cap $c'$ has a cancelling pair of intersections with $W_I$ (due to
$p$, the intersection $c$ had with $W_I$), which can be paired with a Whitney disk $W_{(c',I)}$
formed from the old cap $d$ by attaching a thin band as pictured in the upper right of
Figure~\ref{surgered-cap-1and2-with-trees-fig}. The cap $c'$ supports a split subtower $T_{c'}$
and, as illustrated in the lower part of Figure~\ref{surgered-cap-1and2-with-trees-fig}, there is
no change in $t(g^T)$ since the effect of the surgery is just to isotope (in $X$) the unitrivalent
vertex of $t(g^T)$ that was in $S$ up into $W_{(c',I)}$. (Here we are still denoting the modified
grope subtower by $g^T$.) Repeated application of this construction eventually eliminates all dual
pairs of caps so that each $g_i^c$ in $g^T$ has class 1.
\end{proof}


%======================================================================

\section{Proof of
Corollary~\ref{cor:height}}\label{sec:height-cor}

The idea of the proof of Corollary~\ref{cor:height} in the introduction is to use
Theorem~\ref{grope-to-tower-thm} of Subsection~\ref{grope-to-w-tower-subsec} to convert the
symmetric grope into a Whitney tower whose associated trees are all symmetric and then use
Lemma~\ref{subtower-lemma} to create the desired Whitney tower by moving the unpaired intersection
points appropriately. The gropes in this section are disk-like and assumed to be dyadic as can
always be arranged by Krushkal's splitting procedure \cite{Kr}.


\subsection{Symmetric gropes and trees}\label{sym-grope-tree-subsec}
Let $Y^1$ denote the degree 2 rooted $Y$--shaped tree that corresponds to a punctured torus viewed
as a grope of class 2 (with the bottom univalent vertex designated as the root). For integers
$n\geq 1$, define $Y^{(n+1)}$ to be the rooted product $Y^n\ast Y^n$ (see
Figure~\ref{Y-trees-fig}). A grope $g$ is {\em symmetric} and has {\em height} $n$ if all the trees
in $t(g)$ are of the form $Y^n$. For $n\geq 1$, define the tree $Y^{(n.5)}$ to be the rooted
product $Y^{(n-1)}\ast Y^n$ (where $Y^0$ is the rooted chord). A grope $g$ has {\em height} $n.5$
if all the trees in $t(g)$ are of the form $Y^{(n.5)}$. Note that a grope of height $n$ (resp.
$n.5$) has class $2^n$ (resp. $2^n+2^{(n-1)}=(1.5)(2^n)$).

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.75]{Y-trees-fig.eps}}
        \caption{From left to right: The trees $Y^1$, $Y^2$, $Y^3$ and
        (a punctured) $Y^n$.}
        \label{Y-trees-fig}

\end{figure}

\subsection{The height of a Whitney
tower}\label{w-tower-height-def} Translating the definition of height given in \cite{COT} into our
language we have: An order~$(2^n-2)$ Whitney tower $T$ has {\em height} $n$ ($n\geq 1$) if the
interiors of all Whitney disks in $T$ only intersect surfaces of the same order. Thus, the lowest
order unpaired intersections in a Whitney tower of height $n$ are of order
$(2^{(n-1)}-1)+(2^{(n-1)}-1)+1=2^n-1$ and occur among the highest order Whitney disks (of order
$2^{(n-1)}-1$).

A Whitney tower of {\em height} $n.5$ ($n\geq 1$) is a Whitney tower of height $n$, together with
order $2^n-1$ Whitney disks pairing all order $2^n-1$ intersection points with the requirement that
the interiors of these order $2^n-1$ Whitney disks may intersect each other and the order
$2^{(n-1)}-1$ Whitney disks but are disjoint from all surfaces of lower order. A Whitney tower of
height $n.5$ has order $2^n+2^{(n-1)}-1$.

\begin{proof}(of Corollary~\ref{cor:height}) Applying
Theorem~\ref{grope-to-tower-thm} of Section~\ref{sec:thms} to a disk-like grope $g$ of height $n$
(resp. $n.5$) yields a Whitney tower $T$ of order $2^n-1$ (resp. $2^{(n-1)}+2^n-1$) with $t(T)$
isomorphic to $t(g)$ so that all the connected trees in $t(T)$ are of the form $Y^n$ (resp.
$Y^{(n.5)}$). After splitting $T$ (Lemma~\ref{split-tower-lem}) and applying
Lemma~\ref{subtower-lemma}, we may arrange that the punctured edge in each punctured tree
$t^{\circ}(p)$ in $t^{\circ} (T)$ is adjacent to (what was) a root vertex, that is, the only
unpaired intersection points of $T$ occur between the order zero 2--disk and Whitney disks whose
associated trees are of the form $Y^n$ (resp. $Y^{(n.5)}$) as in the far right of
Figure~\ref{Y-trees-fig}. The Whitney disks of $T$ correspond to the trivalent vertices of $t(T)$
and one can check by examining the shape of the $Y^n$ trees which make up $t^{\circ} (T)$ that $T$
satisfies the above definition of height $n$ (resp. $n.5$). In fact, $T$ satisfies the stronger
condition that its intersections of order $2^n-1$ (between Whitney disks of order $2^{(n-1)}-1$),
which are allowed to be unpaired in \cite{COT}, are in fact all paired by order $2^n-1$ Whitney
disks (each of which corresponds to the trivalent vertex adjacent to the root of a $Y^n$). This is
as expected by Theorem~\ref{class-order-thm} since $T$ should have order $2^n-1$. However, these
order $2^n-1$ Whitney disks intersect the order zero 2--disk so that $T$ does {\em not} have height
$n+1$. The case of half-integer height $n.5$ is checked similarly.
\end{proof}







%======================================================================


\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.75]{simple-trees-fig.eps}}
        \caption{From left to right: The non-simple tree of lowest degree (degree
        5) and the simple trees of degree 5, 6
        and $7+n$.}
        \label{simple-trees-fig}

\end{figure}

\section{Proof of
Corollary~\ref{cor:half-grope} and the Whitney move IHX construction}\label{sec:half-grope-cor}
This section contains a proof of Corollary~\ref{cor:half-grope} which is based on a geometric
realization of the IHX Jacobi relation in the setting of Whitney towers (Lemma~\ref{IHX-lemma}
below).

A dyadic (capped) $A$--like grope $g$ is a {\em half-grope} if all the trees in $t(g)$ are {\em
simple} (right- or left-normed) as illustrated in Figure~\ref{simple-trees-fig}; note that the
roots (shown pointing down in the figure) are required to be at an ``end'' of the tree.
\begin{proof} %(of Corollary~\ref{cor:half-grope})
To prove Corollary~\ref{cor:half-grope}, choose caps for the class $n$ $g_i$ (which we may assume
are dyadic) and use Theorem~\ref{grope-to-tower-thm} of Section~\ref{sec:thms} to convert the
$g_i^c$ into a Whitney tower $T$ of order $n-1$. By the following
Proposition~\ref{simple-tower-prop}, $T$ can be modified (rel boundary) to a Whitney tower $T'$ (of
the same order) with $t(T')$ consisting of only simple trees. Then converting $T'$ back into a
grope via Theorem~\ref{tower-to-grope-thm} of Section~\ref{sec:thms} yields the desired half gropes
$h_i$ in $X$ bounded by $\gamma_i$. (Here we are using (v) of Theorem~\ref{tower-to-grope-thm} to
send chosen univalent ``end vertices'' to roots, while preserving trees.)
\end{proof}

\begin{prop}\label{simple-tower-prop}
Let $T$ be any order $n$ Whitney tower on order 0 surfaces $A_i$. Then, after a regular homotopy
(rel $\partial$), the $A_i$ admit an order $n$ {\em simple} Whitney tower $T'$ contained in a
neighborhood of $T$, that is, $t(T')$ consists of only simple trees.
\end{prop}
The proof of Proposition~\ref{simple-tower-prop} uses the geometric IHX Lemma~\ref{IHX-lemma} below
to follow the algebraic proof that the usual group of unitrivalent trees occurring in finite type
theory is spanned by simple trees as given in e.g., \cite {B2}, \cite{CT1}:
\begin{proof}The simple trees are characterized by the property that any
maximal length chain of edges contains every trivalent vertex. Let $t(p)=t(T_p)\in t(T)$ be a tree
associated to a split subtower $T_p\subset T$ contained in a Whitney tower $T$ (which we may
assume is split by Lemma~\ref{split-tower-lem}). If $t(p)$ contains a trivalent vertex $v_1$ which
is of distance 1 away from a trivalent vertex $v_0$ contained in some maximal chain of edges not
containing $v_1$, then we may assume, by Lemma~\ref{subtower-lemma}, that $v_0$ corresponds to
$W_{(I,J)}$ and $v_1$ corresponds to $W_{((I,J),K)}$ which is incident to some other sheet $W_L$ in
$T_p$ as in left hand side of Figure~\ref{IHX-trees-fig}. Lemma~\ref{IHX-lemma} below shows how to
modify $T$ near $T_p$ so as to replace $t(p)\in t(T)$ by the two trees on the righthand side of
Figure~\ref{IHX-trees-fig} having the same degree as $t(p)$ but with longer length edge chains. By
iterating this modification (for all components of $t(T)$) we eventually arrive at the desired $T'$
with all components of $t(T')$ simple trees.
\end{proof}
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.55]{IHX-trees-fig.eps}}
        \caption{The IHX relation for Whitney disks in a split subtower
        replaces a split subtower whose tree looks locally like the one on the left
        with a pair of nearby disjoint split subtowers whose trees
        look locally like the trees on the right.}
        \label{IHX-trees-fig}

\end{figure}
\begin{lem}[Geometric IHX--Whitney move version]\label{IHX-lemma}
Let $T_p$ be a split subtower in a split Whitney tower $T$. Let $W_{((I,J),K)}$ be a Whitney disk
in $T_p$ so that $t(T_p)$ looks locally like the leftmost tree in Figure~\ref{IHX-trees-fig}. Then
$T$ can be modified in a regular neighborhood $\nu(T_p)$ of $T_p$ yielding a split Whitney tower
$T'$, on the same order 0 surfaces, with $T_p$ replaced by disjoint split subtowers $T_{p'}$ and
$T_{p''}$ contained in $\nu(T_p)$ such that the trees $t(T_{p'})$ and $t(T_{p''})$ are as pictured
on the right hand side of Figure~\ref{IHX-trees-fig}.
\end{lem}
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{IHX-W-move1and2-fig.eps}}
        \caption{The IHX construction starts with a $W_{(I,J)}$ Whitney move
        on $W_I$. Note that intersections between $W_{((I,J),K)}$ and $W_L$ are {\em not} shown in this
        figure (and are suppressed in subsequent figures as well).}
        \label{IHX-W-move1and2-fig}

\end{figure}
The modification involves Whitney moves, finger moves and taking parallel copies of some of the
Whitney disks in $T_p$. The reader familiar with the orientation and sign conventions of \cite{ST2}
can check by inserting signs and orientations in the figures that the following construction
actually replaces an ``I'' tree with the difference ``$\mathrm{H}-\mathrm{X}$'' as in the usual IHX
relation of finite type theory. Note that this differs from the closely related 4--dimensional IHX
construction in \cite{CST} which creates the trees $\mathrm{I}-\mathrm{H}+\mathrm{X}$ for a Whitney
tower on 2--spheres in 4--space by modifying the boundaries of Whitney disks.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{IHX-transfer-move1and1A-fig.eps}}
        \caption{The intersection point $W_J\cap W_{(I,K)}'$ is
        `transferred' via a finger move (top)
        to create a cancelling pair $W_J\cap
W_{(I,K)}$ paired by $W_{(J,(I,K))}$ at the cost of also creating $W_J\cap W_K$ paired by
$W_{(J,K)}$ and $W_I\cap W_{(J,K)}$ paired by $W_{(I,(J,K))}$ (bottom).}
        \label{IHX-transfer-move1and1A-fig}

\end{figure}
\begin{proof}
The first step in the modification is to do the $W_{(I,J)}$ Whitney move on $W_I$ (see
Figure~\ref{IHX-W-move1and2-fig}) and disregard, for the moment, the Whitney disks in the part of
$T$ corresponding to the sub-tree $L$. This eliminates the cancelling pair of intersections between
$W_I$ and $W_J$ at the cost of creating two cancelling pairs of intersections between $W_I$ and
$W_K$ which we pair by Whitney disks $W_{(I,K)}$ and $W_{(I,K)}'$ as illustrated in
Figure~\ref{IHX-W-move1and2-fig}. The new Whitney disks $W_{(I,K)}$ and $W_{(I,K)}'$ each have a
single interior intersection with $W_J$ and the next step is to ``transfer'' (as illustrated in the
upper part of Figure~\ref{IHX-transfer-move1and1A-fig}) the intersection point $W_J\cap W_{(I,K)}'$
to create a cancelling pair $W_J\cap W_{(I,K)}$ paired by $W_{(J,(I,K))}$ at the cost of also
creating $W_J\cap W_K$ paired by $W_{(J,K)}$ and $W_I\cap W_{(J,K)}$ paired by $W_{(I,(J,K))}$ (as
illustrated in the lower part of Figure~\ref{IHX-transfer-move1and1A-fig}). Note that
Figure~\ref{IHX-transfer-move1and1A-fig} differs from Figure~\ref{IHX-W-move1and2-fig} by a
rotation of coordinates which brings the sheet of $W_K$ into the ``present'' slice of 3--space.

\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{IHX-transfer-move2and3-fig.eps}}
        \caption{The transferring finger move is guided by an arc $a$ (top)
        which can be taken to run along what used to be the part of the boundary arc of
        $W_{((I,J),K)}$ lying in $W_K$ (bottom). Indicated in the bottom picture is where
         $W_{((I,J),K)}$ used to be. }
        \label{IHX-transfer-move2and3-fig}

\end{figure}
This transfer move was described in \cite{Y} (see also \cite{ST1} and \cite{S3}) and is just a
(non-generic) finger move applied to $W_J$. The important thing to note here is that the finger
move is guided by an arc $a$ (see Figure~\ref{IHX-transfer-move2and3-fig}) from $\partial
W_{(I,K)}'$ to $\partial W_{(I,K)}$ in $W_K$ and we can take this arc to run along what used to be
the part of $\partial W_{((I,J),K)}$ lying in $W_K$. This is illustrated in the lower part of
Figure~\ref{IHX-transfer-move2and3-fig} which gives a better picture of the situation before the
finger move is applied.
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{IHX-transfer-move3B-fig.eps}}
        \caption{Before the transfer move: New Whitney disks $W_{(I,(J,K))}$
        and $W_{(J,(I,K))}$, whose boundaries are the unions of arcs $a'\cup b'$
        and $a''\cup b''$ (see also Figure
        ~\ref{IHX-W-move1and2B-fig}),
         will be created from parallel copies of the old $W_{((I,J),K)}$.}
        \label{IHX-transfer-move3B-fig}
\end{figure}
\begin{figure}[ht!]
        \centerline{\includegraphics[scale=.6]{IHX-W-move1and2B-fig.eps}}
        \caption{Applying the transfer move to the right-hand side will create
        new Whitney disks $W_{(I,(J,K))}$
        and $W_{(J,(I,K))}$, whose boundaries are the unions of arcs $a'\cup b'$
        and $a''\cup b''$ (see also Figure
        ~\ref{IHX-transfer-move3B-fig}),
         from parallel copies of the old $W_{((I,J),K)}$ shown on the left.}
        \label{IHX-W-move1and2B-fig}

\end{figure}

The Whitney disks $W_{(I,(J,K))}$ and $W_{(J,(I,K))}$ can be taken to be parallel copies of the old
$W_{((I,J),K)}$ as follows: The boundary of $W_{(I,(J,K))}$ (resp. $W_{(J,(I,K))}$) consists of
arcs $a'$ and $b'$ (resp. $a''$ and $b''$) where $a'$ and $a''$ are tangential push-offs of $a$ in
$W_K$ and $b'$ and $b''$ are normal push-offs of what was the boundary arc $b$ of $W_{((I,J),K)}$
in $W_{(I,J)}$. This is shown in both Figure~\ref{IHX-transfer-move3B-fig} and
Figure~\ref{IHX-W-move1and2B-fig}, where again it is easier to picture things before the
transferring finger move. Since $W_{((I,J),K)}$ was framed and embedded, $W_{(I,(J,K))}$ and
$W_{(J,(I,K))}$ can be formed from two disjoint parallel copies of $W_{((I,J),K)}$ which each
intersect $W_L$ as $W_{((I,J),K)}$ did. Two parallel copies of each Whitney disk that was in the
part of $T_p$ corresponding to $L$ can be used to recover the order of the original Whitney tower
(by pairing the new intersections ``over'' $W_{(I,(J,K))}$ and $W_{(J,(I,K))}$ corresponding to
$L$), which means that exactly two new unpaired intersection points $p'$ and $p''$ have been
created with corresponding trees $t(p')$ and $t(p'')$ as shown locally in the right hand side of
Figure~\ref{IHX-trees-fig}. After the transferring finger move, the $W'_{(I,K)}$ Whitney move can
be done (on either sheet) without affecting anything else. Finally, $W_I$, $W_J$ and $W_K$ will
need to be split since they now each contain two boundary arcs of Whitney disks. Splitting $W_I$,
$W_J$ and $W_K$ down into the lower order Whitney disks (as in Lemma~\ref{split-tower-lem}) yields
the two split subtowers $T_{p'}$ and $T_{p''}$.

\end{proof}






%====================================================================


\section{Proof of Corollary~\ref{cor:k-slice}\label{sec:k-slice}}
In this section the main theorems together with the geometric IHX construction of the previous
section are used to prove Corollary~\ref{cor:k-slice} in the introduction. We refer the reader to
\cite{C1} or \cite{IO} for the formal definition of {\em $k$-null-cobordism} or {\em $k$-slice}. It
is enough to show that the link components in $S^3=\partial B^4$ bound disjointly embedded surfaces
$A_i$ such that each $A_i$ contains a symplectic basis of circles which bound continuous maps of
class $k$ gropes in $B^4\setminus \bigcup_iA_i$. In fact, we will find such $A_i$ with symplectic
bases of circles bounding {\em embedded} class $k$ gropes in $B^4\setminus \bigcup_iA_i$. The same
proof shows that class $2k$ grope concordant links are $k$-cobordant.
\begin{figure}[h]
        \centerline{\includegraphics[scale=.6]{k-slice-tree-fig.eps}}
        \caption{A simple tree associated to an order $2k$ (or greater)
        intersection point in an order $2k-1$ Whitney tower $T$ with
        a preferred univalent vertex
        (at least) $k$ trivalent vertices away from both ends. Using
        Theorem~\ref{tower-to-grope-thm} of Section~\ref{sec:thms}
        to convert $T$ to disjoint gropes
        $g_i$ with all such preferred univalent vertices going to roots in $t(g_i)$
        yields $k$-slicing surfaces (the bottom stages of the $g_i$).}
        \label{k-slice-tree-fig}
\end{figure}
\begin{proof}
Apply Krushkal's grope-splitting procedure \cite{Kr} and Theorem~\ref{grope-to-tower-thm} of
Section~\ref{sec:thms} to the class $2k$ (disk-like) gropes (as in the hypotheses of
Corollary~\ref{cor:k-slice}) to get an order $2k-1$ Whitney tower on immersed 2--disks bounded by
the link components in $B^4$. Then apply Proposition~\ref{simple-tower-prop} to get a Whitney tower
$T$ with $t(T)$ consisting of only simple trees. Since the order of $T$ is $2k-1$, the Vassiliev
degree of each tree in $t(T)$ is at least $2k$. This means that we can specify a preferred
univalent vertex in each simple tree in $t(T)$ such that the trivalent vertex adjacent to the
preferred univalent vertex is at least $k$ trivalent vertices away from both ends of the simple
tree, as illustrated in Figure~\ref{k-slice-tree-fig}. Now use Theorem~\ref{tower-to-grope-thm} of
Section~\ref{sec:thms} to convert back to disjointly embedded (disk-like) gropes $g_i$ with the
preferred univalent vertices in $t(T)$ going to the root vertices in the $t(g_i)$. As is seen in
the shape of the trees $t(g_i)$ (Figure~\ref{k-slice-tree-fig}), the desired $A_i$ are the class 2
sub-gropes formed by the bottom (0th and 1st) stages of the $g_i$ and the higher ($\geq 2$) stages
form class $n-1$ gropes which are attached to symplectic bases on the $A_i$ and disjointly embedded
in $B^4\setminus \bigcup_iA_i$.
\end{proof}




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






\begin{thebibliography}{HS1}

%\bibitem{B1} D. Bar-Natan. { \em On the Vassiliev knot invariants.
%}\\ Topology 34 (1995) 423--472.

\bibitem{B2} D. Bar-Natan. { \em Vassiliev homotopy string link invariants.
}\\ J. Knot Theory Ramifications 4 (1995) 13--32.

\bibitem{C1} T. Cochran. { \em $k$-cobordism for links in $S^3$. }\\
Trans. A.M.S. Vol. 327 No. 2 (1991) 641--654.

\bibitem{C2} T. Cochran. { \em Geometric invariants of link cobordism. }\\
Comment. Math. Helv. Vol. 60 (1985) 291--311.

\bibitem{C3} T. Cochran. { \em Derivatives of links, Milnor's concordance invariants
and Massey products. }\\
Mem. Amer. Math. Soc. Vol. 84 No. 427 (1990).

\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 (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 (2002).


\bibitem{F}  M. H. Freedman. { \em The topology of four-dimensional manifolds.}\\
J. Diff. Geom. 17 (1982) 357--453.


\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{IO} K. Igusa and Kent Orr. { \em Links, pictures and the
homology of nilpotent groups. }\\ Topology 40 (2001) 1125--1166.



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

\bibitem{KQ}  V. Krushkal and Frank 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 Combinatoral group theory},
Dover Publications Inc. (1976).


\bibitem{O1} K. Orr. { \em Homotopy invariants of links. }\\
Invent. Math. Vol. 95 (1989) 379--394.

\bibitem{Q}  Frank Quinn. { \em Embedding towers in 4--manifolds.}
\\ Four--manifold Theory AMS Contemporary Math 35 (1984) 461--472.

\bibitem{S3} R. Schneiderman { \em Simple Whitney towers, half-gropes and the Arf invariant of a
knot.}\\ Preprint (2002).


\bibitem{ST1} R. Schneiderman and Peter 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 Peter Teichner.
{ \em Whitney towers, Milnor invariants and the Kontsevich integral.}\\
In preparation.


\bibitem{ST3} R. Schneiderman and Peter Teichner.
{ \em Pulling apart 2--spheres in 4--manifolds.}\\
In preparation.
\bibitem{ST4} R. Schneiderman and P. Teichner. {\em Grope concordance of classical
links.}\\
In preparation.



\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}

math0310303.extracted_bib
\begin{thebibliography}{HS1}

%\bibitem{B1} D. Bar-Natan. { \em On the Vassiliev knot invariants.
%}\\ Topology 34 (1995) 423--472.

\bibitem{B2} D. Bar-Natan. { \em Vassiliev homotopy string link invariants.
}\\ J. Knot Theory Ramifications 4 (1995) 13--32.

\bibitem{C1} T. Cochran. { \em $k$-cobordism for links in $S^3$. }\\
Trans. A.M.S. Vol. 327 No. 2 (1991) 641--654.

\bibitem{C2} T. Cochran. { \em Geometric invariants of link cobordism. }\\
Comment. Math. Helv. Vol. 60 (1985) 291--311.

\bibitem{C3} T. Cochran. { \em Derivatives of links, Milnor's concordance invariants
and Massey products. }\\
Mem. Amer. Math. Soc. Vol. 84 No. 427 (1990).

\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 (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 (2002).


\bibitem{F}  M. H. Freedman. { \em The topology of four-dimensional manifolds.}\\
J. Diff. Geom. 17 (1982) 357--453.


\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{IO} K. Igusa and Kent Orr. { \em Links, pictures and the
homology of nilpotent groups. }\\ Topology 40 (2001) 1125--1166.



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

\bibitem{KQ}  V. Krushkal and Frank 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 Combinatoral group theory},
Dover Publications Inc. (1976).


\bibitem{O1} K. Orr. { \em Homotopy invariants of links. }\\
Invent. Math. Vol. 95 (1989) 379--394.

\bibitem{Q}  Frank Quinn. { \em Embedding towers in 4--manifolds.}
\\ Four--manifold Theory AMS Contemporary Math 35 (1984) 461--472.

\bibitem{S3} R. Schneiderman { \em Simple Whitney towers, half-gropes and the Arf invariant of a
knot.}\\ Preprint (2002).


\bibitem{ST1} R. Schneiderman and Peter 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 Peter Teichner.
{ \em Whitney towers, Milnor invariants and the Kontsevich integral.}\\
In preparation.


\bibitem{ST3} R. Schneiderman and Peter Teichner.
{ \em Pulling apart 2--spheres in 4--manifolds.}\\
In preparation.
\bibitem{ST4} R. Schneiderman and P. Teichner. {\em Grope concordance of classical
links.}\\
In preparation.



\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}