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\topmatter
\title Classical Link Invariants and the Hawaiian Earrings Space
\endtitle
\author Tim D. Cochran\endauthor
\affil \eightpoint{Mathematics Department, Rice University}\\
\eightpoint{P.O. Box 1892, Houston, Texas 77251--1892}
\\
\\
\\
\\
\endaffil
\abstract We show that, in search of link invariants more
discriminating than Milnor's $\ov\mu$-invariants, one is
{\it naturally\/} led to consider seemingly pathological objects
such as links with an infinite number of components and the join of
an infinite number of circles (Hawaiian earrings space). We define
an infinite homology boundary link, and show that any finite
sublink of an infinite homology boundary link has vanishing
Milnor's invariants. Moreover, all links known to have vanishing
Milnor's invariants are finite sublinks of infinite homology
boundary links. We show that the exterior of an infinite homology
boundary link admits a map to the Hawaiian earrings space, and that
this may be employed to get a factorization of K.E\.~Orr's
omega-invariant through a rather simple space.
\bpage
\noindent Keywords: link, Milnor's invariants, Massey products,
boundary link, homology boundary link, concordance
\endabstract
\endtopmatter
\document
\baselineskip=13pt
\sub{Chapter 1}
\mpage
This research was motivated by the desire to classify links of
circles in $S^3$ (or $\BR^3$) up to concordance. Recall that a
{\it link\/} $L=\{K_1,\dots, K_m\}$ of $m$-components is an ordered
collection of disjointly embedded, oriented, smooth submanifolds of
$S^3$ where each $K_i$ is homeomorphic to $S^1$. The links $L_0$
and $L_1$ are {\it concordant\/} if there exists a disjoint
collection of oriented annuli $\coprod\lm^m_{i=1}S^1\x[0,1]$
embedded as a smooth submanifold of $S^3\x[0,1]$ such that the
image of $\coprod\lm^m_{i=1}S^1\x\{0\}$ is $L_0$ and
$\coprod\lm^m_{i=1}S^1\x\{1\}$ is $L_1$. The most powerful
invariants of link concordance which do not depend strongly on the
knot type of the individual components (and hence are more purely
invariants of {\it linking\/}) are {\it Milnor's\/}
$\ov\mu${\it -invariants\/} or, equivalently,
{\it Massey products\/} in the link exterior (\cite{16} \cite{12},
see \cite{20} for a precise statement of the equivalence,
\cite{13}). These have also been related to linking numbers among
derived links \cite{4}. The driving question in this field has
been and is, ``if Milnor's $\ov\mu$-invariants vanish for $L$, what
can be said of the concordance class of $L$?'' Milnor's invariants
are now known to vanish for {\it boundary links\/} and more
generally for sublinks of {\it homology boundary links\/} \cite{12}
\cite{4, Thm\.~8.10} \cite{7} \cite{5}. Recall that $L$ is a
{\it boundary link\/} if there exists a collection
$\{V_1,\dots,V_m\}$ of compact, oriented surfaces (called Seifert
surfaces) disjointly embedded in the {\it link exterior\/}
$E(L)\equiv\ov{S^3-(L\x D^2)}$, such that $\p V_i$ is
$K_i\x\{*\}\sbq L\x\p D^2$ (a {\it longitude\/} of $K_i$). An
{\it homology boundary link\/} is defined similarly except that
$\p V_i$ is allowed to be any disjoint union of longitudes of
components of $K$ such that $[\p V_i]$ in $H_1(\p E(L))$ is equal
to $[K_i\x\{*\}]$, that is to say that $\p V_i$ is merely
``algebraically'' equal to a longitude of $K_i$. A construction of
Pontryagin yields equivalent definitions which shall be crucial to
the theme of our paper. By choosing tubular neighborhoods of the
$V_i$, one obtains a map from $E(L)$ to $\bigvee\lm^m_{i=1}S^1$
by sending each $[-1,1]$ fiber of the $i^{\text{th}}$ tubular
neighborhood around the $i^{\text{th}}$ circle and all other points
to the wedge point, In fact, $L$ is a boundary link (respectively
homology boundary link) if and only if there is a continuous map
$f: E(L)\to\bigvee\lm^m_{i=1}S^1$ which sends the $i^{\text{th}}$
meridian to the $i^{\text{th}}$ circle (respectively to a loop
homologous to the $i^{\text{th}}$ circle) \cite{21}.
In the present paper, we will show that, in search of invariants
more discriminating than Massey products, and in search of new
classes of links with vanishing Massey products, one is
{\it naturally\/} led to consider infinite analogues of the above
objects which might in other contexts be seen as ``pathological''.
In particular we define an {\it homology boundary link with an
infinite number of components\/} and show that any finite sublink
of an infinite homology boundary link has vanishing Milnor's
invariants (Thm\.~2.6). Conversely, we show that a link which
satisfies a geometric condition which {\it seems\/} to be (but may
not be) equivalent to the vanishing of the $\ov\mu$-invariants, is
indeed a sublink of an infinite homology boundary link.
Building on this work, Kent Orr has announced such a converse which
assumed only the vanishing of the $\ov\mu$-invariants. However his
definition of an infinite homology boundary link seems to be weaker
than ours. Since this work has not yet appeared, the status of a
complete converse is, in any case, in doubt.
We propose that this class, sublinks of infinite homology boundary
links, is interesting and potentially new. Unfortunately, as of
this writing, we do not know if this class is strictly larger than
the class of sublinks of finite homology boundary links!
Since the present work, the author and Kent Orr have shown that
most sublinks of (finite) homology boundary links are not
concordant to any boundary link \cite{8, 9}. Therefore the
pressing question has become ``If Milnor's invariants vanish for
$L$, is $L$ concordant to a sublink of an homology boundary link
(or even an homology boundary link)?'' Indeed it may be that
$\ov\mu(L)=0$ if and only if $L$ {\it is\/} a sublink of an
homology boundary link! Consequently our proposed new class of links
becomes even more relevant.
Moreover, to complete the analogy of the infinite case to the
finite case, we show that if $L_\infty$ is an infinite homology
boundary link then there is an induced map from $E(L_\infty)$ to
the countably infinite wedge of circles with the ``Hawaiian
Earrings'' topology, i.e., to
$H=\{(x,y)\in\BR^2\mid(x-2^{-n})^2+y^2=(2^{-n})^2n\in\BZ^+\cup\{0\}
\}$ in the subspace topology.
Finally, we will show that this line of attack may shed some light
on other existing invariants. In particular, Kent Orr has defined,
for any based link $(L,b)$ which has vanishing $\ov\mu$-invariants,
an element $\th_\om(L,b)\in\pi_3(\ov K_\om)$ where $\ov K_\om$ is the
cofiber of the map of classifying spaces $K(F,1)\to K(F^\land,1)$
induced by the natural embedding $F\to F^\land$ of the free group
on $m$-letters in its nilpotent completion \cite{18, 19}. This is
invariant under a ``based concordance''. It is of great interest
therefore to decide whether or not $\ov\mu=0$ implies $\th_\om=0$.
Activity as regards this important question has lanquished due to
the seeming difficulty of analyzing $\pi_3(\ov K_\om)$ and to the
total lack of possible counter-examples (sublinks of homology
boundary links also have vanishing $\th_\om$ \cite{5;
Corollary~2.5}) (however see \cite{14} for important progress). The
class of sublinks of infinite homology boundary links may provide
such counter-examples. We shall show that if $L$ is a sublink of an
infinite homology boundary link $L_\infty$, then
$\th_\om(L): S^3\to\ov K_\om$ factors through a ``simpler'' space
which contains $H$ and which is the inverse limit of finite
$2$-complexes each of which is homotopy-equivalent to a wedge of
$2$-spheres. This space is essentially combinatorially defined from
$H$ using the data of which longitudes $\p V_i$ contains (compare
the ``pattern'' of \cite{7}).
It is our point of view that these spaces should be viewed as
constituting part of the $2$-skeleton of the mysterious space
$\ov K_\om$. If this class of spaces could be shown to have $\pi_3=0$
then $\th_\om$ would vanish for all sublinks of infinite homology
boundary links. On the other hand, if even one of these ``simpler''
spaces is (constructively) shown to have non-zero $\pi_3$ then this
might lead to its realization by a link $L$, shedding light on the
possible gap between $\th_\om$ and the $\ov\mu$-invariants. We
remark that it is known that $\pi_2(\ov K_\om)$ is uncountable making
it likely that $\pi_3(\ov K_\om)$ is uncountable but that many
elements are not realizable as $\th_\om$ of a link \cite{2,
Prop.~4.4}.
\vskip.5cm
\sub{Chapter 2}
\mpage
\definition{Definition 2.1} {\it A link $L$ of $J$-components
in\/} $S^3$ is an indexed collection $(\SL)$ such that:
\roster
\item"a)" for each $\a\in J$, $\SL_\a$ is a pair
$(L_\a, \phi_\a)$ where $\phi_\a$ is a smooth
orientation-preserving embedding of $S^1\x D^2$ into $S^3$,
$\phi_\a(S^1\x\{0\})=K_\a$ (with orientation),
$\phi_\a(S^1\x\{1\})=l_\a$ an oriented longitude of the knot $K_\a$;
and
\item"b)" if $\a\neq\b\ \ \im\phi_\a\cap\im\phi_\b$ is empty.
\endroster
\enddefinition
\bpage
We also allow the degenerate case that $K_\a$ is empty. The
{\it exterior\/} $E(L)$ of the link $L$ is the complement in $S^3$
of $\bigcup\lm_\a\phi_\a(S^1\x\ ^2)$. Then $E(L)$ is a compact
metric space but is {\it not\/} a manifold at cluster points of
$\bigcup\lm_\a\phi_\a(S^1\x S^1)$.
\bpage
\definition{Definition 2.2} An infinite homology boundary link
$L_\infty$ is a link $\{K_n\}$ indexed by the positive integers,
together with a collection $V=\{V_n\bigm| n\in\BZ_+\}$ of connected,
oriented, compact $2$-dimensional submanifolds of $S^3$ with tubular
neighborhoods $\Psi_n: V_n\x[0,1]\hookrightarrow S^3$ whose images
lie in $E(L_\infty)$ and are pairwise disjoint, such that:
\roster
\item"1)" $\Psi_n(\p V_n\x[0,1])\subset\p E(L_\infty)$
\item"2)" $\p V_i$ is a finite union of longitudes of the $\{K_n\}$,
\item"3)" $V_i$ is a {\bf generalized Seifert surface} for $K_i$,
$i=1,\dots,m$ that is, $\p V(x_i)$ is a union of longitudes
which is homologous in $H_1(\p E(L))$ to a longitude of $K_i$.
\endroster
\enddefinition
\bpage
We are obliged to convince the reader that links with an infinite
number of components, in particular infinite homology boundary
links, arise quite naturally from consideration of Massey products
in the exterior of ordinary (finite) links. We assume the reader is
acquainted with Massey products \cite{13, 4}. Recall that the
vanishing of {\it all\/} Massey products for the space $E(L)$
guarantees the existence of an infinite indexed collection of
$1$-cochains whose coboundaries are related, in a specific
inductive fashion, to their cup-products. By Poincar\'e duality,
there ought then to exist an infinite collection of surfaces whose
boundaries are related, in an inductive fashion, to their
intersections. As a trivial example, if the linking number
$lk(K_1,K_2)=0$ then there exist Seifert surfaces $V_1$, $V_2$ for
$K_1$, $K_2$ in $E(K_1\cup K_2)$. Then $V_1\cap V_2$ is a framed
one manifold called say $K_{(1,2)}$ and the length 4 Massey
products vanish iff $lk\(K_{(1,2)}, K^+_{(1,2)}\)=0$ which is to say
$K_{(1,2)}=\p V_{(1,2)}$ where $V_{(1,2)}$ lies in
$E\(K_1\cup K_2\cup K_{(1,2)}\)$. In \cite{4} the philosophy was
espoused that, if these {\it surface systems\/} were assumed to be
``nicely'' embedded then much intuition could be gained. It was
shown that existence of such a system of surfaces implied the
vanishing of the Massey products (but is a priori stronger because
of the conditions imposed on the surface systems).
We formalize the latter assumptions below and retain the
nomenclature Technical Condition (TC) which was introduced in
\cite{6; p\.~130}. Here $L=\{K_1,\dots,K_m\}$. A
$1$~{\it bracket\/} is one of the letters $\{x_1,\dots,x_m\}$. A
$n$~{\it -bracket\/} is a symbol $[b_1,b_2]$ where $b_1$ is an
$i$-bracket and $b_2$ is a $j$-bracket with $i+j=n$. These brackets
are used to index the infinite collection of surfaces in $E(L)$.
\bpage
\sub{Technical Condition on $L$ (TC)} There is a set $\BV$ of
compact, connected, oriented (possibly empty) surfaces in $E(L)$,
indexed by the set of brackets of positive weight such that:
\roster
\item"(1)" $V(x_i)$ is a generalized Seifert surface for $K_i$.
\item"(2)" $V(b_1)$ and $V(b_2)$ are distinct and meet transversely
unless they coincide modulo orientation.
\item"(3)" $\p V([b_1,b_2])$ is $V(b_1)\cap V(b_2)$ which is
{\it connected\/} and oriented according to a convention not
relevant to our present discussions. If $V(b_1)$ and $V(b_2)$
coincide, we understand that their intersection is empty.
\item"(4)" $V(b_1)\cap V(b_2)\cap V(b_3)$ is empty unless, for some
permutation $(i,j,k)$ of $(1,2,3)$ $\p V(b_k)$ is a component of
the intersection of $V(b_i)$ and $V(b_j)$.
\endroster
For our current study we must also impose a further condition 5),
that stipulates that distinct surfaces have tubular neighborhoods
which are disjoint except where they meet transversely. This
implies, for example, that the subspace of all circles $\{\p V(b)\}$
has the topology of a disjoint union of circles.
\bpage
\proclaim{Theorem 2.3} If $L$ satisfies TC above then $L$ is a
sublink of an infinite homology boundary link $L_\infty$.
\endproclaim
\vskip.5cm
\sub{Proof of 2.3} Index the set of unoriented surfaces $V(b)$, by
the positive integers in a way that is compatible with increasing
bracket length. Then define $L_\infty$ by setting $\SL(b_i)=K_i$
$i=1,\dots,m$ and $\SL([b_1,b_2])=\p V(b_1)\cap\p V(b_2)$. Since
$V([b_1,b_2])$ is a Seifert surface for $\SL([b_1,b_2])$, it follows
that $L_\infty$ is an infinite homology boundary link. Thus $L$ is a
sublink of a link which is ``grown'' from $L$ by intersecting
Seifert surfaces $V_i\cap V_j$ to get new link components and
proceeding {\it ad infinitum\/}. In \cite{6, pp\.~131--132} it was
shown that if this procedure ever terminates or even repeats in the
sense that $L_\infty$ is merely an infinite union of parallels of a
finite sublink $L'$ of $L_\infty$, then $L'$ is an homology boundary
link and hence $L$ is a sublink of a {\it finite} homology boundary
link. The only links known to have vanishing Massey products,
namely ``fusions of boundary links'' \cite{4, 8.10} do indeed
admit surface systems with this self-referencing behavior
\cite{6, Thm\.~2.4}.
\midinsert
\vspace{3.3in}
\botcaption{Figure 2.4}\endcaption
\endinsert
Since this self-referencing behavior for fusions of boundary links
has not been shown in print, let us illustrate with the
$2$~component ribbon link $L=\{K_1,K_2\}$ (a fusion of a
$3$~component trivial link) in Figure~2.4. The Seifert surfaces
$V(x)$, $V(y)$ for $L$ are shown in 2.4. Their intersection $K_3$ is
shown as a dashed circle in 2.4 and {\it its\/} Seifert surface
$V([x,y])$ is shown in 2.4. Then $V([x,y])\cap V(y)=K_4$ is shown
dashed in 2.4. It can be seen that $K_4$ and $K_3$ cobound an
annulus in the complement of interior $(V(x)\cup V(y)\cup V$
$([x,y]))$.
Therefore a Seifert surface for $K_4$ may be constructed using this
annulus and $V([x,y])$. Thus it is seen that this may be continued
indefinitely in such a way that $L_\infty$ is merely $\{K_1, K_2,
K_3\}$ together with parallels of $K_3$. For a general fusion of a
boundary link an identical situation holds: there exist Seifert
surfaces such that $L_\infty$ is
$L\cup$~\{parallels of meridians to bands\}. If $L'$ is $L$
together with one ``meridian to each band'' then it follows almost
by definition that $L'$ is an homology boundary link. For each
component $K'_i$ of $L'_i$ admits a generalized Seifert surface
$V'_i$ such that $V'_i\cap V'_j$ consists only of parallels of
$\{K'_i\mid i\in I\}$. Therefore a larger tubular neighborhood of
the $K'_l$ will engulf these intersections, leaving generalized
Seifert surfaces $V''_j$ which are pairwise disjoint (see
Figure~2.5)!
\midinsert
\vspace{1.2in}
\botcaption{Figure 2.5}\endcaption
\endinsert
Several remarks are in order about TC as it compares to what one
might expect to conclude from the hypothesis of vanishing Massey
products. First, condition 4) rules out triple points in the
surface systems. Indeed if all Massey products vanish one expects
to see triple points occuring in algebraically cancelling pairs,
but whether or not it is always possible to achieve this
geometrically is not clear (see Chapter~9 of \cite{4}). Similarly,
(3) requires that the intersections of the surfaces are connected.
This is a very inconvenient condition in practice. Even for the
simplest examples one wants to allow these intersections to be at
least a union of parallels of a fixed circle. If one attempts to
allow arbitrary disconnectedness, one gets into trouble very quickly
as will be seen in Example~2.7.
\bpage
\proclaim{Theorem 2.6} If $L$ is a finite sublink of an infinite
homology boundary link $L_\infty$, then the $\ov\mu$-invariants of
$L$ vanish.
\endproclaim
\vskip.5cm
\sub{Remarks} The theorem is false under several seemingly
reasonable weakenings of the hypothesis. For example it fails if
the ``components'' of an infinite homology boundary link are
allowed to be disconnected, that is if we allow $K_n$ to be a union
of circles. In fact a failure occurs even for such a finite link.
Suppose $L=\{K_1,\dots, K_5\}$ where $K_1$ to $K_4$ are shown in
Figure~2.7a. Then each
\midinsert
\vspace{1.5in}
\botcaption{Figure 2.7}\endcaption
\endinsert
\flushpar $K_i$, $i\le 4$ bounds a punctured torus $V_i$ as in 2.7b
such that $V_1\cap V_2$ is a circle $A$, $V_3\cap V_4$ is a circle
$B$ which link as shown in 2.7c. Let $K_5=A\cup B$, so $K_5$ bounds
an annulus disjoint from $\bigcup\lm^4_{i=1}V_i$. Then $L$ is an
``homology boundary link'' except that $K_5$ is disconnected. It is
easy to calculate that $\ov\mu(1212)=\pm1$ for $\{K_1, K_2\}$
which is a Whitehead Link.
More interestingly, the theorem is false if the surfaces of an
infinite homology boundary link are allowed to have an infinite
number of boundary components! Consider Figure~2.8 of a cross
section of a link with components labelled $\{K_1,K_2,\dots\}$ and
surfaces labelled $\{1,2,3,\dots\}$. The components converge to a
circle labelled $*$. The desired link $L_\infty$ is
\midinsert
\vspace{1.5in}
\botcaption{Figure 2.8}\endcaption
\endinsert
\flushpar obtained by crossing the figure with $S^1$, embedding the
resulting link in a solid torus then embedding the solid torus in
$S^3$ with a full twist. Thus $\{K_1,K_2\}$ has linking number one
so the conclusion of 2.6 fails. Yet each surface $V_i$ is an infinite
union of annuli and is a generalized Seifert surface for the $K_i$
(except that it has an infinite number of boundary components).
This example is especially interesting because meridians can be
chosen which go to conjugates of generators under maps to free
groups!!
\vskip.5cm
\sub{Proof of 2.6} Assume $L=\{K_1, \dots, K_m\}$. Let
$G=\pi_1(E(L))$ with respect to some basepoint and let
$u: F\left\to G$ be defined by $u(x_j)$
equals the $j^{\text{th}}$ meridian (joined to the basepoint). It
suffices to show that there is a homomorphism $G\to F^\land$, the
nilpotent completion of $F$, which is an isomorphism on
abelianizations. This will be a by-product of our proof of the
factorization of Orr's invariant so we begin an introduction to the
latter.
Following Orr, let $K_\om$ denote an Eilenberg-Maclane space
$K(F^\land,1)$ of $F^\land\equiv\varprojlim F/F_n$ associated to
$F\left$. Let $K_n$ denote $K(F/F_n,1)$. Then
there are fibrations $\pi'_n: K_n\to K_{n-1}$ such that
$\varprojlim K_n$ may be identified with $K_\om$. If $L$ is a based
link with group $G$, inducing the meridional map $u: F\to G$, then
the $\ov\mu$-invariants of $L$ vanish iff $u$ induces an isomorphism
on each quotient $F/F_n\to G/G_n$. Thus $u$ induces a factorization
of the canonical map $F\overset i\to\lra F^\land$ as follows
$F\overset u\to\lra G\overset\pi\to\lra G^\land
\overset(\hat u)^{-1}\to\lra F^\land$ and consequently induces maps
$K(F,1)\overset u_{\#}\to\lra E(L)\overset\phi_{\#}\to\lra K_\om$,
where $\phi=\hat u^{-1}\circ\pi$. Then $\phi_{\#}$ is extended to
$S^3$ yielding an element $\th_\om(L,u)$ lying in $\pi_3$ of the
cofiber of $\phi_{\#}\circ u_{\#}=i_{\#}$ \cite{19}.
\bpage
\proclaim{Theorem 2.9} Suppose $L$ is a finite sublink of the
infinite homology boundary link $L_\infty$ and suppose $u$ is any
basing which factors through $E(L_\infty)$. Then there is a compact
metric space $E$, which is an inverse limit of finite
$2$-complexes, and continuous maps
$E(L)\overset\tl f\to\lra E\overset g\to\lra K_\om$ inducing a
factorization of Orr's invariant $\th_\om(L, u)$ through the cofiber
$\ov E$ of the map
$\bigvee\lm^m_{i=1} S^1\overset\tl f\circ u\to\lra E$. $\ov E$ is the
inverse limit of finite $2$-complexes each of which is homotopy
equivalent to a wedge of $2$-spheres.
\endproclaim
\bpage
Our strategy to factor Orr's invariant is to construct the following
commutative diagram of spaces.
$$
\define\toparrow{@>\pretend f\haswidth
{\text{Clifford mult}}>>}
\split
E(&L_\infty)\toparrow H\\
u \nearrow\quad&\Biggl\downarrow i\hskip60pt\swarrow\ngth i\
\searrow g_H=\la\\
\bigvee\lm^m_{i=1} S^1\overset u\to\lra E&(L)\ \ \
\overset\tl f\to\dashrightarrow\quad E\ \ \
\overset g\to\dashrightarrow \qquad K_\om
\endsplit\tag2.10
$$
This will occupy the rest of the chapter. Note that
$(\tl f\circ g)_*: G\to F^\land$ as desired to show the
$\ov\mu$-invariants vanish. Suppose we are given a based link
$(L,u)$ such that image~$(u)\subset E(L_\infty)$. First we construct
the map $f$, generalizing the well-known result that the exterior of
an homology boundary link maps to $\bigvee\lm^m_{i=1} S^1$. Then we
define the space $E$ in which $H$ is embedded, and the map $g$.
Lastly we define $\tl f$ and compute the Orr invariant.
Recall that to any finite collection of disjointly embedded framed
surfaces in $S^3$
$\{\Psi_i: V_i\x[-1,1]\hookrightarrow S^3, 1\le i\le m\}$ whose
boundaries are copies of the longitudes of a link $L$, the
Pontryagin procedure associates a continuous map
$p_m: E(L)\to\bigvee\lm^m_{i=1} S^1$ by sending the complement of
the images of $\Psi_i$ to the wedge point and sending $\Psi_i(x,t)$
to $(\cos\pi t, \sin\pi t)$ in the $i^{\text{th}}$ circle. Then the
inverse image of $(1,0)$ on the $i^{\text{th}}$ circle is $V_i$.
Thus if $L$ is an $m$-component homology boundary link, $L$ admits
such a map and the images of some meridian set normally generates
$\pi_1\(\bigvee\lm^m_{i=1} S^1\)$. The infinite case yields:
\bpage
\proclaim{Theorem 2.11} Let $L_\infty$ be an infinite homology
boundary link. Let $H$ be the Hawaiian earrings space, that is
$\{(x,y)\mid\(x-\f1n\)^2+y^2=\(\f1{n^2}\)n\in\BZ_+\}$ in the
subspace topology. The infinite surface system $\{V_n\}$ induces a
continuous map $f: E(L_\infty)\to H$ such that
$f^{-1}\(\(\f2n,0\)\)=V_n$ and such that the composition
$E(L_\infty)\to H\to\bigvee\lm^k_{i=1} S^1$ (the largest $k$
circles) sends an $i^{\text{th}}$ meridian to $x_i$ or $0$ in
$H_1\(\bigvee\lm^k_{i=1} S^1\)$ according as $i\le k$ or $i>k$.
Moreover, for any $n$, there is $u(n)\in\BZ_+$ such that if
$i>u(n)$, the restriction of $f$ to an unbased $i^{\text{th}}$
meridian has image disjoint from $\(\f2n,0\)$.
\endproclaim
\vskip.5cm
\sub{Proof of 2.11} The following will be used throughout.
\bpage
\sub{Lemma 2.12} If $L$ is an $m$-component $\infty$ homology
boundary link then for all $j\in\BZ_+$ there exist positive
integers $n(j)$ and $u(j)$ such that $u(j-1)0$ there is a positive integer $j$ such that
$n(j)\ge n$, and so
$\p V_n\subset\bigcup\lm^{n(j)}_{i=1}\p V_i\subset\p
E(K_1\cup\dots\cup K_{u(j)})$. Thus if $i>u(j)$ the image of an
unbased $i^{\text{th}}$ meridian under $f$ does not go around the
$n^{\text{th}}$ circle.\qed
\bpage
One wonders about the converse to 2.11. Note that the link of 2.7
possesses a map $f: E(L_\infty)\to\bigvee\lm^5_{i=1}S^1$ while the
link of 2.8 does {\it not\/} admit a continuous map
$f: E(L_\infty)\to H$.
We now construct the space $E$ in which $H$ will be naturally
embedded. First we establish some notation. Let $M_i$ stand for
both an unbased $i^{\text{th}}$ meridian map
$M_i: S^1\to E(L_\infty)$ $(M_i=\phi_i\bigm|_{*\x\p D^2})$ and
for the same map with range $E(K_1\cup\dots\cup K_s)$ when defined.
For $i\le m$, we can base $M_i$ to agree with the given basing $u$ of
$L$. Let $M_i(n)$ denote the composition
$p_n\circ M_i=h_n\circ f\circ M_i$. Note that if $i>u(j)$ then
$M_i(n(j))$ is the constant map to the wedge point (see 2.11). Let
$m_i$ (respectively $m_i(n)$) denote an element in the conjugacy
class of $M_i$ (respectively $M_i(n)$) in $\pi_1(E(L_\infty))$
(respectively $\pi_1\(\bigvee\lm^n_{i=1}S^1\)$). Note that
$\pi_j\circ M_i(n(j))=M_i(n(j-1))$ and
$\pi_*(m_i(n(j)))=m_i(n(j-1))$. Identify
$\pi_1\(\bigvee\lm^n_{i=1}S^1\)$ with $F\left$
with the standard generators. Then we remark for later use that
for any $n$ and $i\le n$, $m_i(n)$ is homologous to $x_i$ whereas for
$i>n$, $m_i(n)$ is homologous to $0$.
Now define the space $E_j$ $j\ge 1$ by adjoining to
$\bigvee\lm^{n(j)}_{i=1}S^1$ the $2$-cells
$\{e^2_i\mid m*u(j-1)$,
$M_i(n(j-1))$ is the constant map to the wedge point so let
$\pi(e^2_i)=*$ in this case. Setting $E=\varprojlim E_j$ we see that
$H$ is a natural subspace of $E$.
We shall now inductively construct a family of maps
$g_j: E_j\to K_{j+1}=K\(F/F_{j+1},1\)$ such that
$g_{j-1}\circ\pi=\pi'\circ g_j$ where $\pi': K_{j+1}\to K_j$, and
this will induce $g: E\to K_\om$ as desired. In addition we shall
arrange that $g\bigm|_H\circ f\circ u$ induces the canonical map
$F\overset i\to\lra F^\land$ on fundamental group.
\bpage
\sub{Lemma 2.14} There are continuous maps
$\la_i: (S^1,*)\to(K_\om,*)$ $i\in\BZ_+$ such that
\roster
\item"1." If $i>n(j)$ then the composition
$S^1\overset\la_i\to\lra K_\om\lra K_j$ is the constant map to $*$.
Consequently the $\{\la_i\}$ define $\la: H\lra K_\om$ equal to
$\la_i$ on the $i^{\text{th}}$ circle.
\item"2." If $i\le m$, $\la\circ f\circ M_i$ represents the
image of $x_i$ under the map $F\to\wh F$.
\item"3." If $i>m$, $\la\circ f\circ M_i$ is null-homotopic rel $*$.
Therefore so is each $\bigvee\lm^n_{i=1}\la_i\circ M_i(n)$ (refer
to diagram~2.15.)
\endroster
$$
\split
S^1\overset M_i\to\lra E(L_\infty)\overset f\to\lra &H\quad
\overset\la\to\lra K_\om\overset\th_j\to\lra K_j\\
p_n\searrow\ \ \ \ &\Bigl\downarrow h_n\\
&\bigvee\lm^n_{i=1} S^1\qquad \bigvee\lm^n_{i=1}\la_i\\
\ngth\ngth\ngth\ngth\ngth\ngth M_i(n)&\qquad
\endsplit
$$
\centerline{Diagram 2.15}
\bpage
Given 2.14 we show how to define $\{g_j\}$. Let $\th_j$ denote the
map $K_\om\to K_j$ and $\la^j_i=\th_j\circ\la_i$. By induction, we
will define $g_j: E_j\to K_{j+1}$ $j\ge 1$ such that
$g_{j-1}\circ\pi=\pi\circ g_j$ and such that $g_j$ restricted to
$\bigvee\lm^{n(j)}_{i=1}S^1$ is $\bigvee\lm^{n(j)}_{i=1}\la^{j+1}_i$
which is of course merely $\th_{j+1}\circ\la$ restricted. Recall that
$E_1\equiv\bigvee\lm^m_{i=1}S^1\cup e^2_{m+1}\cup\dots\cup
e^2_{u(1)}$ where the two-cells are attached along $M_i(m)$ for
$m+1\le i\le u(1)$. By 2.13, $m_i(m)$ lies in the commutator
subgroup of $F\left$ so if we define $g_1$
restricted to $\bigvee\lm^m_{i=1}S^1\subset E_1$ to be
$\th_2\circ\(\bigvee\lm^m_{i=1}\la_i\)=\bigvee\lm^m_{i=1}\la^2_i$
then $g_1$ extends to $E_1$ since $K_2=K\(F/F_2,1\)$ has abelian
fundamental group. Now suppose $g_{j-1}$ is defined and satisfies
the properties stated above. We define $g_j: E_j\to K_{j+1}$ as
follows. Set $g_j$ restricted to $\bigvee\lm^{n(j)}_{i=1}S^1$ to be
$\bigvee\lm^{n(j)}_{i=1}\la^{j+1}_i$.
Now we check that $g_{j-1}\circ\pi=\pi'\circ g_j$ on
$\bigvee\lm^{n(j)}_{i=1}S^1$ where $\pi: E_j\to E_{j-1}$ and
$\pi': K_{j+1}\to K_j$. We see that $\pi'\circ g_j$ is
$\pi'\circ\bigvee\lm^{n(j)}_{i=1}\la^{j+1}_i=\pi'\circ\th_{j+1}
\circ\bigvee\lm^{n(j)}_{i=1}\la_i=\th_j\circ\bigvee\lm^{n(j)}_{i=1}
\la_i=\(\th_j\circ\bigvee\lm^{n(j-1)}_{i=1}\la_i\)\vee
\(\bigvee\lm_{i>n(j-1)}\text{(constant map)}\)$ by 2.14.1. But the
latter is clearly $g_{j-1}\circ\pi$ since $\pi$ crushes the circles
$n(j-1)**m$), so there is {\it some\/}
extension $h_j: e_i\to K_{j+1}$. Since $\pi_2(K_j)=0$,
$\pi'\circ h_j$ is homotopic rel $\p e^2_i$ to $g_{j-1}\circ\pi$.
Since $\pi'$ is a fibration, this homotopy lifts to a homotopy
rel $\p e^2_i$ from $h_j$ to a map called $g_j: e_i\to K_{j+1}$.
This then is the desired $g_j$ and covers $g_{j-1}\circ\pi$.
\bpage
\sub{Proof of 2.14} We shall first produce {\it homotopy\/} classes
$\la_i\in\pi_1(K_\om,*)$ that satisfy 2.14.2 and 2.14.3; then
choose particular representatives to achieve 2.14.1.
Let $x_i: (S^1,*)\to(H,*)$ be the class in $\pi_1(H,*)$ represented
by the $i^{\text{th}}$ circle. Since $\pi_1(H,*)$ modulo the normal
subgroup generated by $\{x_k\mid k>n(j)\}$ is merely
$\pi_1\(\bigvee\lm^{n(j)}_{i=1}S^1\)$ detected by the map
$\(h_{n(j)}\)_*$, the image $f_*(m_i)$ of a {\it based\/} meridian
of the $i^{\text{th}}$ component of $L_\infty$ can be written as a
word $r^j_i$ in $\{x_i,\dots, x_{n(j)}\}$ modulo this same normal
subgroup. Indeed, 2.11 implies that up to conjugation $r^j_i$ is of
the form $x_iw^j_i(x_1,\dots, x_{n(j)})$ if $i\le n(j)$ or
$w^j_i(x_1,\dots, x_{n(j)})$ if $i>n(j)$, where $w^j_i$ is a word
in the commutator subgroup of $F\left$.
Moreover, by 2.12, for any integer $l$, if $i>u(l)$ then $x_k$ does
not occur in $w^j_i$ for any $k\le n(l)$ and any $j$. We remind the
reader that $x_k$ would occur precisely when $V_k$ had a boundary
component on the tubular neighborhood of $K_i$. Finally, since
$\pi\circ m_i(n(j))=m_i(n(j-1))$, $w^{j-1}_i$ is obtained from
$w^j_i$ by setting $x_i=e$ for $n(j-1)**\to F^\land$. Then we consider the
system of equations
$\{r^j_i=e\mid i>m, j\ge 1\}\cup\{r^j_i=y_i\mid i\le m, j\ge 1\}$
over the group $F^\land$ in the variables $x_i$. We claim that
there exist compatible solutions $\la_i\in F^\land$ in the
following sense. If $\la^j_i$ denotes $\th_j(\la_i)$, i.e., the
image of $\la_i$ in $F/F_{j+1}=\pi_1(K_j)$ then we shall establish
the following by induction:
\bpage
$\underline{P(j)}$: There exist elements $\la^j_i\in F/F_{j+1}$ such
that
\roster
\item"a)" $\la^j_i\equiv\la^{j-1}_i$ in $F/F_j$,
\item"b)" if $i>n(j)\ge m$ then $\la^j_i=e$,
\item"c)" if $i>m$ then
$\la^j_i w^j_i(\la^j_1,\dots,\la^j_{n(j)})=e$ in $F/F_{j+1}$,
\item"d)" if $i\le m$ then $r^j_i=\la^j_i$ in $F/F_{j+1}$.
\endroster
Once having shown this, by a), the collection $\{\la^j_i\}$ defines
$\la_i\in F^\land$ which can be realized by maps
$\la_i: S^1\to K_\om$. Using b) and the fact that $K_i\to K_{i-1}$
are fibrations, the $\la_i$ may be modified by based homotopy to
achieve 2.14.1 ``on the nose''. Since
$K_\om=\varprojlim(\pi': K_{j+1}\to K_j)$, 2.14.1 is precisely what
is needed to ensure that the maps $\la_i$ induce a continuous map
from $H\equiv\varprojlim\vee S^1$ to $K_\om$. Now, for any $i>m$ and
$j\ge 1$, the map $\th_j\circ\la\circ f\circ M_i$ is null-homotopic
rel basepoint, because it is represented by a conjugate of the
word $\la^j_i w^j_i(\la^j_1,\dots,\la^j_{n(j)})$ (note $\la^j_i=e$
if $i>n(j)$). Similarly if $i\le m$, it is $y^j_i$. Thus
$\la\circ f\circ M_i$ is null-homotopic if $i>m$ establishing
2.14.2 and 2.14.3 and completing the construction of
$g: E\lra K_\om$. It remains only to establish $P(j)$ by induction.
Now $P(1)$ holds with $\la^j_i=e$ for $i>m$ and $\la^j_i=y^1_i$ for
$i\le m$ because $n(1)=m$ and the $w^j_i$, being commutators,
vanish in the abelian group $\pi_1(K_1)\cong F/F_2$. Now assume
$P(j-1)$. Consider the system of $n(j)$ equations
$\{e=x_i w^j_i(x_1,\dots,x_{n(j)})\mid m** n(j-1)$ since $w^{j-1}_i$ and $\d^{j-1}_i$ are obtained from
$w^j_i$ and $\d^j_i$ by setting $x_k=e$ for $k> n(j-1)$. Since
these solutions are also unique, $\la^j_i\equiv\la^{j-1}_i$ in
$F/F_j$. If $i>n(j)$ set $\la^j_i=e$. It remains only to show that
if $i>n(j)$, $\la^j_i w^j_i(\la^1_i,\dots,\la^j_{n(j)})=e$ in
$F/F_{j+1}$. since $i>n(j)>u(j-1)$ (by 2.12), $w^j_i$ contains no
occurences of $x_k$ for $k\le n(j-1)$. Therefore
$w^j_i(\la^j_1,\dots,\la^j_{n(j)})$ is a product of conjugates of
commutators $[\la^j_\a,\la^j_\b]$ where $n(j-1)<\a$, $\b\le n(j)$.
Since $\la^{j-1}_\a=e=\la^{j-1}_\b$ in $F/F_j$ for such $\a$, $\b$,
$\la^{j-1}_\a$ and $\la^{j-1}_\b$ lie in $F_j/F_{j+1}$ and so their
commutator vanishes in $F/F_{j+1}$. This concludes the proof of
$P(j)$ and hence 2.14 and the definition of $g: E\lra K_\om$.\qed
\bpage
Now, referring to 2.10, we construct the map
$\tl f: E(L)\lra E$ such that $i\circ f=\tl f\circ i$. Let
$\tl f=\varprojlim\tl f_j$ where $\tl f_j: E(L)\lra E_j$ is as
follows. On $E(L_\infty)$ let $\tl f_j$ agree with $f$ (for this
proof let $f$ stand for the composite
$E(L_\infty)\overset f\to\lra H\overset i\to\lra E\lra E_j$). On
the tubular neighborhood $\phi_i(S^1\x D^2)$ of $K_i$ ($i>m$), let
$\tl f_j$ be the constant map to $*$ if $i>u(j)$; and if $i\le u(j)$
let it send $\phi_i(S^1\x D^2)$ to $\phi_i(\{1\}\x D^2)$ and identify
this with the $2$-cell $e^2_i$ of $E_j$. One checks that
$\tl f_{j-1}=\pi'_j\circ\tl f_j$. To show $\tl f_j$ is continuous
it suffices to show it is continuous on the closed sets
$A = E_L - \bigcup\lm^{u(j)}_{i=1}$ int $(\phi_i(S^1\x D^2))$ and
$B=\bigcup\lm^{u(j)}_{i=1}\phi_i(S^1\x D^2)$. The latter
restriction is clearly continuous. Suppose $x_0\in A$. The
restriction to $A$ is clearly continuous at $x_0$ if
$x_0\in E_{L_\infty}-\bigcup\lm^\infty_{u(j)}\phi_i(S^1\x D^2)$
since the latter is open in $E(L_\infty)$ and $\tl f_j$ agrees with
$f$ there. It is also clearly continuous at $x_0$ if $x_0$ lies in
some int $(\phi_i(S^1\x D^2))$. Suppose $x_0\in\phi_i(S^1\x S^1)$
and $x_n\to x_0$ where $x_n\in A$. If $x_{n_j}$ is the
subsequence of elements of $E(L_\infty)$ then $x_{n_j}\to x_0$
in $E(L_\infty)$ so
$\tl f_j(x_{n_j})=f(x_{n_j})\lra f(x_0)=\tl f_j(x_0)$.
Thus given any neighborhood $U$ of $f_j(x_0)$ in $E_j$, there is
$N$ s.t\. if $n>N$ {\it and\/} $x_n\in E(L_\infty)$ then
$\tl f_j(x_n)\in U$. Since
$A=E(L_\infty)\bigcup\lm_{i>u(j)}\phi_i(S^1\x D^2)$, if
$x_n\in A-E(L_\infty)$ then $x_n\in\phi_i(S^1\x D^2)$ where
$i>u(j)$ so $\tl f_j(x_n)=*=\tl f_j(x_0)$ by definition. Thus
$\tl f_j(x_n)\lra\tl f_j(x_0)$. Since $A$ is metrizable,
$\tl f_j$ is continuous on $A$. Hence $\tl f_j$ itself is
continuous. This completes the construction of
$\tl f: E(L)\to E$ as in 2.10.
We return to the factorization of Orr's $\om$-invariant. We must
suppose that $u: \bigvee\lm^m_{i=1}S^1\lra E(L)$ is a basing which
avoids $L_\infty$. Then we have constructed the entire diagram~2.10.
Moreover if $G=\pi_1(E(L))$ and
$\Psi=(g\circ\tl f)_*: G\lra F^\land$ then, since we have arranged
that $\Psi\circ u: F\lra F^\land$ is $i: F\lra F^\land$, it follows
that $\Psi^\land\circ\mu^\land=i^\land=$ identity. Consequently
$\Psi^\land=(\mu^\land)^{-1}$ and the Orr invariant factors through
the cofiber of the map
$\tl f\circ u: K(F,1)\lra E$ (since $\Psi^\land_\#=g\circ\tl f$).
Let $\ov E$ stand for this cofiber. More specifically we may take
$\ov E$ to be $E\bigcup\lm^m_{i=1}e^2_i$ where the $2$-cell $e^2_i$
is attached via $i\circ f\circ M_i$ (the image of the meridian of
the $i^{\text{th}}$ component). Then we see that
$\ov E\equiv\varprojlim\ov E_j$ where
$\ov E_j=E_j\bigcup\lm^m_{i=1}e^2_i$ where $e^2_i$ is attached along
$\pi\circ i\circ f\circ M_i$ which is merely $M_i(n(j))$. Thus
$\ov E$ is the inverse limit of finite $2$-complexes which we claim
are simply-connected. Since
$\ov E_j=\bigvee\lm^{n(j)}_{i=1}S^1\cup\{e^2_i\mid 1\le i\le u(j)\}$
using the maps $M_i(n(j))$, it suffices to show that
$\pi_1\(\bigvee\lm^{n(j)}_{i=1}S^1\)$ is normally generated by
$\{M_i(n(j))\mid 1\le i\le u(j)\}$. Recalling the Pontryagin map
$p: E(K_1\cup\dots\cup K_{u(j)})\lra\bigvee\lm^{n(j)}_{i=1}S^1$,
since $\pi_1\(E(K_1\cup\dots\cup K_{u(j)})\)$ is normally generated
by the meridians $M_i$, $1\le i\le u(j)$, it suffices to show that
$p_*$ is surjective on $\pi_1$. We now argue that
$\bigcup\lm^{n(j)}_{i=1}V_i$ does not separate
$E(K_1\cup\dots\cup K_{u(j)})$. Let $\wh V_i$ be the space obtained
by identifying each component of $\p V_i$ which is a longitude of
$K_s$ to a single circle, for each $1\le s\le u(j)$. Let $\wh V$ be
$\bigcup\lm^{n(j)}_{i=1}V_i$ with similar identifications. Then
$\wh V$ is a natural subspace of $S^3$ and
$E(K_1\cup\dots\cup K_{u(j)})-\bigcup\lm^{n(j)}_{i=1}V_i$ is
homotopy-equivalent to $S^3-\wh V$. By Alexander Duality,
$\ov H^0(S^3-\wh V)\cong H_2(\wh V)$. Since the $V_i$ are
(connected!) generalized Seifert surfaces, $\wh V_i$ is obtained
from a surface with a single boundary component by identifying sets
of homologically linearly independent circles. Thus it follows that
$H_2(\wh V_i)$ is zero and that $H_2(\wh V)\cong 0$ follows from an
extension of this reasoning, since $\wh V$ is a quotient of
$\coprod\lm^{n(j)}_{i=1}\wh V_i$. Therefore,
$\bigcup\lm^{n(j)}_{i=1}V_i$ does not separate
$E(K_1\cup\dots\cup K_{u(j)})$. For each $1\le i\le n(j)$, choose an
interval fiber of the tubular neighborhood of $V_i$ and choose a
path disjoint from $\bigcup\lm^{n(j)}_{i=1}V_i$ connecting its
endpoints. This loop then maps under $p_*$ to a generator of $\pi_1$
of the $i^{\text{th}}$ circle. Hence $\ov E$ is an inverse limit of
finite $2$-complexes each of which is homotopy equivalent to a wedge
of $2$-spheres. This completes the proofs of 2.6 and 2.9.
\vskip.5cm
\sub{Chapter 3: Remarks on the spaces $E$ and $\ov E$}
\mpage
We have seen that $\ov E$ is the inverse limit of finite
$2$-complexes, each of which is homotopy-equivalent to a wedge of
$2$-spheres. Certainly not every such inverse limit space is
contractible. For example suppose we define
$X_j=\bigvee\lm^j_{i=1}S^1\cup e^2_1\cup e^2_2\cup\dots\cup e^2_j$
where $e^2_i$ $i\neq j$ attaches to $\bigvee\lm^j_{i=1}S^1$ along a
path traversing the $i^{\text{th}}$ circle counter-clockwise and
$e^2_j$ attaches along the $j^{\text{th}}$ circle clockwise. Then
let $\ov X_j=X_j\cup e^2_0$ where $\p e^2_0$ is the 1st-circle
clockwise, and define $\ov X_j\overset\pi\to\lra\ov X_{j-1}$ to
send $e^2_j$ to the wedge point, $e^2_i$ $0\le i < j-1$ to itself,
and $e^2_{j-1}$ in $\ov X_j$ to its companion $e^2_{j-1}$ in
$\ov X_{j-1}$. Then clearly
$\varprojlim\ov X_j\simeq\ov X_j\simeq S^2$ and $\pi_*$ is an
isomorphism on $\pi_2$. Thus
$\varprojlim H_2(\ov X_j)=H_2(\varprojlim\ov X_j)\cong\BZ$.
Therefore the way the $2$-cells are attached to $\ov E_j$ is very
important. In particular in our situation we at least have the
following:
\bpage
\proclaim{Theorem 3.1} For any $j\in\BZ^+$, there exists a positive
integer $d=d(j)$ such that the map $\ov E_d\to\ov E_{j-1}$ is
null-homotopic. In particular
$\varprojlim H_*(\ov E_j)=\varprojlim\pi_*(\ov E_j)=0$ if $*\ge 1$.
\endproclaim
\bpage
\sub{Proof} Since each $\ov E_j$ is a wedge of $2$-spheres it
suffices to show $H_2(\ov E_d)\to H_2(\ov E_{j-1})$ is zero. Recall
that
$\ov E_d=\bigvee\lm^{n(d)}_{i=1}S^1\cup\{e^2_i\mid 1\le i\le u(d)\}$
where $n(d)$, $u(d)$ are as in 2.12 and $e^2_i$ is attached along
$M_i(n(d))$ as defined above 2.14. By those remarks $M_i(n(d))$ is
homologous to $x_i$ if $1\le i\le n(d)$ and $0$ if $i>n(d)$. Thus
the image of
$H_2(\ov E_d)\to H_2\(\ov E_d, \bigvee\lm^{n(d)}_{i=1}S^1\)$ is
generated by $\{e^2_i\mid n(d) < i\le u(d)\}$. By 2.12, given $j$,
there is an integer $d$ such that $n(d)>u(j-1)$. Then under the map
$\pi: \ov E_d\to\ov E_j$ a cell $e^2_i$ above maps either to the
wedge point or to the cell $e^2_i$ attached along $M_i(n(j))$ in
$\ov E_j$. But since $i>n(d)>u(j-1)$ the cell $e^2_i$ of $\ov E_j$
maps to the wedge point in $\ov E_{j-1}$ by the definition above
2.14. Therefore the image $H_2(\ov E_d)\to H_2(\ov E_{j-1})$ is
zero. \qed
\mpage
This may be compared with the result of Bousfield and Kan that
$\varprojlim H_q(F/F_n)=0$ for $q\ge 2$ which implies the results
of 3.1 for the tower $\to\ov K_j\to\ov K_{j-1}$ (where
$\varprojlim\ov K_j=\ov K_\om$) (see Theorem~1.2 of \cite{5}). Thus
3.1 merely reflects our claim that the tower $\{E_j\}$ is basically
a 2 skeleton of the tower $\{K(F/F_n,1)\}$. If this is precisely true
then Bousfield's result that $H_2(F^\land ; Q)$ is uncountable is
discouraging since then, by cardinality considerations, most of
these elements could not contribute to a non-zero Orr invariant
\cite{2; Prop\.~4.4}. Still, the concreteness of the spaces
$\ov E$ leads us to hope that they can be understood and perhaps
even facilitate an understanding of $H_*(F^\land)$. Understanding
even a single non-zero element of $\pi_3(\ov E)$ would very likely
lead to its realization and a new type of linking phenomenon.
We close with some interesting remarks and questions.
We do not know if $\ov E$ is simply-connected (as $K_w$ is) but
expect this to be the case. For the surface systems that arise in
2.3 and especially TC part~3, one sees that the most fundamental
example of a space $E$ arises from imagining an $m$-component link
with Seifert surfaces each pair of which intersects in a circle
and that each of these circles possesses a (true) Seifert surface
which intersects each previous one in a single circle, etcetera.
The surfaces of this imagined system and hence the circles of the
Hawaiian Earrings are indexed by the simple commutators of
$F\left$ which are ordered compatibly with
commutator length. Each component $c(b_1,b_2)=\p(V(b_1,b_2))$ of
this hypothetical $L_\infty$ arises precisely as
$V(b_1)\cap V(b_2)$ ($=-(V(b_2)\cap V(b_1)$) and so the
corresponding meridian $f_*(M_{(b_1,b_2)})\in\pi_1(H)$ is merely
$x_{(b_1,b_2)}[x_{(b_1)}, x_{(b_2)}]$. In this standard situation the
map $\la: H\to K_\om$ is defined by sending $\la_i$ to the
$i^{\text{th}}$ simple commutator in the ordering (compare 2.14). It
is then easy to show that $\la$ induces an epimorphism $\la_*:
\pi_1(H)\twoheadrightarrow F^\land$. The group $\pi_1(H)$ is itself
fascinating. It is $\om$-milpotent like $F^\land$ but is locally
free, unlike $F^\land$. Moreover if $\{g_i\}$ is any sequence of
elements of $\pi_1(H)$ which converges to $1$ in the obvious
inverse-limit topology, then there is an endomorphism of $\pi_1(H)$
sending $x_i\to g_i$ \cite{11}. It would be nice to factor Levine's
refinement $\ov\th_\om$ of Orr's invariant, but one can see from the
proof of 2.14 that an infinite link leads to an infinite number of
equations whereas Levine's algebraic closure $\ov F\subset F^\land$
is defined using sets of finite number of equations. Still the
similarities are striking and bear further study.
\spage
\sub{Acknowledgments}
Partially supported by N.S.F. grants DMS-8903514,
DMS-9100254.
First announced in special session of American Mathematical
Society, Chicago, Illinois, May 1989
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*