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\topmatter
\title Homology Boundary Links and Blanchfield Forms: Concordance
Classification and New Tangle-Theoretic Constructions
\endtitle
\author Tim D. Cochran* and Kent E. Orr*\endauthor
\abstract Seifert forms and Blanchfield forms are defined for
homology boundary links. New tangle constructions are used to show
that any pair (pattern, Blanchfield form) can be realized by a
homology boundary link. A classification theorem is proved for
homology boundary links of fixed pattern modulo homology boundary
link concordance. This is done from the points of view of Seifert
matrices, Blanchfield forms and $\G$-groups. The analogous notions
for links in $\BZ_p$-homology spheres are discussed.
\endabstract
\thanks *partially supported by the National Science
Foundation\endthanks
\endtopmatter
\document
\subhead{\bf\S0. Introduction}\endsubhead
\mpage
This work forms part of our on-going effort to classify the set of
concordance classes of links. Recall that a {\it link\/}
$L=\{K_1,\dots,K_m\}$ in $S^{n+2}$ is a locally flat
piecewise-linear, oriented submanifold of $S^{n+2}$ of which each
component $K_i$ is homeomorphic to $S^n$. The {\it exterior\/}
$E(L)$ of a link $L$ is the closure of the complement of a small
regular neighborhood $N(L)$ of $L$. A {\it longitude\/} of a
component $K_i$ is a parallel of $K_i$ lying on the boundary of the
tubular neighborhood (untwisted if $n=1$). A {\it meridian\/}
$\mu_i$ is a path from a basepoint to $\p N(L)$ which traverses a
fiber of $\p N(L)$ and returns. A {\it Seifert Surface\/} for $K_i$
is a connected, compact, oriented, $(n+1)$-manifold $V_i\sbq E(L)$
such that $\p V_i$ is a longitude of $K_i$. Links $L_0$, $L_1$ are
concordant (or cobordant) if there is a smooth, oriented submanifold
$C=\{C_1,\dots,C_m\}$ of $S^{n+2}\x[0,1]$ which meets the boundary
transversely in $\p C$, is piecewise-linearly homeomorphic to
$L_0\x[0,1]$, and meets $S^{n+2}\x\{i\}$ in $L_i$ for $i=0$,$1$. In
the mid-60's M\.~Kervaire and J\.~Levine gave an algebraic
classification of knot concordance groups $(m=1)$ in high dimensions
$(n>1)$ \cite{L3}. For even $n$ these are trivial and for
odd $n$ they are infinitely generated, being isomorphic to certain
Witt groups obtained from information garnered from the Seifert
surface.
Extending Levine's knot cobordism classification to links is
difficult for several reasons. Firstly, if $m>1$, the natural
operation of connected-sum is not well-defined on concordance classes
so there is no obvious group structure. Secondly, the Seifert
surfaces for different components of a link may intersect,
obstructing at least the naive generalization of the Seifert form
information.
However the techniques do extend well to the class of
{\it boundary links\/}. A boundary link is one which admits a
collection of $m$ {\it disjoint\/} Seifert surfaces, one for each
component. In fact, S\.~Cappell and J\.~Shaneson classified
boundary links modulo {\it boundary link cobordism\/} in 1980 using
their homology surgery groups, followed later by Ki Ko and W\.~Mio
who accomplished this via Seifert surfaces \cite{CS1, Ko, Mi}. A
{\it boundary link cobordism\/} is a cobordism $C$ between $L_0$
and $L_1$ for which there exist disjointly embedded $2n$-manifolds
$IV=\{IV_1,\dots,IV_m\}$ in $E(C)$ such that
$IV\cap(S^{n+2}\x\{i\})$ is a system of Seifert surfaces for the
boundary link $L_i$, $i=0$,$1$, and such that
$$
(\p N(C), IV\cap\p N(C))\cong(\p N(L_0)\x[0,1], (IV\cap\p N(L_0))
\x[0,1]).
$$
These successes focussed intense scrutiny on the question of
whether or not every link were concordant to a boundary link (if
$n=1$, Milnor's $\ov\mu$-invariants were known obstructions). If
this had been the case, the concordance classification of links (at
least if $n>1$) would have been almost complete.
Unfortunately the situation was not so simple. In 1989 the present
authors defined new invariants which showed that many
odd-dimensional {\it homology boundary links\/} are not concordant
to any boundary link \cite{CO1, CO2}. This development focussed
attention on the previously obscure class of homology boundary
links, first defined by N\.~Smythe in 1965 \cite{S}. To define an
homology boundary link, let us first define a more general notion
of Seifert surface which we use for the remainder of this paper.
Let $F$ be the free group on $m$ letters $\{x_i\}$. Consider the
subset of $F^m$ consisting of those $(w_1,\dots,w_m)$ for which
$w_i\equiv x_i$ in the abelianization and for which
$\{w_1,\dots,w_m\}$ normally generates the free group. Consider the
equivalence relation on this subset where
$(w_1,\dots,w_m)\sim(w'_1,\dots,w'_m)$ if and only if there are
elements $\eta_i\in F$ and an automorphism $\phi$ of $F$ such that
$w'_i=\phi(\eta_i w_i\eta^{-1}_i)$ for all $i$. An element
$(w_1,\dots,w_m)$ of this set of equivalence classes is called a
{\it pattern\/} $P$. A {\it system of Seifert surfaces of pattern\/}
$P$ {\it for\/} $L$ is a collection $\SV=\{V_1,\dots,V_m\}$ of
pairwise-disjoint, connected, compact, oriented, $(n+1)$-dimensional
submanifolds of $E(L)$ such that $\p V_i$ consists of a union of
longitudes (up to orientation) of various $K_j$ in such a way that
if one traverses $\mu_i$ and ``reads out'' $x_j$ (or $x^{-1}_j$) as
one transversely encounters $V_j$ (or $-V_j$), then one spells
out the word $w_i$ {\it and\/} such that the homomorphism
$\phi: \pi_1(E(L))\to F$ associated to the system (by the
Thom-Pontryagin construction mapping $E(L)$ to a wedge of $m$
circles \cite{Ko; 2.1}) is {\it surjective\/}. An {\it homology
boundary link\/} of $m$ components with pattern $P$ may then be
defined as one which possesses such a system of Seifert surfaces. In
\cite{CL} it is shown that the pattern is an invariant of the
isotopy class of $L$. Boundary links are, of course, those with
pattern $(x_1,\dots,x_m)$. Therefore a homology boundary link is
seen to be a sort of ``algebraic'' boundary link since $\p V_i$ is
{\it homologous\/} to a single longitude of $K_i$. The class of
homology boundary links first received attention (from the point of
view of link concordance) when the first author observed in
\cite{C1, C2} that fusions of boundary links gave examples of
non-boundary, non-ribbon links with vanishing Milnor's
$\ov\mu$-invariants and that these were in fact {\it sublinks\/} of
homology boundary links. Confirmation that sublinks of homology
boundary links was the {\it correct\/} class upon which to focus
study was provided by \cite{CL}, \cite{L4} and \cite{LMO} where it
was shown that, the classes of sublinks of homology boundary links
and fusions of boundary links are identical up to concordance, and
that the vanishing of Jean Le~Dimet's homotopy invariant of (disk
link) concordance was essentially equivalent to being concordant to a
sublink of a homology boundary link. This culminated in the
above-mentioned result of the authors that, in fact, many homology
boundary links are {\it not\/} concordant to boundary links. It is
unknown whether or not every link (with vanishing
$\ov\mu$-invariants if $n=1$) is concordant to a homology boundary
link. Therefore we turn to the project of classifying concordance
classes of {\it homology\/} boundary links.
Recall that there were two ingredients to the invariants of
\cite{CO1, CO2}. The first was ``complexity'' which was there
explained to be purely a function of the {\it pattern\/} $P$. The
second was a function of the universal Blanchfield form $B$ of the
homology boundary link, which may also be viewed in terms of
``cobordism classes of Seifert matrices.'' The primary aim of this
paper is to show that any such pair $(P,B)$ may be realized by an
explicit geometric construction. An important secondary goal is to
classify homology boundary links modulo a suitable cobordism
relation.
Recall the group $G(m,\e)$ of
{\it cobordism classes of Seifert matrices\/} of type $(m,\e)$
defined as in \cite{Ko; \S3}. If $(L,\SV)$ is an $m$-component
link in $S^{2q+1}$, $q>1$, with system $\SV$ with pattern $P$ then
one can associate to $(L,\SV)$ an element of $G(m,(-1)^q)$ by taking
the ``Seifert form''
$\f{H_q(\SV)}{\left<\text{torsion}\right>}\x
\f{H_q(\SV)}{\left<\text{torsion}\right>}\overset\th\to\lra\BZ$
given by $\th(x,y)=lk(x,y^+)$. If $q=1$ we must consider
$H_q(\SV)/H_q(\p V)$ instead of $H_q(\SV)$, and the restriction on
the pattern $(w_1,\dots,w_m)$ guarantees that $\th$ descend to a
form on the quotient. Note that $\th$ can be defined from any set
of disjoint codimension-$2$ oriented submanifolds of $S^{2q+1}$
each component of which is labeled by an element of
$\{1,\dots,m\}$, as long as, when $q=1$, the boundaries of the
surfaces have zero linking numbers with all elements of $H_1(\SV)$.
Specifically, our main theorem will be a stronger form of the
following.
\bpage
\proclaim{Theorem 3.6} Given any pattern $P$, any $q\ge 1$ and
$\a\in G(m, (-1)^q)$, there is an $m$-component homology boundary
link in $S^{2q+1}$ with system of Seifert surfaces of pattern $P$
and Seifert form equivalent to $\a$. (If $q=2$, $\a$ must lie in
the index $2^m$ subgroup of $G(m,1)$ described by \cite{Ko} to
account for Rochlin's theorem).
\endproclaim
\bpage
In \cite{CL} it was shown how to construct a link with arbitrary
$P$ and $\a\cong 0$ (actually a ribbon link) although it would be
nice to have a more constructive algorithm. At the other extreme,
it is relatively easy to construct a boundary link with
$P=(x_1,\dots,x_m)$ and arbitrary $\a$ (see Theorem~3.4 of
\cite{Ko} for a proof generalizing Seifert's proof for $q=1$). The
general idea of Seifert's method (for $q=1$) is that, given a
Seifert matrix $A=(a_{ij})$ of type $(m,\e)$, one can take $m$
disjoint wedges of appropriate numbers of circles and modify them
so that the linking number between the $i\supth$ circle and
$j$\supth circle is $a_{ij}$. Then one can ``thicken'' the wedges
of circles to create punctured surfaces in such a way that the
``self-linking'' of the $i\supth$ circle is $a_{ii}$. These will be
the Seifert surfaces of a boundary link whose Seifert matrix is $A$
with respect to those surfaces. This procedure always produces a
boundary link (as opposed to an arbitrary homology boundary link).
No such simple-minded procedure has been found for homology boundary
links.
Theorem 3.6 will be a corollary of a new and interesting
construction for links that is useful in creating homology boundary
links with prescribed properties. This method was employed in
\cite{CO2} to generate examples. On the other hand our work will be
helpful in {\it calculation\/} as well, since many examples in knot
theory consist of a simple knot or link with some bands of its
Seifert surface tied into knots. Therefore the Seifert forms of
such links are easily computed by our techniques.
We also recover a classification theorem for homology boundary
links which parallels the classification theorem for boundary
links but is much more complicated. Unlike boundary links, the set
of homology boundary links is not closed under connected sum.
Consequently we fix the pattern $P$ and consider only those homology
boundary links with pattern $P$. Just as in \cite{Ko}, we must
consider pairs $(L,\SV)$ where $L$ is such an homology boundary link
in $S^{2q+1}$ and $\SV$ is a system of Seifert surfaces for $L$.
However for homology boundary links we must narrow our focus
further by only considering $\SV$ whose surfaces meet the link
components in a fixed combinatorial {\it scheme\/} $S$. For any
pattern $P$ many schemes are possible, and only links with
identical schemes may be summed in such a way that their Seifert
forms also naively sum. We define the set of {\it scheme cobordism\/}
classes, $C(m,q,S)$, of such pairs where two are scheme-cobordant
if there is a concordance between the links and an embedded
cobordism between the Seifert surface systems, that preserves the
scheme (is a product on its boundary). We show in \S5 that
$C(m,q,S)$ is naturally a group isomorphic to $G(m,(-1)^q)$, if
$q>1$, where this isomorphism is given by the Seifert form. We also
interpret this as a relative $L$-group and a $\G$-group by using the
Blanchfield form. This much is perfectly analagous to the previous
work on boundary links.
When we analyze the effect on the Seifert form of changing $\SV$
for a fixed $L$, we begin to see some surprising complications in
the case of general homology boundary links of pattern $P$. In \S7
we define two such links $L$, $L'$ to be
{\it homology-boundary-link-concordant\/} if they are concordant in
such a way that for {\it some\/} $\SV$, $\SV'$ the pairs $(L,\SV)$,
$(L,\SV')$ are scheme-cobordant. The set of these equivalence
classes is denoted $\SP(m,q,P)$. We then analyze this set and find
that:
\bpage
\proclaim{Theorem 6.3} For any fixed pattern $P$ and any
representative $(w_1,\dots,w_m)$ of $P$, this is a bijection
$\ov\th: \SP(m,q,P)\to G(m, (-1)^q)/\Aut_{w_i}F$ given by taking
the Seifert form of a system of Seifert surfaces with scheme
$S=(w_1,\dots,w_m)$. Similarly the Blanchfield form induces a
bijection $\ov B: \SP(m,q,P)\to L^{(-1)^{q+1}}(\BZ[F],\Sigma)/
\Aut_{w_i}F$. Here $\Aut_{w_i}F$ is the subgroup of automorphisms
of the free group which send $w_i$ to a conjugate of $w_i$ (the
actions are given in 3.4 and 4.5). (If $q=2$ we need to take certain
index $2^m$ subgroups to account for Rochlin's theorem).
\endproclaim
\bpage
A very surprising aspect of 6.3 is that $\SP(m,q,P)$ depends on $P$
(whereas $C(m,q,S)$ is independent of $S$)! A translation of 6.3 in
terms of $\G$-groups yields the following.
\bpage
\proclaim{Theorem 6.4} Suppose $q>2$. For any fixed pattern $P$
and any representative\linebreak
$(w_1,\dots,w_m)$ of $P$, there are functions
$\wt\G_{2q+2}(\BZ F\lra\BZ)/\Aut_{w_i}F\overset\phi_S\to\lra
\SP(m,q,P)\overset\pi\to\lra L_{2q+1}(F)$ such that $\pi$ is
surjective and $\phi_S$ is an injection with image $\pi^{-1}(0)$.
(Here $\wt\G$ is the gamma group modulo the image of $L_{2q+1}$).
\endproclaim
\bpage
Finally, in \S7, we discuss the analogues in the case of links in
$\BZ_p$-homology spheres to establish some claims made in
\cite{CO1, CO2}.
\newpage
\subhead{\bf\S1. Generalized basings and tangle sums}\endsubhead
\mpage
The method of construction we shall presently detail is perhaps
best described as an ``action'' of the set of boundary links on the
set of homology boundary links with pattern $P$. There are actually
many actions depending on various initial data. Given an homology
boundary link $L$, and a sort of generalized basing which
effectively decomposes $L$ into two tangles, one of which is
trivial; we ``act'' on $L$ by removing the trivial tangle and
inserting a boundary disk link (suitably modified for this
purpose). To be more specific, we need to set up some notation.
\bpage
\sub{Definition} A {\it generalized basing\/} $b$ of a link $L$ is an
embedding $b$ of the $2$-disk $\Delta=I\x I$ into $S^{n+2}$ such
that, with regard to the subdivision of the $2$-disk shown in
Figure~1
\midinsert
\vspace{1.5in}
\botcaption{Figure 1}\endcaption
\endinsert
\roster
\item"i)" $b$ is transverse to $L$
\item"ii)" $(\oper{image}b)\cap L$ lies interior to
$\bigcup\lm_{i=1}^m\Delta_i$ along the line $b(I\x\{1/2\})$.
\endroster
Now suppose $\Delta_i\cap L$ is
$\{K_{i_1},\dots,K_{i_{n_i}}\}\cap\Delta_i$ reading left to right.
Then this will be called a generalized basing of type
$(b_1,\dots,b_k)$ where $b_i=x^{\pm1}_{i_1}\dots
x^{\pm1}_{i_{n_i}}$ and the plus sign is used if $K_{i_j}\cap\Delta$
is $+1$. Note that a basing of type $(x_1,\dots,x_m)$ is the usual
(strong) basing that decomposes $L$ as the closure of a disk link.
Also note that it is not necessary that $k=m$.
A generalized basing of $L$ may be slightly thickened to given an
embedding of $\Delta\x D^n=I\x I\x D^n$ whose intersection with $L$
is the product $(L\cap\Delta)\x D^n$. Therefore $b$ induces a
``tangle'' decomposition of $L$ along $\p(I\x I\x D^n)$, one
``summand'' of which is a standard trivial disk link of type
$(b_1,\dots,b_k)$. Since a ``strand'' of one of these tangles
inherits a label $i$ if it was part of the $i^{\text{th}}$
component knot of $L$, these tangles are unusual as ordered links
in that the set of strands labelled $i$ may be disconnected.
Suppose $L$ and $L'$ are endowed with basings $b$, $b'$ of the same
type. Then we can define $L_b\op_{b'}L$ by deleting the induced
``trivial'' tangles of type $(b_1,\dots,b_k)$ from each and
identifying along the common boundaries of the remainders by the
(orientation-reversing) restriction of the homeomorphism
$r: I\x I\x(I\x D^{n-1})\circlearrowleft$ given by
$r(x,y,z,w)=(x,y,-z,w)$. If $n=1$ this {\it tangle-sum\/} may not
yield a true $m$-component link because the ``$i^{\text{th}}$
component knot'' of the result might be disconnected as is seen in
Figure~3 for $m=1$, $L=L'$, $b=(x_1x_1^{-1})$. Even if $n>1$, this
tangle-sum may have components homeomorphic to the connected sum of
copies of $S^1\x S^{n-1}$.
\midinsert
\vspace{1.5in}
\botcaption{Figure 3}\endcaption
\endinsert
However if $b'$ separates $L'$ into {\it pure\/} tangles (like pure
braids) then the tangle sum will be a true $m$-component link.
Specifically, a labelled oriented tangle is called a {\it pure\/}
tangle if each connected component of the strands labelled $i$ is
homeomorphic to the $n$-disk. A ``link'' is called a {\it true\/}
link if the union of the spheres labelled $i$ is connected. In what
follows, we shall require situations where $L'$ is {\it not\/} a
true link but whose components are parallel copies of the
components of a true link. As long as $L$ is a true link and $b'$
separates $L'$ into 2 pure tangles, the sum will be a true link.
\midinsert
\vspace{1.5in}
\botcaption{Figure 4}\endcaption
\endinsert
\midinsert
\vspace{1.5in}
\botcaption{Figure 5}\endcaption
\endinsert
Moreover if the above links have Seifert surface system which are
``compatible'' then we ought to be able to ``add'' these as well.
Here the situation is slightly more complicated. If $L$ has a
system $\SV$, we may and shall assume that $\Delta$ has been
isotoped, relative to $L$, so it meets $\SV$ transversely in one of
a fixed set of standard schemes as shown by example in Figure~4.
This is possible because $\Delta$ may be isotoped to look like
Figure~5a and hence the intersections with $\SV$ may be assumed to
be as in 5b, for example. The set of possible intersection schemes
is larger than the set of possible
$((b_1,\dots,b_k), (w_1,\dots,w_m))$ where the latter is the
pattern. For example, the schemes in Figure~6 are different
although both have $b=b_1=x_1x_1^{-1}x_2x_2^{-1}$ and pattern ($x_1$,
arbitrary) with respect to the meridian $\mu_1$.
\midinsert
\vspace{1.5in}
\botcaption{Figure 6}\endcaption
\endinsert
Instead we need the extra data of the words $\g_1,\dots,\g_k$ in
the alphabet $\{x_1,\dots,x_m\}$ obtained by traversing $\p\Delta_i$
$i=1,\dots,k$, in a counter-clockwise fashion, and reading
$x^{\pm1}_i$ upon encountering $\pm V_i$. In fact, precisely what
we need is the factorization of $\g_i$ as $\xi_i r_i\xi^{-1}_i$
where the letters of $r_i$ correspond only to those components of
$\SV\cap\Delta$ which have boundary on $\p(E(L))$. Therefore, given
any $m$-tuple of words $(\xi_i r_i\xi^{-1}_i)$ we shall say that
$(L,\SV,b)$ has scheme $S=(\xi_i r_i\xi_i^{-1})$ if the words
$\g_i$ are identical to the words $\xi_i r_i\xi_i^{-1}$ such that the
letters $r_i$ correspond precisely to those components of
$\SV\cap\Delta$ which have boundary on $\p E(L)$. A scheme is called
{\it reduced\/} if each $\xi_i$ is empty. $S$ determines $P$, or more
specifically, $S$ determines $w_i$ up to conjugacy for those $i$
which have strands intersecting $\Delta$.
Therefore $b$ induces a tangle decomposition of $(L,\SV)$, one of
which is a standard trivial disk link of type $b$, with standard
trivial Seifert surface system of some scheme $S$.
\bpage
\proclaim{Proposition 1.1} The tangle sum
$(L,\SV)\bigoplus\lm_{bb'}(L',\SV')$ of two links of basings $b$,
$b'$ of the same type and scheme $S$ may be added to yield
$(L\op L', \SV\op\SV')$. Here $L$ is a true link but $L'$ may have
disconnected components. If $b'$ separates $L'$ into 2 pure tangles
then the sum is a true link of as many components as $L$.
\endproclaim
\bpage
\sub{Proof} Since the boundaries of $(L,\SV)$ and $(L,\SV')$, after
deleting the standard trivial tangle of type $S$, are standard of
type $S$, the result is clear. The orientation-reversing nature of
the gluing map $r$ ensures that the orientations extend. \qed
\newpage
\subhead{\bf\S2. The geometric actions of boundary links
on links of pattern $P$}\endsubhead
\mpage
Now, suppose $(L,\SV)$ is an $m$-component homology boundary link
of pattern $P$ and $b=(b_1,\dots,b_k)$ is a generalized basing of
$(L,\SV)$. We describe how to ``twist'' $(L,\SV)$ by a
$k$-component boundary link $(B,\SW)$, resulting in a new
$m$-component homology boundary link of pattern $P$ but whose
Seifert form has been altered. Here we explain the action and give
examples. In the next section we describe the effect on the
Seifert form.
First, given $b=(b_1,\dots,b_k)$ we describe how to alter the
$k$-component boundary link $(B,\SW)$ to get a boundary link of
more components with a natural basing of type $(b_1,\dots,b_k)$.
This is done merely by forming parallel copies of the components of
$\SW$ dictated by the $b_i$. Specifically if
$b_1=x^{\e_1}_{i_1}\dots x^{\e_n}_{i_n}$ where $\e_i\in\{\pm1\}$
then form $n$ parallel copies of $W_1$, so that the $j^{\text{th}}$
copy is oriented ``oppositely'' to $W_1$ if $\e_j=-1$. Proceeding
around a positively oriented unbased meridian of $K_1=\p W_1$, one
encounters these copies in succession. Label the $j^{\text{th}}$
copy with the number $i_j$ as it appears in $b_1$. Similarly do the
same for $\{W_2,\dots,W_k\}$. Let $W'_i$ be the union of all copies
appearing with the label $i$. Thus we have formed a new boundary
link $(B',\SW')$ where we shall say $\SW'=(b_1,\dots,b_k)^\#(\SW)$.
This boundary link has many components as were involved in
$\Delta\cap L$. Note that $B'$ is very likely not a {\it true\/}
link since for any fixed $i$, more than one of its components may
have the label $i$. Also note that since $(B,\SW)$ has a basing
$b'$ of type $(x_1,\dots,x_k)$ this basing becomes a generalized
basing $b'$ of type $(b_1,\dots,b_k)$ for $(B',\SW')$, by
construction. Therefore we may form $L_b\op_{b'}B'$ (remember that
identical basings is sufficient to enable tangle addition of
{\it links\/}, whereas tangle sum of {\it Seifert surfaces\/}
requires identical schemes). Since $b'$ separates $B'$ into 2
{\it pure\/} tangles, $L_b\op_{b'}B'$ is indeed a true link with $m$
connected components. The result may be denoted $(b,b',B)^\#(L)$. The
definition of this action is independent of the pattern $P$ of $L$.
Indeed $L$ need not have been an homology boundary link. In the next
section, we see how to endow $B'$ with a Seifert system with scheme
$S$ and calculate the effect on $\th$. For now we consider examples
of this action.
\midinsert
\vspace{2.2in}
\botcaption{Figure 7}\endcaption
\endinsert
\newpage
\noindent{\bf Example 1}: {\bf Connected Sum}: If
$b=(x_1,\dots,x_m)$ then $\SW'=\SW$ and $(b,b',B)^\#(L)$ is merely
the usual connected-sum $L\#B$.
\bpage
\noindent{\bf Example 2}: {\bf Tying a local knot in $\BL$}: If
$b=(b_1)$ then $B$ is a knot $K$ and $(b,b',K)^\#(L)$ is obtained by
``seizing some strands'' of $L$ according to $b_1$ and tying the
whole thing in the knot $K$ as shown in Figure~7 for a link in $S^3$
and $b_1=x_1x^{-1}_1x_2$.
\vskip.7cm
\subhead{\bf\S3. Effect of Action on the Seifert form}\endsubhead
\mpage
Suppose $(L,\SV,b)$ is an $m$-component homology boundary link
with pattern $P$ and generalized basing $b=(b_1,\dots,b_k)$
inducing a scheme $S$. Suppose that $(B,\SW)$ is a boundary link of
$k$-components. In the previous section we described how to use
parallel copies of $\SW$ to form $(B',\SW',b')$ where $b'$ is of
the same type as $b$. This allowed us to form the tangle sum. Now
we endow $B'$ with a new system $\SW''$ of Seifert surfaces with
scheme $S$ so that the tangle sum can be performed on the surfaces
as well. This will endow the tangle sum with a surface system of
pattern $P$. Consider $\Delta_1$ as in Figure~1 induced by $b$. The
word $\g_1$ obtained by traversing $\p\Delta_1$ counter-clockwise
is necessarily a product of conjugates
$\g_1=\prod\lm_{j=1}^n\xi_j w^{\pm1}_{i_j}\xi^{-1}_j$ where
$P=(w_1,\dots,w_m)$ and
$b_1=x^{\pm1}_{i_1}x^{\pm1}_{i_2}\dots x^{\pm1}_{i_n}$ as shown by
example in Figure~8a (see Figure~5 and surrounding discussion). The
corresponding (trivial) scheme for the boundary link is shown in
8b. Now merely form parallel copies of $V_{i_j}$, changing
orientations and relabeling to achieve the identical $\g_1$ as in
8a. This is shown in 8c. Note that since $B'$ is a boundary link,
$\p V_{i_j}$ is connected so these relabellings will not be
inconsistent. The (reduced) scheme of 8c is not quite the same as
the (perhaps unreduced) scheme of 8a so we must join the oppositely
oriented copies of Seifert surfaces that correspond to the
conjugating elements in the word
$\g_1=\prod\lm_{j=1}^n\xi_j w^{\pm1}_{i_j}\xi^{-1}_j$. Note that this
is done by attaching an ``annulus'' $S^{2q-1}\x[-1,1]$ from
$\p V_{\xi_1}$ to $\p(-V_{\xi_1})$. Note that this last process does
not change $H_q$ so does not alter the Seifert form. Doing similar
modifications for $\Delta_i$, $1\le i\le k$, completes the
description of $(B',\SW'',b')$.
\midinsert
\vspace{2.4in}
\botcaption{Figure 8}\endcaption
\endinsert
\mpage
\sub{Definition 3.1} Given $b$, the result of acting on
$(L,\SV)$ by $(B,\SW,b')$ is the tangle sum $\(L_b\op_{b'}B',
\SV\op\SW''\)$ which is an $m$-component homology boundary link of
pattern $P$.
\mpage
To calculate the effect of this action on the cobordism class of
the Seifert form, first we will investigate the additivity of the
Seifort form under tangle sum of links in $S^{2q+1}$. We find that
this additive if $q\neq 1$ but, surprisingly, that additivity fails
in general for $q=1$. Fortunately, since $B'$ is a boundary link
the additivity will hold for $L\op B'$.
\mpage
\proclaim{Theorem 3.2} Suppose $(L,\SV,b)$, $(L',\SV',b')$ are links
in $S^{2q+1}$ with generalized basings of identical type and scheme
$S$. Suppose $L$ is a true link of $m$ components, image $b'$
intersects every component of $L'$, the non-trivial tangle
associated to $b'$ is pure and suppose that $\{j_1,\dots,j_n\mid
j_1<\dots$ and that it satisfies certain
``functorial'' properties.
\bpage
\proclaim{Proposition 3.4} Suppose
$f: F\left\lra F\left$ is
a homomorphism such that $f(x_i)$ is represented by the word $w_i$,
$1\le i\le k$. Then $f$ induces a homomorphism $f_*: G(k,\e)\lra
G(m,\e)$ which is geometrically defined by choosing a simple
boundary link with surface system $\SV$ representing $\a\in G(k,\e)$
then letting $f_*(\a)\equiv\th((w_1,\dots,w_k)^\#(\SV))$. In addition
$(\id)_*=\id$ and $(g\circ f)_*=g_*\circ f_*$.
\endproclaim
\bpage
\sub{Remark} Since $G(m, (-1)^q)$ has essentially been identified
with S\.~Cappell and J\.~Shaneson's $L$-theoretic group
$\G_{2q+2}(\BZ F\overset\e\to\lra\BZ)$, 3.4 reflects the
functoriality of the $\G$-groups. In section~5 we shall discuss
these connections.
\bpage
\sub{Proof} Although the matrix representing $f_*(\a)$ may be
described in algebraic terms, it is more intuitive to use Seifert
surfaces. Suppose $(L, \{V_1,\dots,V_k\})$ is a simple boundary link
in $S^{2q+1}$ with $\th(\SV)=\a$ \cite{Ko; Thm\.~3.4}. We form
$(w_1,\dots,w_k)^\#(\SV)$ as described earlier. Specifically, if
$w_1=x^{\e_1}_{i_1}\dots x^{\e_n}_{i_n}$, consider $n$ parallel
copies of $V_1$, the $j^{\text{th}}$ of which is oriented
oppositely to $V_1$ if $\e_j=-1$. Proceeding around a
positively-oriented meridian to $\p V_1$, one encounters these
copies in succession and assigns the label $i_j$ to the
$j^{\text{th}}$ copy. Do the same for $V_2$ through $V_k$ to
complete the definition of $(w_1,\dots,w_k)^\#(\SV)$. Finally set
$f_*(\a)$ equal to $\th((w_1,\dots,w_k)^\#(\SV))$. \qed
\bpage
We need to show $f_*(\a)$ is independent of the representatives
$w_i$ and of $(V_1,\dots,V_k)$. For simplicity let
$(w_1,\dots,w_k)^\#(\SV)$ be abbreviated $w^\#(\SV)$. First,
suppose $(J, \{W_1,\dots,W_k\})$ is another such representative of
$\a$. We may form a connected sum of $L$ with the concordance
inverse of $J$ in such a way that $L\#-J$ is a simple boundary link
admitting the system $\SV\natural -\SW$, and $\th$ of this system is
$\a-\a=0$ by 3.3. But it is easy to see that
$w^\#(\SV\natural-\SW)=w^\#(\SV)\natural w^\#(-\SW)$, so that the
block sum of $\th(w^\#(\SV))$ and $-\th(w^\#(\SW))$ is represented
by $\th(w^\#(\SV\natural -\SW))$. It suffices to show the latter is
zero. Since $\th(\SV\natural -\SW)=0$, there is a choice of basis
of $H_q$ of each component of $\SV\natural -\SW$ with respect to
which the Seifert matrix is composed of blocks $N_{ij}$ each of the
form
$$
\pmatrix
0 & C_{ij}\\
D_{ij} &E_{ij}
\endpmatrix
$$
as described in \cite{Ko; p.668}. Thus, with respect to the
``same bases'', the $(i,j)$ block of the Seifert matrix for
$w^\#(\SV\natural -\SW)$ will consist of sub-blocks each of which
is some $\pm N_{s,t}$. But such a block is congruent to one of the
form
$$
\pmatrix
0 &A\\
B &C
\endpmatrix
$$
by merely re-ordering basic elements. Thus
$\th(w^\#(\SV\natural -\SW))=0$, so $\th(w^\#(\SV))=\th(w^\#(\SW))$
as desired.
Now suppose $w_i$ and $z_i$ are words which are equal in the free
group. It suffices to consider the case that $z_i$ is obtained from
$w_i$ by inserting $x_jx^{-1}_j$ somewhere in $w_i$. Suppose
$(L,\SV)$ and $\SV'=w^\#(V)$ are as above in the definition of
$f_*$. Let $\SV''=z^\#(\SV)$, so $\SV''$ is $\SV'$ together with 2
more copies of $V_i$ (oppositely oriented) which form part of
$\SV''_j$. Consider the product
$(S^{2q+1}\x [0,1], L\x [0,1], \SV'\x [0,1])$. This is the product
concordance from $L$ to $-L$ together with the product
``cobordism'' from $\SV'$ to $-\SV'$. Now in $S^{2q+1}\x\{0\}$
insert the extra manifolds, $V_i\amalg-V_i$, to form $\SV''$ and
in $S^{2q+1}\x[0,1]$ insert the product $V_i\x[0,1]$ in such a way
that $\p(V_i\x[0,1])=V_i\amalg-V_i$. Then the resulting collection
is what we might call a boundary cobordism from $(L, \SV'')$ to
$(-L, -\SV')$. In particular, we may look at the union of $\SV''$
with $-\SV'$ together with $\p\SV'\x[0,1]$ as a collection $\SW$ of
closed $2q$-manifolds. The argument of \cite{Ko; Lemma~3.3 and
page~671} applies to show $\th(\SW)=0=\th(-\SV')\op\th(\SV'')$.
Therefore $\th(\SV')=\th(\SV'')$ showing that $f_*(\a)$ is only
dependent on the class of $w_i$ in the free group.
The ``functorial'' properties of $f_*$ are straightforward to
verify. \qed
\bpage
We can now evaluate the effect of our actions on $\th$.
\bpage
\proclaim{Theorem 3.5} Suppose $(L,\SV,b)$ is an $m$-component
homology boundary link of pattern $P$ with fixed generalized
basing $(b_1,\dots,b_k)$ for which the loops
$\{\p\Delta_1,\dots,\p\Delta_k\}$ intersect $\SV$ in the words
$\{w_1,\dots,w_k\}$ (see section~1). Suppose $(B,\SW)$ is a
$k$-component boundary link. Then the result of acting on
$(L,\SV,b)$ by $(B,\SW)$ has Seifert form equivalent to
$$
\th(\SV)\op f_*(\th(\SW))
$$
where $f_*: G(k, (-1)^q)\lra G(m, (-1)^q)$ is induced by
$f: F\left\lra F\left$
given by $x_i\lra w_i$.
\endproclaim
\bpage
\sub{Proof} The result of the action is
$(L_b\op_{b'}B', \SV\op\SW'')$ as defined previously so, by 3.3,
$\th$ is $\th(\SV)\op\th(\SW'')$. But
$\th(\SW'')=\th((w_1,\dots,w_k)^\#(\SW))$ as remarked below 3.3.
Then apply 3.4. \qed
\midinsert
\vspace{4in}
\botcaption{Figure 10}\endcaption
\endinsert
\bpage
\proclaim{Theorem 3.6} Given any scheme
$S=(\eta_1 w_1\eta^{-1}_1,\dots,\eta_m w_m\eta^{-1}_m)$ inducing
the pattern $P$ and any $\a\in G(m, (-1)^q)$ (subject to the usual
signature restrictions if $q=2$), there is an $m$-component ribbon
link $(R,\SV)$ and an (ordinary) basing $b$ such that $(R,\SV,b)$
has scheme $S$ (and pattern $P$) and $\th(R,\SV)=0$. By acting on
$(R,\SV)$ appropriately by a boundary link with Seifert form $\a$,
one obtains an homology boundary link $(L,\SV',b')$ with scheme
$S$, pattern $P$ and $\th(L,\SV')=\a$.
\endproclaim
\midinsert
\vspace{2.8in}
\botcaption{Figure 11}\endcaption
\endinsert
\midinsert
\vspace{3in}
\botcaption{Figure 12}\endcaption
\endinsert
\midinsert
\vspace{3in}
\botcaption{Figure 13}\endcaption
\endinsert
\bpage
\sub{Proof} According to \cite{CL; Thm\.~2.3}, every pattern
$P$ is the pattern of a ribbon homology boundary link in $S^{2q+1}$.
More precisely, for every $m$-tuple of words
$(\eta_i w_i\eta^{-1}_i)$ representing a pattern $P$, there is a
ribbon homology boundary link $R$, a map
$g_*: \pi_1(ER)\to F\left$ and a basing
$b=(u_1,\dots,u_m)$ such that $g_*([u_i])=[\eta_i w_i\eta^{-1}_i]$.
But then there exists a map $g: ER\to\bigvee\lm_{i=1}^m S^1$
inducing $g_*$ such that pulling back points under $g$ yields (via
the Thom-Pontryagin construction) $\SV=(V_i)$ a system of Seifert
surfaces (perhaps disconnected) for $R$ such that, with respect to
$b$, $(R,\SV)$ has scheme
$(\xi_1r_1\xi^{-1}_1,\dots,\xi_mr_m\xi^{-1}_m)$ where
$\xi_ir_i\xi^{-1}_i=\eta_i w_i\eta^{-1}_i$ in the free group $F$.
We shall now {\it alter\/} $\SV$ by moves called elementary
reductions and enlargements, until $(R,\SV,b)$ has scheme $S$. To
explain these moves, consider Figures~10--13. A general scheme
$(\Delta_i, \Delta_i\cap\SV)$ is shown in 10a which can be encoded
by the unreduced word $\xi_ir_i\xi^{-1}_i$. The first elementary
reduction (Figure~10b) fuses adjacent copies $V_j$ and $-V_j$
allowing for a potential cancellation of any occurence of
$x_jx^{-1}_j$ or $x^{-1}_jx_j$ in $r_i$. The first elementary
expansion involves adding a small $S^{2q-1}\x[0,1]$ as a new
component of $V_j$ as shown in 11. This allows for the insertion of
$x_jx^{-1}_j$ or $x^{-1}_jx_j$ in $r_i$. The second elementary
reduction and its inverse are shown in Figure~12. Using this move we
may alter $\SV$ to assume $\xi_i=n_i$ and $r_i=w_i$ as elements of
the free group. Using the first moves, we can assume $r_i=w_i$ as
words. Finally, the third elementary reduction (respectively
expansion) is shown in Figure~13a (13b). Using this move we can
assume $\SV$ has precisely the given pattern $S$. These moves do not
change the Seifert form of $\SV$ except for the second elementary
reduction, which changes $V_j$ by an ambient 1-handle addition and
thus do change the cobordism class of the Seifert form. The
resulting $(R,\SV,b)$ may not be satisfactory since $V_j$ may not be
connected. We must alter $\SV$ further to remedy this. However
before proceeding note that the Thom-Pontryagin construction applied
to $(R,\SV)$ yields a map $g'$ homotopic to $g$. If $A$ and $B$ are
two components of $V_j$, choose a path $\d$ in $E(L)$ from the
positive side of $A$ to the negative side of $B$, which meets $\SV$
transversely and misses the basing disk $b$.
Let $*$ denote the wedge point of $\bigvee\lm_{i=1}^m S^1$ and let
$y$ denote the mid-point of the $j\supth$ circle so
$(g')^{-1}(y)=V_j$. The image of $\d$ under $g'$ represents an
element of $\pi_1\(\bigvee\lm_{i=1}^m S^1, y\)$. Since $g'_*$ is
surjective, the path $\d$ can be altered so that its image under
$g'$ represents zero in $\pi_1$. Thus $\d$ hits $\SV$ in a pattern
such that the corresponding word may be reduced to the empty word
by deleting occurences of $x_ix^{-1}_i$ or $x^{-1}_ix_i$. Hence by
tubing of $V_i$ along $\d$, say, we may alter $V_i$ so that it
misses $\d$ until $\d$ is a path in complement of $\SV$ connecting
$A$ to $B$. Then $A$ may be joined to $B$ by tubing. The resulting
$(R,\SV,b)$ is the desired ribbon link with scheme $S$. Moreover,
if $g''$ represents the associated Thom-Pontryagin map, then
$g''_*=g'_*=g_*$ by the same argument as that of \cite{Ko; 2.2}.
Since $\{\eta_iw_i\eta^{-1}_i\mid 1\le i\le m\}$ normally generates
the free group, there are disjointly embedded loops
$\g_1,\dots,\g_m$ in $E(R)$ sharing the common basepoint $*$,
disjoint from $b$ (each of which travels to a component $R_i$,
traverses a meridian, returns along nearly the same path and sets
off again, et~cetera) such that $g_*([\g_i])=x_i$. These loops
$\g_i$ induce a generalized basing $b$ where $\g_i=\p\Delta_i$.
Choose a boundary link $(B,\SW,b')$ with $\th(\SW)=\a$ (and trivial
basing $b'$). Act on $(R,\SV,b)$ by $(B,\SW,b')$. The result, by
3.1, is an $m$-component homology boundary link of scheme $S$ with
Seifert form equivalent to
$\th((R,\SV))\op\id_*(\a)=\th(R,\SV)\op\a$ by 3.5.
We must now see that $\th(R,\SV)=0$. Recall the system of Seifert
surfaces induces, by the Pontryagin construction, a map
$g: E(R)\lra\bigvee\lm_{i=1}^m S^1$ such that the inverse image of
$\{1\}$ on the $i^{\text{th}}$ circle is $V_i$. Now, the proof of
Theorem~2.3 \cite{CL} shows (see the proof of Theorem~3.1 of
\cite{L1} for a more complete argument) that $R$ may be chosen to
possess slice disks $\{D_1,\dots,D_m\}=\SD$ in $B^{2q+2}$ such that
$\pi_1(E(R))\overset j_*\to\lra\pi_1(E(\SD))$ is an epimorphism
(isomorphism if $q\neq 1$). Extend $g$ over the boundaries of the
tubular neighborhoods of the $D_i$ in the obvious way. Since
$H_2(\pi_1(E(\SD)))=0$, a theorem of Stallings \cite{St} ensures
that $j_*$ induces an isomorphism modulo the intersection of the
finite terms of the lower-central series. Since free groups are
$\om$-nilpotent, $g$ extends to
$\wh g: E(\SD)\lra\bigvee\lm_{i=1}^m S^1$. After modifying $\wh g$
by an isotopy rel~$g$, let $W_i$ be the inverse image of $\{1\}$ on
the $i^{\text{th}}$ circle. Then $\p W_i$ is $V_i$ together with
various copies of $D^{2q}$ glued along the components of $\p V_i$.
This collection $\{W_i\}$ shows that $\th(\SV)=0$ as in
\cite{Ko; Lemma~3.3 and page~24}. \qed
\bpage
Recall that we have failed to establish addivity of Seifert form
under tangle sum when $q=1$. The following shows that this will
hold for {\it ordinary\/} connected sum of classical links if those
links are obtained from acting on ribbon links by boundary links and
the connected sum avoids the boundary link tangles. This establishes
details of certain claims of additivity in Chapter~3, section~B of
\cite{CO2}.
\bpage
\proclaim{Theorem 3.7} Suppose $(L,\SV)$ is an homology boundary
link with scheme $S$ in $S^3$ which is obtained from the boundary
link $(B_0,\SW_0)$ acting on the ribbon link $(R_0,\SV_0,b_0)$.
Similarly suppose $(L',\SV')$ is another such obtained from
$(B_1,\SW_1)$ acting on the ribbon link $(R_1,\SV_1,b_1)$. Finally
suppose that $b=(x_1,\dots,x_m)$ is a (normal) basing of $(L,\SV)$
and $b'$ of $(L',\SV')$ (with respect to which $L\op L'$ is the
ordinary connected sum of links) each of which is disjoint from
their respective boundary link tangle summand. Then
$\th(L\op L')=\th(L,\SV)\op\th(L',\SV')$.
\endproclaim
\bpage
\sub{Proof} Since $b$ and $b'$ lie entirely within the ``ribbon
link tangle'' summands of $L$ and $L'$ respectively,
$(L\op L', \SV\op\SV')$ is merely the result of acting on
$(R_0,\SV_0,b)\op(R_1,\SV_1,b')$ first by $(B_0,\SW_0)$ and then by
$(B_1,\SW_1)$ (or vice-versa). By 3.5,
$\th(L,\SV)=\th(R_0,\SV_0)\op f_*(\th(B_0,\SW_0))$ and
$\th(L',\SV')=\th(R_1,\SV_1)\op f'_*(\th(B_1,\SW_1))$ where $f_*$
is defined by the way $\p\Delta_i\subset b_0$ intersects $\SV_0$
and $f'_*$ by the way $\p\Delta_i\subset b_1$ intersects $\SV_1$.
Similarly $\th(L\op L')$ is, using our first remark,
$\th(R_0\op R_1)\op f_*(\th(B_0,\SW_0))\op f'_*(\th(B_1,\SW_1))$.
Since $\th(R_0,\SV_0)=\th(R_1,\SV_1)=0$ by the proof of 3.6, we
need only show that $\th((R_0,\SV_0,b)\op(R_1,\SV_1,b'))$ is zero.
Since $b$ and $b'$ are ordinary basings we may use the well-known
fact that the connected-sum of two ribbon links is a ribbon link.
By 1.1, $R_0\op R_1$ is an homology boundary link with surface
system $\SV_0\op\SV_1$. Then the proof of 3.6 shows that
$\th(R_0\op R_1)=0$. Hence $\th(L\op L')=\th(L)\op\th(L')$ as
desired. \qed
\midinsert
\vspace{2.3in}
\botcaption{Figure 14}\endcaption
\endinsert
\bpage
\sub{Example 3.8} We will show how to construct a $2$-component
homology boundary link in $S^{2q+1}$ with arbitrary Seifert form $\a$
and with pattern $(x,yw)$, where $w$ is an element of the subgroup of
$F\left$ generated by
$\{x, x^{-1}, yxy^{-1}, yx^{-1}y^{-1}\}$ (and also lies in the
commutator subgroup). First we construct a ribbon homology boundary
link with the correct pattern. This link will be a {\it fusion\/}
of a 3 component trivial link \cite{C1, C2} and in fact what has been
called a {\it strong fusion\/} of a 2-component trivial link by
U\.~Kaiser \cite{Ka}. As an aside, we note the fascinating fact
that Theorem~3.15 of \cite{Ka} proves that the patterns $(x,wy)$ of
type above are the {\it only\/} ones possible for a strong fusion
of a 2-component boundary link. Express $w$ as a word in
$\{x, x^{-1}, yxy^{-1}, yx^{-1}y^{-1}\}$ so $w=w_1\dots w_n$.
Form a trivial link of $n$ components in $S^{2q+1}$ by nesting as in
Figure~14a.
\midinsert
\vspace{2.3in}
\botcaption{Figure 15}\endcaption
\endinsert
The ``first'' component is innermost, et cetera. Orient the
$i^{\text{th}}$ component counter-clockwise if $w_i=x$ or
$yxy^{-1}$, otherwise clockwise. Join all components corresponding to
$w_i=yx^{\pm1}y^{-1}$ to the left as in 14b, respecting
orientation, and join all components corresponding to $w_i=x^{\pm1}$
to the right as shown in 14b. The result is a trivial link of 2
components $\{J_1,J_2\}$. Form a ribbon knot $K_1$ by ``fusing''
$J_1$ to $J_2$ using a single ``band'' $b$ (tube if $q>1$) that
originates at $*_1$, dives down through all the nested circles and
terminates at $*_2$ as shown in Figure~15. Lastly add a trivial
component $K_2$ as shown in Figure~15.
\midinsert
\vspace{2.5in}
\botcaption{Figure 16}\endcaption
\endinsert
Then there is a system of Seifert surfaces $\SV=(V_x,V_y)$ for the
homology boundary link $R=\{K_1,K_2\}$ such that $\mu_1$ spells the
word $x$ while $\mu_2$ spells $yw$. The Seifert surface $V_x$ for
$K_1$ is a union of ``disks with holes'' and tubes as shown in
Figure~16. The tubes are nested and run along $b$, each terminating
as a longitude of $K_2$. $V_y$ is a union of ``cocoons with holes''
and tubes as shown in Figure~17. The tubes are nested (with each
other and with the tubes of $V_x$) and run along $b$, terminating in
longitudes of $K_2$. Thus $R$ is the desired ribbon link with
surface system.
\midinsert
\vspace{2.5in}
\botcaption{Figure 17}\endcaption
\endinsert
\midinsert
\vspace{3.1in}
\botcaption{Figure 18}\endcaption
\endinsert
Now, as in the proof of 3.6, we must find paths $\{\g_1,\g_2\}$
which spell $\{x,y\}$. These are shown in Figure~18. Now, for
example, suppose $\a$ were the Seifert form for the {\it split\/}
link $\{J_1,J_2\}$. Then the desired homology boundary link with
pattern $(x,wy)$ and form equivalent to $\a$ would be as shown in
Figure~19.
\midinsert
\vspace{3.1in}
\botcaption{Figure 19}\endcaption
\endinsert
We remark in passing that the links in \cite{CO2; A\. The
Simplest Examples} are of this general type with $w=[x,y^{-1}]^m$,
$\g_1=[x,y^{-1}]$ $\g_2=$ the empty word and $\a$ the form of a
knot $J$. In addition the examples in Section~B, Figure~3.12 of
that paper are of the same family with
$\g_1=y^{-1}[y^{-1},x]^{m-1}$, $\g_2$ being the empty word and $\a$
being the form of a knot~$J$.
\vskip.7cm
\subhead{\bf\S4. Blanchfield forms of Simple Homology
Boundary Links}\endsubhead
\mpage
We follow the development of \cite{Du} where Blanchfield forms
were defined for boundary links and mimic developments of
\cite{H1, 14--15, 122--124} \cite{H2, page~372}. Suppose $L$ is
an $m$-component homology boundary link in $S^{2q+1}$ equipped with
a homomorphism
$\phi: \pi_1(E(L))\lra F\left$. Then
$(L,\phi)$ induces a regular covering space $\wt X$ of $X=E(L)$
whose group of deck translations is identified with the free group.
$\wt X$ is unique up to covering space isomorphism and the
identification is unique up to a global conjugation in the group of
deck translations. If $\phi$ were surjective then $\wt X$ would
merely be the usual connected covering space associated to the
kernel of $\phi$. Any {\it such\/} cover is covered by the
$(\pi_1(E(L)))/(\pi_1(E(L)))_\om\cong F\left$
cover. If $\phi$ is not surjective then $\wt X$ is a disjoint union
of copies of the connected cover associated to
$\phi: \pi_1(E(L))\twoheadrightarrow\image\phi$. Let $A=\BZ[F]$
endowed with the involution
$\ov{\sum n_iw_i}=\sum n_iw^{-1}_i$, let $H_*(X,A)$ denote the right
$A$-module $H_*(\wt X\ ;\ \BZ)$, and $M=H_q(X,A)$. If $q=1$, some
modification is necessary. There seem to be two ways to proceed. The
first is to consider the quotient module $M=H_1(\wt X)/H_1(\p\wt X)$
which is the same as the quotient of $H_1(X,A)$ by the $A$-submodule,
denoted $\SL$, generated by lifts of longitudes which must lie in
kernel $\phi$ if $L$ is an homology boundary link. Later in this
section we shall explicitly investigate this situation and see that
a Blanchfield form can be defined on this module. The second way is
to restrict to those $(L,\phi)$ for which there exists a ribbon
homology boundary link $(R,\psi)$ and a degree one map relative
boundary $f: E(L)\lra E(R)$ such that $\psi\circ f_*=\phi$. If
$\wt{E(L)}$ and $\wt{E(R)}$ are the covering spaces associated to
$\phi$ and $\psi$, let $\wt Z$ be the mapping fiber of
$\tl f: \wt{E(L)}\lra\wt{E(R)}$ \cite{Wh, p\.~43}. Let $M$ denote the
$A$-module $H_*(\wt Z\ ;\ \BZ)$ in this case. In 4.4, we shall show
that these two Blanchfield forms, while not isomorphic, are
equivalent in the relevant Witt group.
Now we return to the general case. Let $\La$ denote the Cohn
localization of $A$ with respect to the augmentation $\e: A\lra\BZ$
(see \cite{Du}). Recall that $A\overset i\to\lra\La$ is an embedding
with the property that any square matrix over $A$ which is
invertible when augmented, is invertible over $\La$. Recent work of
M\.~Farber and P\.~Vogel has identified $\La$ as the ring of
``rational functions'' in non-commuting variables \cite{FV}. We
wish now to restrict ourselves to ``simple'' homology boundary links.
\bpage
\sub{Definition 4.1} (compare \cite{Du, \S6} \cite{Ko; 2.8}) An
homology boundary link $(L,\SV)$ in $S^{2q+1}$ is {\it simple\/} if
each Seifert surface $V_i$ is $(q-1)$-connected.
\bpage
Then we define a $(-1)^{q+1}$-Hermitian ``Blanchfield linking
form'' $B: \ov{H_q(X,A)}\lra\Hom_A(H_q(X,A), \La/A)$ (see $B'$ in
\cite{Du, 624}) as follows. Consider the intersection form
$C_q(\wt X)\ox_\BZ C_{q+1}(\wt X)\lra A$ denoted by $\cd$, inducing
$I: H_{q+1}(X, \La/A)\ox_\BZ H_q(X,A)\lra\La/A$ given by
$I(C\ox\a, y\ox\b)=\bar\b\(\sum\lm_{\la\in F}(C\cd y\la)\la\)\a$
where $\a$, $\b\in\La/A$, $C\in C_{q+1}(\wt X)$, $y\in C_q(\wt X)$.
Consider also $\p_*: H_{q+1}(X, \La/A)\lra H_q(X,A)$. Then set
$B(x,y)=I(\p^{-1}_* x,y)$. In case $A$ were commutative this agrees
with \cite{CO, \S1} and \cite{H1; 120} but differs slightly from
\cite{Du; 624}. The pair $(M,B)$ shall be referred to as
{\it the Blanchfield form associated to\/} $(L,\phi)$. One key point
of \cite{Du} was to ensure that
$\p_*: H_{q+1}(X,\La/A)\lra H_q(X,A)$ be an isomorphism by showing
$H_{q+1}(X,\La)\cong H_q(X,\La)\cong 0$. Suppose $q>1$ and let $W$
be a wedge of $m$ circles. Then $\phi$ induces $\phi: X\lra W$ and
$\tl\phi: \wt X\lra\wt W$. Since $\phi$ is an integral homology
equivalence up to and including dimension $2q-1$, $\tl\phi$ is a
$\La$-homology equivalence in the same range (see page~624 of
\cite{Du}). But $\wt W$ is a $1$-complex so
$H_{q+1}(X,\La)\cong H_q(X,\La)\cong 0$. If $q=1$, since
$f: E(L)\lra E(R)$ is an isomorphism on integral homology, $\tl f$
is an isomorphism on $\La$-homology and so $\ov H_*(X,\La)=0$.
Strictly speaking, the above extension of DuVal serves to define
only the Blanchfield form associated to the ``free'' cover of
$E(L)$ associated to the epimorphism
$\phi: \pi_1(E(L))\twoheadrightarrow F\left$
induced by $V$. An arbitrary homomorphism
$\phi'': \pi_1(E(L))\lra F\left$ factors as
$$
\pi_1(E(L))\overset\phi\to\twoheadrightarrow
\pi_1(E(L))/(\pi_1(E(L)))_\om\overset\psi\to\twoheadrightarrow
F\left\overset f\to\hookrightarrow
F\left
$$
\vskip.5cm
\flushpar where $\psi$ is onto and $f$ is injective. Suppose $(M,B)$
is the Blanchfield form associated to $\phi$. The Blanchfield form
associated to $\phi'$ is defined to be $(M',B')$ where
$M'=M\ox_{\BZ H}\BZ F\left$ where
$H=\pi_1/(\pi_1)_\om$ and if $x$, $y\in M$, $\a$,
$\b\in F\left$,
$B'(x\ox\a, y\ox\b)=\bar\b\psi_*(B(x,y))\a$. Alternatively it is
easily seen that DuVal's work and the definitions above extend
trivially to {\it these\/} ``reduced free covers'' and so the
previous definition may be used and agrees with this one. The
Blanchfield form $(M'',B'')$ associated to $\phi''$ is then given by
$M''=M'\bigotimes\lm_{\BZ F[x_1,\dots,x_k]}\BZ F[x_1,\dots,x_m]$ and
$B''(x\ox\a, y\ox\b)=\bar\b f_*(B(x,y))\a$. Here the covering space
associated to $\phi''$ is a union of disjoint copies of that
associated to $\phi'$. The fact that this definition of $B''$
agrees with the obvious generalization of DuVal (given by our
original formula for $B$), is obtained in a manner precisely like the
proof immediately preceeding Theorem~1.9 of \cite{CO2}.
Another key point for DuVal was that the module on which $B$ is
defined be of type $S$. We shall presently see that this is the
case, also implying that they are $Z$-torsion free
\cite{Du; Proposition~4.1}.
We shall show that the Blanchfield form is determined by the
Seifert matrix for a simple homology boundary link $(L,\SV)$ where
by {\it the Blanchfield form\/} of $(L,\SV)$ we mean that associated
to the map $E(L)\overset\phi\to\lra\bigvee\lm_{i=1}^m S^1$ by the
Pontryagin construction applied to $\SV$.
For such a simple homology boundary link let
$Y=E(L)-\bigcup\lm_{i=1}^m W_i$ where $W_i$ is an open tubular
neighborhood $V_i\x[-1,1]$ of $V_i$. Let $Z$ be the complex
obtained by identifying all those boundary components of $\SV$ which
are an $i^{\text{th}}$ longitude, $i=1,\dots,m$. Then
$H_j\(\bigcup\lm_{i=0}^m W_i\)\cong H_j(Z)$ if $j\neq 0$, $1$,
$2q-1$. By Alexander Duality,
$H_q(Y)\cong H_q(S^{2q+1}-Z)\cong H^q(Z)\cong\Hom(H_q(Z),\BZ)\op
\Ext(H_{q-1}(Z),\BZ)\cong\Hom(H_q(\SV)\ ;\ \BZ)$ if $q\neq 1$
(note if $q=2$, $H_{q-1}(Z)$ is torsion-free). Therefore
$H_q(\SV)\cong H_q(Y)$ are free abelian of the same rank. Choose a
basis $\{\a_{ik}\mid 1\le k\le r(i)\}$ for $H_q(V_i)$,
$1\le i\le m$.
Since the isomorphism above is detected by ordinary linking number
in $S^{2q+1}$, we may choose a basis $\{\hat\a_{jn}\}$ for $H_q(Y)$
such that $lk(\a_{ik},\hat\a_{jn})=\d_{ij}\d_{kn}$.
Suppose now that $(L,\SV)$ is an $m$-component simple homology
boundary link in $S^{2q+1}$. Then there is a continuous map
$f: E(L)\lra\bigvee\lm_{i=1}^m S^1$ and points $p_i$ on the
$i^{\text{th}}$ circle such that $f^{-1}(p_i)=V_i$. Such an $f$
induces homomorphism $f_*=\phi$ as above. If $f_*$ is onto then the
covering space $\wt X$ so induced may be constructed as in
\cite{H1, page~14} by splitting $E(L)$ open along $V$. Then there
is a Mayer-Vietoris sequence:
$$
A\ox H_q(\SV)@>d>> A\ox H_q(Y) @>i>> H_q(X ; A)
@>\p>> A\ox H_{q-1}(\SV)
$$
where $d(\g\ox\a_j)=\g x_j\ox(i_{j+})(\a_j)-\g\ox(i_{j-})(\a_j)$
for $\a_j\in H_q(V_j)$ and $i_{j\pm}$ the two inclusions
$V_j\lra Y$. Since $L$ is simple, $H_{q-1}(\SV)=0$. By our remarks
above, if $q\neq 1$ then with respect to the bases $\{\a_{ik}\}$,
$\{\hat\a_{jn}\}$, the matrix of $(i_+)_*: H_q(\SV)\lra H_q(Y)$ is
merely $\th$ where $\th$ is the Seifert matrix for $\SV$ relative
to $\{\a_{ik}\}$. Moreover the map
$d: A^{\sum r(i)}\lra A^{\sum r(i)}$ is given by the matrix
$\Delta=\G\th+\e\th^T$ where $\e=(-1)^q$, $\G$ is the block
diagonal matrix $(x_1 I_{r(1)},\dots, x_m I_{r(m)})$ with
$I_{r(i)}$ the identity matrix of rank $r(i)$. Therefore $\Delta$
yields a presentation matrix for the module $H_q(X\ ;\ A)$. Since
$\th+\e\th^T$ is unimodular, $\Delta$ is invertible when augmented.
Therefore, {\it by definition\/} of the Cohn localization $\La$,
$\Delta$ is invertible in the larger ring $\La$. In particular $d$
and $\Delta$ are injective, establishing that $H_q(X,A)$ is of type
$S$ when $q>1$.
We may now compute the Blanchfield form, mirroring
\cite{H1; 122--123}. Suppose $C_{ik}$ denotes a fixed translate of
the $(q+1)$-chain $[-1,1]\x\a_{ik}$ in $\wt X$. Note that
$\p C_{kn}=x_k\ox\a^+_{kn}-1\ox\a^-_{kn}=d(\a_{kn})$, so for any
$w\in H_q(\SV)\ox A$, $w=\sum w_{kn}\a_{kn}$ and
$\p(\sum w_{kn}C_{kn})=dw$. Now, to compute $B(z,y)$ where
$z=ir=i(\sum r_{kn}\hat\a_{kn})$ and
$y=i(\sum s_{jm}\hat\a_{jm})$, set $w=d^{-1}r$ or
$w_{kn}=(\Delta^{-1}\cd r)_{kn}$. Then one sees that
$z=i\circ\p(\sum(\Delta^{-1}\cd r)_{kn}C_{kn})$. Thus
$$
\split
B(z,y) &= I\(i\(\sum(\Delta^{-1}\cd r)_{kn}C_{kn}\),
i\(\sum s_{jm}\hat\a_{jm}\)\)\\
&= \sum\lm_{j,m}\bar s_{jm}\(\(\sum\lm_{k,n} I(C_{kn},
i(\hat\a_{jm}))\) (\Delta^{-1}\cd r)_{kn}\).
\endsplit
$$
But $I(A_{kn}, i(\hat\a_{jm}))=\d_{jk}\d_{mn}(1-x_k)$ so
$$
\boxed{B(z,y) = \bar s^T(I - \G)\Delta^{-1} r\quad\mod A}
$$
where $r$ and $s$ are here viewed as column vectors. Summarizing,
we have shown the following for $q>1$. The proof for $q=1$, using
our first definition of Blanchfield forms, is immediately below.
\bpage
\proclaim{Theorem 4.2} If $(L,\SV)$ is a simple homology boundary
link in $S^{2q+1}$, then with respect to the generators
$i(\hat\a_{kn})$ as defined above, the Blanchfield form is
represented by the square matrix
$(I-\G)(\G\th+(-1)^q\th^T)^{-1}$ where $\th$ is the Seifert
matrix with respect to $\a_{kn}$ and $\G$ is the block diagonal
matrix defined above, and $I$ is the identity matrix. The module
$H_q(X,A)$ is presented by the matarix $\G\th+(-1)^q\th^T$.
\endproclaim
\bpage
While the terminology is fresh in the reader's mind, we turn to the
case $q=1$. We shall show that there is a Blanchfield pairing on
the quotient module $H_1(\wt X)/H_1(\p\wt X)\cong H_1(X,A)/\SL$.
Since $H_1(\p\wt X)$ is generated by lifts of longitudes, the
inclusion-induced map $H_1(\p X-V)\ox A\lra H_1(\p\wt X)$ is onto.
The argument of \cite{H2; p\.~373} works almost word for word even
though that argument concerned the Blanchfield form on the
universal {\it abelian\/} covering space. One special argument is
necessary to establish that the map
$i: H_1(X-V)\ox A\lra H_1(\wt X)$ of the Mayer-Vietoris sequence is
onto in our case. For this consider the map
$\phi: X\lra\bigvee\lm_{i=1}^m S^1$ such that
$\phi^{-1}(\{p_i\}\in S^1)=V_i$, which induces
$\tl\phi: \wt X\lra\wt W$ where $W$ is the wedge. Therefore there
is a map of chain complexes as below
$$
\define\toparrow{@>\pretend\e\haswidth
{\text{Clifford multidc}}>>}
\define\atoparrow{@>\pretend\p\haswidth
{\text{Cliffo}}>>}
\define\dtoparrow{@>\pretend d_0\haswidth
{\text{Cliffo}}>>}
\split
H_1&(\wt X)\atoparrow H_0(V)\ox A\dtoparrow H_0(Y)\ox
A\quad \toparrow\quad\BZ\\
&\Biggl\downarrow \tl\phi\hskip50pt\Biggl\downarrow\cong\hskip60pt
\Biggl\downarrow\cong\hskip107pt\Biggl\downarrow\cong\\
H_1&(\wt W)\atoparrow H_0(\cup\{p_i\})\ox A\overset(d_0)'\to\lra
H_0\(W - \textstyle{\bigcup\lm_{i=1}^m\{p_i\}}\)\ox A\lra\BZ.
\endsplit
$$
Since $\wt W$ is contractible, $(d_0)'$ is injective. Since the
middle vertical maps are isomorphisms, $d_0$ is also injective
implying that $i$ above is onto.
Hillman's arguments result in the exact sequence
$$
\f{H_1(V)}{H_1(\p V)}\ox A\overset d\to\lra
\f{H_1(Y)}{H_1(\p X-V)}\ox A\overset i\to\lra
\f{H_1(\wt X)}\SL\lra 0
$$
where the first two terms are shown to be free $A$-modules of the
same rank $(\rank H_1(V)-\rank H_1(\p V)+m)$. Moreover, if
$\{\a_{ik}\}$ is a basis for $H_1(V)/H_1(\p V)$ represented by
loops on $V$ and $\hat\a_{ik}$ the corresponding elements in
$H_1(Y)$ such that $lk(\a_{ik},\hat\a_{jn})=\d_{ij}\d_{kn}$, then
clearly $\{[\hat\a_{jn}]\}$ generates $H_1(Y)/H_1(\p X-V)$ since
each $V_i$ is homotopy equivalent to a $1$-complex. Furthermore
this set is linearly independent because if
$\sum n_{jn}\hat\a_{jn}=\g\in H_1(\p X-V)$ then
$0=lk(\a_{ij},\g)=n_{ij}$ since $H_1(\p X-V)$ is generated by
longitudes and the $V_i$ give null-homologies for the longitudes
(disjoint from $\a^+_{ij}$). Therefore the matrix of $d$ is given
by the same square matrix as in the case $q>1$ and all of our
conclusions for that case apply. In particular $H_1(\wt X)/\SL$ is of
type $S$ and is $\BZ$-torsion-free. In this way we recover 4.2 for
the case $q=1$, at least under our first definition of the
Blanchfield form. \qed
By \cite{Du; Prop\.~4.1, 4.2, 4.3} the Blanchfield forms defined
herein are $(-1)^{q+1}$-linking forms $(M,B)$ in the sense of
\cite{V1}. A ``Witt'' group of such $\e$-linking forms is then
defined by DuVal \cite{Du; \S8} which we shall denote by
$L^\e(A,\Sigma)$ where $\e=(-1)^{q+1}$,
$A=\BZ[F\left]$ and $\Sigma$ is the group of
square matrices which, when augmented, are invertible over $\BZ$.
Then it is not difficult to see that
\bpage
\proclaim{Corollary 4.3} The matrix correspondence of 4.2 induces a
homomorphism $G(m, (-1)^q)\mathbreak
\overset\psi\to\lra L^\e(A,\Sigma)$,
$\e=(-1)^{q+1}$, which sends a representative of the Seifert matrix
of an homology boundary link $(L,\SV)$ to the class of its
Blanchfield linking form (when $q=2$ we have taken an index $2^m$
subgroup of the usual $G(m,-1)$ so the definition of
$L^\e(A,\Sigma)$ would need to be similarly restricted in this case).
\endproclaim
\bpage
We can now show that the two Blanchfield forms defined in case
$q=1$ are ``cobordant'' (equal in $G(m,-1)$). They are certainly
not isomorphic, for, in the case that $L$ were itself a ribbon
homology boundary link in $S^3$, our second Blanchfield form could
be taken to be defined on the trivial module, whereas the first
Blanchfield form would, in general, be non-trivial.
\bpage
\proclaim{Theorem 4.4} In case $q=1$, the Blanchfield form $B$,
defined on $H_1(\wt X)/H_1(\wt{\p X})$, is equivalent in $G(m,-1)$
to the Blanchfield form $B'$, defined on the kernel
$H_1(\wt X)\overset\tl f_*\to\lra H_1(\wt{E(R)})$ (see the
beginning of this section for terminology).
\endproclaim
\bpage
\sub{Proof of 4.4} We are given that
$f: (E(L), \p E(L))\lra(E(R), \p E(R))$ is a degree 1 map of simple
Poincar\'e pairs in the sense of Wall \cite{W; \S2}. Let $X=E(L)$
and $Y=E(R)$. By Lemma~2.2 of \cite{W} the horizontal short exact
sequence below is split exact, and since $f$ is a homeomorphism on
$\p X$, the upper map is an isomorphism
$$
\define\toparrow{@>\pretend\cong\haswidth
{\text{Clifford }}>>}
\define\tparrow{@>\pretend\tl f_*\haswidth
{\text{Clifford }}>>}
\split
H_1&(\p X ; A) \toparrow H_1(\p Y, A)\\
&\Big\downarrow i_X \hskip70pt \Big\downarrow i_Y \\
0\lra M\overset j\to\lra H_1&(X ; A) \tparrow
H_1(Y ; A)\lra 0.\\
&\hskip47pt g_*
\endsplit
$$
It follows that the following is exact
$$
0\lra\ker i_X\lra\ker i_Y\lra M\lra\oper{cok} i_X\lra
\oper{cok}i_Y\lra 0.
$$
But since $H_2(X ; \p X ; A)\overset\tl f_*\to\lra H_2(Y, \p Y ; A)$
is onto, it is easily seen that $\ker i_X\lra\ker i_Y$ is
surjective. Therefore
$0\lra M\lra\oper{cok} i_X\lra\oper{cok} i_Y\lra 0$ is exact, and
in fact split exact.
The latter observation necessitates showing that $g_*$, when
restricted to the image of $H_1(\p Y ; A)$, is an inverse to $f_*$,
that is to say, if $\a\in H_1(\p Y ; A)$ then
$g_*i_Y(\a)=i_Xf_*^{-1}(\a)$. This may be shown directly using the
fact that $g_*(\b)$ is given by the Poincar\'e dual of $f^*$ of
the Poincar\'e dual of $\b$. Thus
$g_*i_Y(\a)=(f^*(i_Y\a)^\land)\cap\G_X$ where $(\ )^\land$ denotes
Poincar\'e dual and $\G_X$ is the fundamental class. But
$(i_Y\a)^\land=\d_Y(\hat\a)$ (\cite{GH; 28.18}), and
$f^*\d_Y(\hat\a)=\d_Xf^*\hat\a$. By the same fact,
$(\d_Xf^*\hat\a)\cap\G_X=i_X(f^*\hat\a\cap\G_{\p X})$. Finally
$f_*(f^*\hat\a\cap\G_{\p X})=\hat\a\cap f_*\G_{\p X}$ by \cite{GH;
24.14}, which in turn equals $\a$ since $f$ is a homeomorphism on
$\p X$. Therefore there is an isomorphism
$$
H_1(X ; A)/H_1(\p X ; A)\longleftarrow
M\op H_1(Y ; A)/H_1(\p Y ; A)
$$
given by $(m,y)\lra m+g_*(y)$. Since we have already established
that $H_1(X ; A)/H_1(\p X ; A)$ and $H_1(Y ; A)/H_1(\p Y ; A)$ are
of type $S$, it follows that $M$ is $\BZ$-torsion-free and of type
$L$ \cite{Du; 3.1i}, hence of type $S$ \cite{Du; 4.1}. Consider the
intersection forms $I_X$, $I_Y$ used to define the Blanchfield
forms. It is a small exercise to show that
$I_X(\a,g_*\b)=I_Y(f_*\a,\b)$ using the fact that $f$ is degree~1.
Thus $B_Y(f_*\a,\b)=B_X(\a,g_*\b)$. It follows that $B_X(m,g_*y)=0$
for all $m\in M$ and $y\in H_1(Y ; A)/H_1(\p Y ; A)$, and that
$B_X(g_*y_1, g_*y_2)=B_Y(y_1,y_2)$. Hence $B_X$ is isomorphic to
$B_Y$ (on $H_1(Y ; A)/H_1(\p Y ; A)$) plus the Blanchfield form on
$M$ (which we have called $B'$). But $B_Y$ is trivial in $G(m,-1)$
as shown in the proof of Theorem~3.6 (a ribbon $S$-link is scheme
null-cobordant). Hence $B_X\cong B'$ in $G(m,-1)$. \qed
\bpage
\proclaim{Proposition 4.5} Suppose
$f: F\left\lra F\left$ is
an homomorphism. Then there is a commutative diagram
$$
\CD
G(k,\e) @>\psi_k>>
L^{-\e}(\BZ[F\left],\Sigma)\\
@VVf_*V @VVf_*V\\
G(m,\e) @>\psi_m>>
L^{-\e}(\BZ[F\left],\Sigma)
\endCD
$$
where the left-hand $f_*$ is defined in 3.4 and the right-hand
$f_*$ is the usual homomorphism induced by an
augmentation-preserving, involution-preserving ring homomorphism
$f$, namely
$f_*((M,B))=\Bigl(M\bigotimes\lm_{\BZ F\left}
\BZ F\left, B'\Bigr)$ where
$B'(x\ox\a, y\ox\b)=\bar\b f_*(B(x,y))\a$ for $x$, $y\in M$ and
$\a$, $\b\in F\left$.
\endproclaim
\bpage
\sub{Proof of 4.5} We know that $f_*$ is realized by taking
parallel copies of Seifert surfaces for a boundary link of
$k$-components and labelling them appropriately as in 3.4.
Therefore we go from the Blanchfield form associated to the
standard epimorphism
$\phi: \pi_1(E(L))\twoheadrightarrow F\left$
defining the usual free cover of the exterior of the boundary link,
to one associated to
$f\circ\phi: \pi_1(E(L))\lra F\left$. Thereby
the result is reduced to showing that the one definition of the
Blanchfield form, namely that given by 4.3, is the same as the
other one we gave. We leave the details to the reader. \qed
\bpage
This allows us to re-state our major theorems 3.5 and 3.6 in terms
of Blanchfield linking forms. To do so we need the algebraic fact
that $\psi$ (see 4.2) is onto. In our exposition this is postponed
until just before Theorem~5.7. We beg the reader's indulgence.
In summary, any pattern and any linking form may be realized by
acting on a (simple) ribbon link with a simple boundary link. The
following, in particular, justifies Theorem~3.16 of \cite{CO1} which
was there used for several computations.
\bpage
\proclaim{Theorem 4.6} (see Theorem 3.5) Under the hypotheses
of Theorem 3.5, the result of acting on $(L,\SV,b)$ by the boundary
link $(B,\SW)$ has Blanchfield linking form equivalent to the sum of
the linking form of $(L,\SV,b)$ and the image under $f_*$ of the
linking form of $(B,\SW)$. Here we also assume that $(L,\SV)$ and
$(B,\SW)$ are simple.
\endproclaim
\bpage
\proclaim{Theorem 4.7} (see Theorem 3.6) Given any pattern
$P$, any $q\ge 1$ and any
$\la\in L^\e(\BZ[F\mathbreak
\left],\Sigma)$, $\e=(-1)^{q+1}$,
(subject to the usual restriction if $q=2$), there is a simple
$m$-component homology boundary link $(L,\SV)$ in $S^{2q+1}$ with
pattern $P$ and Blanchfield linking form equivalent to $\la$. This
link is obtained by acting on a ribbon link with pattern $P$ by a
simple boundary link with linking form $\la$.
\endproclaim
\vskip.7cm
\subhead{\bf\S5. Scheme Cobordism classes of homology boundary
links}\endsubhead
\mpage
\sub{Definition 5.1} Suppose $(L,\SV)$ and $(L',\SV')$ are
$m$-component homology boundary links in $S^{2q+1}$ which ``have
the same scheme'' $S$ in the sense that there exist basings $b$,
$b'$ of type $(x_1,\dots,x_m)$ (i.e., ordinary basings) inducing the
scheme $S=(w_1,\dots,w_m)$. Then we say that $(L,\SV)$ is
{\it scheme-cobordant\/} to $(L',\SV')$ if in $S^{2q+1}\x[0,1]$ there
is a link concordance
$g:\coprod\lm_{i=1}^mS^{2q-1}\x[0,1]\hookrightarrow S^{2q+1}\x[0,1]$
from $L$ to $L'$ and a set $I\SV=\{IV_1,\dots,IV_m\}$ of connected
compact, oriented $(2q+1)$-dimensional manifolds embedded in the
exterior of the concordance such that
$\p(IV_i)=V_i\cup(-V'_i)\cup(\p V_i\x[0,1])$ for $i=1,\dots,m$ and
such that the intersection of $I\SV$ with a tubular neighborhood of
the concordance is a {\it product\/} of its intersection with
$\p E(L)$ (or $\p E(L')$) by $[0,1]$.
\bpage
In the case that the scheme is $(x_1,\dots,x_m)$ (boundary links)
this agrees with \cite{Ko; \S2}. This is clearly an equivalence
relation, abbreviated $L\sim L'$. We have already sketched a proof
that any ribbon homology boundary link with scheme $S$ is
scheme-cobordant to a trivial link with scheme $S$. It is known that
any even-dimensional homology boundary link is scheme-cobordant to
the trivial one \cite{C3} \cite{De2}.
\bpage
\proclaim{Proposition 5.2} If $q>1$, the addition
$(L,\SV)\op(L',\SV')$ of two $m$-component homology boundary links
of reduced scheme $S$ ($S$-links) given by the tangle sum using {\it
any\/} basings $b$, $b'$ of type $(x_1,\dots,x_m)$ which induce $S$,
is a well-defined, commutative and associative operation on
scheme-cobordism classes of $S$-links. Any ribbon homology boundary
link with scheme $S$ acts as identity.
\endproclaim
\bpage
\sub{Proof} Firstly, the tangle sum using a basing of type
$(x_1,\dots,x_m)$ and {\it reduced\/} scheme is just the usual
connected-sum along arcs which do not intersect $\SV$ as defined in
\cite{Ko; \S2}, together with the boundary-connected-sum along the
same arcs to join up each sheet of the Seifert surfaces. The proof
of \cite{Ko; Prop\.~2.11} works to show that $\op$ is well-defined up
to scheme-cobordism since the present situation is so clearly
related. The commutativity and associativity are clear from the
``connected-sum along arcs'' definition. Any ribbon homology
boundary link with scheme $S$ will serve as identity. \qed
\bpage
\proclaim{Theorem 5.3} (compare \cite{De, 5.2 and 6.2}) Any homology
boundary link $(L,\SV)$ with scheme $S$ is scheme-cobordant to a
simple homology boundary link with scheme $S$.
\endproclaim
\bpage
\sub{Proof of 5.3} The proof in \cite{Ko; 2.8} generalizes to these
generalized Seifert surfaces, but our Lemma~6.10 is needed to get the
scheme-cobordism. \qed
\bpage
\sub{Definition} The set of scheme-cobordism classes of homology
boundary links $(L,\SV)$ in $S^{2q+1}$ with scheme $S$ will be
denoted $C(m,q,S)$ (or sometimes merely $C(S)$ or $C(q,S)$). (We
will shortly see that, if $q>1$, this is an abelian group and will
use the same symbol for the group).
\bpage
\proclaim{Proposition 5.4} The cobordism class of the Seifert form
$\th: C(m,q,S)\lra G(m,(-1)^q)$ is a well-defined and, if $q>1$,
additive function sending the identity to the identity.
\endproclaim
\bpage
\sub{Proof} We have shown additivity in 3.2. The well-definedness
is proved as in (\cite{Ko}; see just prior to Theorem~3.4).
\bpage
\proclaim{Theorem 5.5} If $\th((L,\SV))=0$ then $(L,\SV)\sim 0$.
\endproclaim
\bpage
After proving 5.5 we will get immediately that $C$ is a group.
\bpage
\proclaim{Corollary 5.6} If $q>1$ and $S$ is reduced, $C(m,q,S)$ is a
group and $\th_S$ is an isomorphism. Thus the group of
scheme-cobordism classes of homology boundary links with reduced
scheme $S$ is isomorphic to $G(m,(-1)^q)$.
\endproclaim
\bpage
\sub{Proof of 5.6} We showed $\th$ surjective in 3.6. Define the
inverse of $L$ to be an element in the inverse image of $-\th(L)$.
Then $\th(L\op-L)=\th(L)\op-\th(L)=0$ so $L\op(-L)\sim 0$ by 5.5.
Therefore $C$ is a group. But $\th$ has been shown to be additive,
injective and surjective so it is an isomorphism. \qed
\bpage
\sub{Proof of 5.5} It suffices to assume that $(L,\SV)$ is a simple
$m$-component homology boundary link in $S^{2q+1}$ where $q>1$ and
$\th(\SV)=0$. We shall first show that $(L,\SV)$ is ``$S$-slice'',
that is that the components of $L$ bound disjoint $2q$-dimensional
disks $\Delta=\{\Delta_1,\dots,\Delta_m\}$ in $B^{2q+2}$ and there
is a collection of $(2q+1)$-manifolds $\SW$ embedded disjointly in
the exterior of $\Delta$ such that
$$
\p W_i = V_i\cup(\p W_i\cap N(\Delta))
$$
and the intersection of $\p\SW$ with the boundary of a tubular
neighborhood $(S^1\x\Delta)$ of $\Delta$ is a product
$(\SV\cap(S^1\x\{p\}))\x\Delta$ for $p\in\p\Delta$. The desired
result follows easily from this.
Suppose $\phi: E(L)\lra\bigvee\lm_{i=1}^m S^1$ is induced by $\SV$.
Let $\SS(L)$ be the result of stably-framed surgery on the
components of $L$. Thus
$\SS(L)=E(L)\bigcup\lm_{\p E(L)}\(\coprod\lm_{i=1}^m D^{2q}\x S^1\)$
and we can extend $\phi$ to $\SS(L)$ by
$\hat\phi\bigm|_{D^{2q}\x S^1}=\phi\bigm|_{p\x S^1}$ for
$p\in\p D^{2q}$. To show that $(L,\SV)$ is ``$S$-slice'' it
suffices to show that the triple ($\SS(L)$, stable framing,
$\hat\phi_*$) is the boundary of ($Y^{2q+2}$, stable framing,
$\psi_*$) where $\psi_*: \pi_1(Y)\lra F\left$,
$H_*(Y)\cong H_*(\natural S^1\x D^{2q+1})$ and $\pi_1(Y)$ is
normally generated by the meridians of $L$ (their images in
$\pi_1(\SS(L))$). For then $(Y,\p Y)$ is transformed to
$(B^{2q+2}, S^{2q+1})$ by attaching $m$ 2-handles along the
meridians and thus $W$ is seen to be the exterior in $B^{2q+2}$ of
a null-concordance $\Delta=\{\Delta_1,\dots,\Delta_m\}$ for $L$.
Since $\hat\phi_*$ extends, the reverse of the Pontryagin
construction applied to $\psi$ yields the necessary $W_i$.
To produce $Y$, begin with $B^{2q+2}$ and attach $m$ $2q$-handles
$h_1,\dots,h_m$ along the components of $L$ in such a way that the
resulting $(2q+2)$-manifold $Z$ is stably-parallelizable. Then
$\p Z\equiv\SS(L)$. Note
$H_*(Z)\cong H_*(\natural^m_{i=1}D^2\x S^{2q})$. It only remains to
find disjointly embedded $2q$-spheres representing a basis for
$H_{2q}(Z)$ (which have trivial normal bundle since $q\neq 1$) and
perform framed surgery on these, resulting in the desired $Y$. We
also need to ensure that $\hat\phi_*$ extends to the exterior (in
$Z$) of these $2q$-spheres and $\pi_1(\p Y)\lra\pi_1(Y)$ is a
``normal surjection''. Consider the Seifert surface $V_i$ capped
off along each of its boundary $(2q-1)$-spheres by copies of the
$2q$-disk which are parallels of cores of the handles
$\{h_1,\dots,h_m\}$. Then these capped-off manifolds, $\wh V_i$ may
be ambiently surgered along $q$-spheres to yield the desired
$2q$-spheres, exactly as in the injectivity part of the proof of
Theorem~3.5 of \cite{Ko}. This necessitates $q>1$. Finally note
that $\pi_1$ of the complement in $B^{2q+2}$ of a set of Seifert
surfaces pushed-in slightly is a free group on a set of
``meridians'' $x_i$ to these surfaces. Since $\phi_*$ is onto, each
of these is in the normal closure of the meridians of $L$. In fact
we may take $\psi_*$ to be what amounts to the identity map. Note
that the ambient surgeries on $q$-spheres are of high codimension
and irrelevant to $\pi_1$. \qed
\bpage
We may now summarize all of these relationships in Diagram~20. We
assume $q>1$. When $q=2$, the index $2^m$ subgroups must be used as
previously discussed.
$$
\define\ctoparrow{@>\pretend\th_S\haswidth
{\text{Cliffo}}>>}
\define\dtoparrow{@>\pretend\psi\haswidth
{\text{Cliffo}}>>}
\split
C&(m, q, (x_1,\dots,x_m))\\
T\hskip45pt &\th\Biggl\downarrow\hskip80pt B\\
C(m, q, S)\ctoparrow G&(m, (-1)^q)\dtoparrow
L^{(-1)^{q+1}}(\BZ[F], \Sigma)\\
\\
&\hskip75pt B_S
\endsplit
$$
\centerline{Diagram~20}
\mpage
\flushpar Here $C(m, q, (x_1,\dots,x_m))$ can be seen to be
identical to Ko's group, $C_{2q-1}(B_m)$, of boundary cobordism
classes of boundary links with chosen Seifert surface systems. Here
$T(\a)$ is defined to be the result of a simple boundary link with
Seifert form $\a$ acting on a ribbon homology boundary link with
scheme $S$. Both $\th$ and $\th_S$ are isomorphisms by
\cite{Ko, Thm\.~3.5} and by 5.6, so $T$ is also an isomorphism. The
map $B$ is an isomorphism for $q\ge 3$ by Theorem~9.1 of \cite{Du}
and Theorem~2.7 of \cite{Ko}. It follows that $\psi$ and $B_s$ are
isomorphisms for $q\ge 3$. But since, if $q\neq 2$, the domain and
range of $\psi$ depend only on the parity of $q$, $\psi$ is an
isomorphism for $q=1$. Since the index $2^m$ subgroup of $G(m,+1)$
used when $q=2$ is merely the subgroup of matrices $A$ such that the
blocks $A_{ii}$ have signatures multiples of 16, and this is carried
over naturally to the $L$-group, $\psi$ is seen to be an isomorphism
in all cases, with the understanding, when $q=2$, that we restrict
to the subgroup of $L$. It follows that $B$ and $B_s$ are
isomorphisms onto this subgroup for $q=2$. Thus all maps are
isomorphisms if $q\neq 1$.
When $q=1$, $C(m,q,S)$ and $C(m,q,(x_1,\dots,x_m))$ are not groups
but merely sets of equivalence classes, all maps are defined, $\psi$
is a isomorphism and $\th$, $\th_s$, $B$, $B_s$ are surjective.
Therefore we have
\bpage
\proclaim{Theorem 5.7} (compare \cite{Du; Thm\.~9.1}) If $q>1$, the
group $C(m,q,S)$ of scheme-cobordism classes of $m$-component
homology boundary links (with surface systems of reduced scheme $S$)
in $S^{2q+1}$ is isomorphic to $L^{(-1)^{q+1}}(\BZ [F],\Sigma)$ (when
$q=2$, replace $L$ by the appropriate index $2^m$ subgroup). This
isomorphism is given by the Blanchfield form associated to the free
cover associated to the system of Seifert surfaces. Hence
$C(m,q,S)\cong C(m,q,\{x_1,\dots,x_m\})$ for all reduced schemes
$S$.
\endproclaim
\bpage
Let $C(m,q)$ stand for $C(m,q,\{x_1,\dots,x_m\})$. Let $\SF$ stand
for $\BZ [F\left]\overset\e\to\lra \BZ$. Let
$\G_{2q+2}(\SF)$ stand for the homology-surgery group of Cappell and
Shaneson \cite{CS2} and $\wt\G_{2q+2}(\SF)$ be its quotient by the
image of
$L_{2q+2}(F\left)\overset i\to\lra\G_{2q+2}
(\SF)$. Recall that Cappell, Shaneson and DuVal established the
exact sequences below \cite{Du; p\.~633--634}.
$$
\define\etoparrow{@>\pretend\ \haswidth
{\text{Clifford}}>>}
\split
0\lra\wt\G_{2q+2}&(\SF)\overset\phi\to\lra C(m,q)\etoparrow
\ \ L_{2q+1}(F)\lra 0\\
&\Big\| \hskip50pt \Biggl\downarrow B \hskip70pt \Big\| \\
0\lra\wt\G_{2q+2}&(\SF)\lra L^{(-1)^{q+1}}(A,\Sigma)\lra L_{2q+1}(F)
\lra 0
\endsplit
$$
Therefore we can conclude the following using 5.6 and 5.7. The
first exact sequence below was (essentially) obtained by DeMeo
(unpublished) in \cite{De; Thm\.~7.2}. There he deals with a group
analogous to the $F_m$-cobordism classes of Cappell and Shaneson
but the equivalence to scheme-cobordism classes is not hard to
deduce (see 6.10).
\bpage
\proclaim{Theorem 5.8} If $q>2$, there are exact sequences for any
reduced scheme $S$
$$
\define\etoparrow{@>\pretend\ \haswidth
{\text{Clifford}}>>}
\split
0\lra\wt\G_{2q+2}&(\SF)\overset\phi_S\to\lra C(m,q,S)\etoparrow
\ \ L_{2q+1}(F)\lra 0\\
&\Big\| \hskip50pt \Biggl\downarrow B_S \hskip70pt \Big\| \\
0\lra\wt\G_{2q+2}&(\SF)\lra L^{(-1)^{q+1}}(A,\Sigma)\lra L_{2q+1}(F)
\lra 0
\endsplit
$$
where $B_S$ is the Witt class of the Blanchfield linking form
associated to the free covering space dictated by the system of
Seifert surfaces. Moreover $\phi_S=T\circ\phi$ (see 5.6) so that
$\phi_S(\a)$ is obtained by allowing the boundary link $\phi(\a)$
to act on a ribbon homology boundary link of scheme $S$.
\endproclaim
\bpage
\sub{Proof of 5.8} Merely replace $C(m,q)$ in the
Cappell-Shaneson-DuVal sequence by\linebreak
$C(m,q,S)$ using the isomorphism $T$ of Diagram~20. \qed
\vskip.7cm
\subhead{\bf\S6. Classification of Homology Boundary Links Modulo
Homology Boundary Link Concordance}\endsubhead
\mpage
In this section we investigate the question of when two homology
boundary links of pattern $P$ are concordant respecting that
pattern. In the proof of 6.3, we shall see that this is the same
as asking that for {\it some\/} Seifert surface systems the links
are scheme-cobordant, indicating that this is the proper analogue of
boundary link cobordism of boundary links, and justifying the
equivalent name of homology boundary link cobordism. This
necessitates computing the effect on Seifert form of choosing
{\it different\/} Seifert surface systems. This mirrors the
analysis of Ko in the case of boundary links, but is much more
complicated.
\bpage
\sub{Definition 6.1} Two $P$-links (links of pattern $P$) $L$ and
$L'$, are $P${\it -cobordant\/}, or {\it pattern-cobordant\/} or
{\it homology boundary link cobordant\/} if there is a concordance
$C$ from $L$ to $L'$ and an epimorphism
$g: \pi_1(E(G))\twoheadrightarrow F$ such that
$g\circ i: \pi_1(E(L))\twoheadrightarrow F$ and
$g\circ i': \pi_1(E(L'))$ are epimorphisms.
\bpage
It follows that the ``pattern'' of the concordance is $P$. Let
$\SP(m,q,P)$ denote the set of $P$-cobordism classes of
$m$-component homology boundary links in $S^{2q+1}$ with pattern
$P$.
Suppose $(w_1,\dots,w_m)$ is in the equivalence class of the fixed
pattern $P$.
\bpage
\sub{Definition 6.2} $\Aut_{w_i}F$ is the subgroup of automorphisms
of the free group $F$ on $m$ letters which send $w_i$ to a
conjugate of $w_i$ for $1\le i\le m$.
\bpage
\proclaim{Theorem 6.3} For any fixed pattern $P$ and any
representative $(w_1,\dots,w_m)$ of $P$, there exists a bijection
$\ov\th: \SP(m,q,P)\lra G(m,(-1)^q)/\Aut_{w_i}F$ where the action
is defined as in 3.4 (and if $q=2$ we mean the usual index $2^m$
subgroup of $G$). $\ov\th(L)$ is defined by finding (for any scheme
$S$ compatible with $(w_1,\dots,w_m)$) a system of Seifert surfaces
$\SV$ for $L$ which induces the scheme $S$ (for some basing) and
setting $\ov\th(L)=\th_S(\SV)$. Similarly the map given by the
Blanchfield Form of a simple representative induces a bijection
$\ov B: \SP(m,q,P)\lra L^{(-1)^{q+1}}(\BZ[F], \Sigma)/\Aut_{w_i}F$
where the action is as in 4.5.
\endproclaim
\bpage
A translation of 6.3 in terms of $\G$-groups yields the following.
\bpage
\proclaim{Theorem 6.4} Suppose $q>2$. For any fixed pattern $P$ and
any representative $(w_1,\dots,w_m)$ of $P$, there are functions
$$
\wt\G_{2q+2}(\BZ F\to\BZ)/\Aut_{w_i}F\overset\phi_S\to\lra
\SP(m,q,P)\overset\pi\to\lra L_{2q+1}(F)
$$
such that $\pi$ is surjective and $\phi_S$ is an injection with
image $\pi^{-1}(0)$. Here $\wt\G$ is a gamma group modulo the image
of $L_{2q+2}$.
\endproclaim
\bpage
Most of the rest of this chapter will be devoted to proving 6.3. We
should note that this answer surprised us. We had thought that the
answer would be $G/\Aut_0F$ where $\Aut_0F$ are those automorphisms
inducing the identity on homology. This is tempting to conclude
given the work of Cappell and Shaneson and the following
propositions.
\bpage
\sub{Definition 6.5} A {\it splitting map\/} for the $m$-component
homology boundary link (with basepoint) $(L,*)$ is an epimorphism
$\phi: \pi_1(E(L),*)\lra F$, where $F$ is free of rank $m$, such
that, for some meridional map $\mu: F\lra\pi_1(E(L),*)$ ($\mu(x_i)$
is an $i^{\text{th}}$ meridian), $\phi\circ\mu$ induces the
identity map on abelianizations. Clearly $\phi$ is a splitting map
with respect to some $\mu$ if and only if $\phi$ is a splitting map
with respect to each possible $\mu$.
\bpage
Let $\Aut_0(F)$ be the group of automorphisms of $F$ which induce
the identity on abelianization.
\bpage
\proclaim{Proposition 6.6} If $\phi$ is a splitting map for $(L,*)$
then for any $\psi\in\Aut_0(F)$, $\psi\circ\phi$ is a splitting map
for $(L,*)$. If $\phi$ and $\phi'$ are splitting maps for $(L,*)$
then there exists $\psi\in\Aut_0(F)$ such that
$\phi'=\psi\circ\phi$.
\endproclaim
\bpage
\sub{Proof of 6.6} The first claim is obvious. For the second
claim, let $G=\pi_1(E(L),*)$ and let $\pi$ be the quotient map
$G\lra G/G_\om$. By our remark above, $\phi$ and $\phi'$ are
splitting maps with respect to some $\mu$. By Stallings' theorem,
$\phi$ and $\phi'$ induce isomorphisms $\phi_\om$, $\phi'_\om$ from
$G/G_\om$ to $F$ such that $\phi=\phi_\om\circ\pi$,
$\phi'=\phi'_\om\circ\pi$. Setting
$\psi=\phi'_\om\circ\phi^{-1}_\om$ we see that
$\phi'=\psi\circ\phi$. Moreover, upon abelianization,
$\psi(x_i)\equiv\psi(\phi\mu(x_i))\equiv\phi'_\om\circ\phi^{-1}_\om
\circ\phi\circ\mu(x_i)\equiv\phi'_\om\circ\pi\circ\mu(x_i)\equiv
\phi'(\mu(x_i))\equiv x_i$. \qed
\bpage
Recall that a {\it scheme\/} $S$ of basing type $(x_1,\dots,x_m)$
{\it has pattern\/} $P$ if the circles
$\{\p\Delta_i\mid i=1,\dots,m\}=\{\g_i\mid i=1,\dots,m\}$ trace out
words $(w_1,\dots,w_m)$ in $F\x\dots\x F$ which has pattern $P$.
Recall that if $(w_1,\dots,w_m)$ and $(w'_1,\dots,w'_m)$ have
pattern $P$ then $w'_i=\psi(g_iw_ig^{-1}_i)$ for some
$\psi\in\Aut_0(F)$ and some $g_i\in F$. If $(L,\SV)$ is an homology
boundary link with pattern $P$, and $(r_1,\dots,r_m)$ is an
$m$-tuple of words of $F$ such that $f(x_i)=r_i$ defines an
element of $\Aut_0(F)$, then we can define a new system $(L,\SV'')$,
denoted $f^\#(L,\SV)$, as follows. Assume
$r_i=x_{i_1}^{\e_{i_1}}\dots x_{i_k}^{\e_{i_k}}$. Merely replace
each $V_i$ by a disjoint union $\coprod\lm_{j=1}^k\e_{i_j} V_i$ of
parallel copies of $V_i$, with orientation varying according to
$\e_{i_j}=\pm1$, the relabelling the $n^{\text{th}}$ copy with the
letter $i_n$. Set $V'_i$ equal to the union of the components
labelled with $j$. Since each $V_i$ was connected, $S^{2q+1}-\SV'$
is connected. Thus we can tube components of $V'_i$ together to form
$V''_i$ which is connected. This $(L,\SV'')$ is the desired surface
system. This is very similar to the $f^\#L$ defined in \S3 except
that here we are not changing the link $L$, merely making the
Seifert surfaces more complicated. Note, however that $f^\#L$
depends on more than $f$ and as such involves arbitrary choices. The
following is then immediate from the definitions.
\bpage
\proclaim{Proposition 6.7} Suppose $r_i$ are words such that the
endomorphism defined by $f(x_i)=r_i$ lies in $\Aut_0(F)$. If
$(L,\SV)$ induces the splitting map $\phi$ (by the Pontryagin
construction), then $f^\#(L,\SV)$ induces the splitting map
$f\circ\phi$.
\endproclaim
\bpage
Given 6.6 and 6.7 it is very tempting to think that $\th$ gives,
somehow, a well-defined bijection from $\SP(m,q,P)$ to
$G(m,(-1)^q)/\Aut_0F$. It does not. To correctly analyze the
situation, it is helpful to introduce an intermediate, more
algebraic, notion.
Given $(w_1,\dots,w_m)$ which represents a pattern $P$, consider
pairs $(L,\phi)$ where $L$ is an homology boundary link and
$\phi: \pi_1(EL)\twoheadrightarrow F$ is an epimorphism such that,
for some meridians $\mu_i$, $\phi(\mu_i)=w_i$. Given two such
$(L_0,\phi_0)$, $(L_1,\phi_1)$ we say $(L,\phi_0)\sim(L_1,\phi_1)$
if there is a concordance $C$ from $L_0$ to $L_1$ and an epimorphism
$\psi: \pi_1(EC)\twoheadrightarrow F$ which restricts on
$\pi_1(EL_0)$ to $\phi_0$ and on $\pi_1(EL_1)$ to $\phi_1$ (after
an inner automorphism of $\pi_1(EL_1)$ to change the basepoint of
$\pi_1(EC)$). Let $\SH(m,q,w_i)$ or simply $\SH(w_i)$ represent the
set of equivalence classes. This set was defined by Cappell and
Shaneson in the case $w_i=x_i$ and by De~Meo in general \cite{De}.
Now fix a pattern $P$. The group $\Aut_0F$ acts on the disjoint
union $\amalg H(w_i)$ (taken over all $(w_1,\dots,w_m)$ which are
in the equivalence class of the pattern $P$) as follows:
$f^\#([L,\phi)]=[(L,f\circ\phi)]$.
\bpage
\proclaim{Lemma 6.8} The forgetful map
$F:\f{\amalg H(m,q,w_i)}{\Aut_0F}\lra\SP(m,q,P)$ is a bijection. Here
the disjoint union is over the set of all $m$-tuples $(w_i)$ in the
equivalence class of $P$.
\endproclaim
\bpage
\sub{Proof of 6.8} First note that $F$ is well-defined on $H(w_i)$
since the equivalence relation $\sim$ is stronger than
$P$-cobordism. It is independent of the action of $\Aut_0F$ since
the action changes only the splitting map, not the link.
Since $F$ is obviously surjective, we need only show injectivity.
Suppose $(L,\phi)\in H(w_i)$, $(L',\phi')\in H(w'_i)$ and suppose
that $L$ is homology boundary link concordant to $L'$. Thus there
exists a concordance $C$ from $L$ to $L'$ and an epimorphism
$g: \pi_1(EC)\twoheadrightarrow F$ such that $\psi=g\circ i$ is a
splitting map for $L$ and $\psi'=g\circ\k\circ i'$ is a splitting map
for $L'$ (here $\k$ is an automorphism of $\pi_1(EC)$ to change
basepoints). Since $(L,\phi)\in H(w_i)$, there exist meridians
$\mu_i\in\pi_1(EL)$ such that $\phi(\mu_i)=w_i$.
By 6.6 there exist elements $f$, $f'$ of $\Aut_0F$ such that
$f^\#(L,\phi)=(L,f\circ\phi)=(L,\psi)$ and
$(f')^\#(L',\phi')=(L,\psi')$. Therefore it suffices to show that
$(L,\psi)\sim(L,\psi')$ in $H(f(w_i))$. Note that
$\psi(\mu_i)=f(w_i)$ so indeed $(L,\psi)\in H(f(w_i))$. Now choose
meridians $\mu'_i\in\pi_1(EL')$ such that
$\k\circ i'(\mu'_i)=i(\mu_i)$. Then
$\psi'(\mu'_i)=g\circ\k\circ i'(\mu'_i)=g\circ i(\mu_i)=\psi(\mu_i)=
f(w_i)$ so $(L',\psi')\in H(w_i)$, and the concordance $(C,g)$
shows $[(L,\psi)]=[(L',\psi')]$. \qed
\bpage
Note that if $[(L,\phi)]\in H(w_i)$ and $f\in\Aut_{w_i}F$,
$f^\#(L,\phi)$ is still in $H(w_i)$. For, if
$f(w_i)=\eta_iw_i\eta_i^{-1}$, then choose $\xi_i$ such that
$f\circ\phi(\xi_i)=\eta_i^{-1}$ and observe that
$f\circ\phi(\xi_i\mu_i\xi_i^{-1})=w_i$.
\bpage
\proclaim{Lemma 6.9} For any $m$-tuple $(w_1,\dots,w_m)$ inducing
the pattern $P$, the inclusion map
$\f{H(w_i)}{\Aut_{w_i}F}\overset i\to\lra
\f{\amalg H(w_i)}{\Aut_0(F)}$ is a bijection.
\endproclaim
\bpage
\sub{Proof of 6.9} First we show surjectivity. Suppose
$(L,\phi)\in H(w'_i)$. Since $(w'_i)$ is in the same pattern $P$ as
$(w_i)$, $w_i=f(\eta_i w'_i\eta_i^{-1})$ for some $f\in\Aut_0F$.
Choose meridians $\mu_i$ such that $\phi(\mu_i)=w'_i$. Consider
$f^\#(L,\phi)=(L, f\circ\phi)$. Choose $\xi_i$ such that
$\phi(\xi_i)=\eta_i$. Then
$f\circ\phi(\xi_i\mu_i\xi_i^{-1})=f(\eta_iw'_i\eta_i^{-1})=w_i$,
showing that $(L, f\circ\phi)\in H(w_i)$. Thus $i$ is onto.
Now suppose $(L_0,\phi_0)$ and $(L_1,\phi_1)\in H(w_i)$ and
$i((L_0,\phi_0))=i((L_1,\phi_1))$. It follows that there is a
$g\in\Aut_0F$ such that $g^\#[(L_1,\phi_1)]=[(L_0,\phi_0)]$ in
$H(w_i)$. In particular this implies $(L_1, g\circ\phi_1)$ lies in
$H(w_i)$!! This places strong restrictions on $g$ since
$(L_1,\phi_1)$ also lies in $H(w_i)$. Suppose $\mu_i$ are meridians
such that $\phi_1(\mu_i)=w_i$. Then there must be meridians
$\eta_i\mu_i\eta_i^{-1}$ such that
$g\circ\phi_1(\eta_i\mu_i\eta_i^{-1})=w_i$. But this immediately
implies $g(w_i)$ is conjugate to $w_i$. Therefore
$g\in\Aut_{w_i}(F)$ and $(L_0,\phi_0)$ is equal to $(L_1,\phi_1)$
in the domain of our map $i$, concluding our proof that $i$ is
injective. \qed
\bpage
\proclaim{Lemma 6.10} Suppose $S=(w_1,\dots,w_m)$ is a scheme. The
Pontryagin construction yields a bijection
$p: C(m,q,S)\lra H(m,q,w_i)$. Therefore $H(m,q,w_i)$ is naturally a
group if $q>1$.
\endproclaim
\bpage
\sub{Proof of 6.10} First we show $p$ is well-defined. Suppose
$(L,\SV)$ and $(L',\SV')$ are $S$-links for which the Pontryagin
construction using basings $b$, $b'$ (see 5.1) yields splitting
maps $\phi$ and $\phi'$ respectively where $b$ and $b'$ induce the
scheme $S$. If $(L',\SV')$ is scheme-cobordant to $(L,\SV)$ via $C$
and $I\SV$, then we can show that $(L,\phi)\sim(L',\phi')$ in
$H(w_i)$ by using the basepoint of $b$ and applying the Pontryagin
construction to $I\SV$ to yield a homomorphism
$\psi: \pi_1(EC,b_*)\lra F$ such that $\psi\circ i=\phi$ and
$\psi\circ\k\circ i'=\phi'$ where $\k$ is a change of basepoint
from $b_*$ to $b'_*$. Thus $p$ is well-defined.
The map $p$ is onto by the techniques of the proof of Theorem~3.6,
which shows that given any link $(L,\SV)$ and splitting map $g$
such that $g_*(\mu_i)=[w_i]$ and $S=(w_1,\dots,w_m)$ is any scheme,
$\SV$ can be modified, {\it preserving\/} $g_*$, until $\SV$
induces $S$ precisely.
Now suppose $p((L,\SV))=p((L',\SV'))$. Then there exists a
concordance $C$ and an epimorphism
$\psi: \pi_1(EC)\twoheadrightarrow F$ such that $\psi\circ i_C=\phi$
and $\psi\circ\k\circ i'=\phi'$ as usual. Let $f$, $f'$ be the maps
from $EL$, $EL'$ respectively to $\bigvee\lm^m_{i=1}S^1$ induced by
$\SV$, $\SV'$ as above where $f_*=\phi$, $(f')_*=\phi'$. Under an
identification $\p_+EC\equiv\p N(L)\x[0,1]=\p N(L')\x[0,1]$, we can
extend $f$ and $f'$ to $F: \p EC\lra\bigvee\lm^m_{i=1}S^1$ by
letting $F=f\circ p_1$ ($p_1=$ projection onto 1st factor) on
$\p N(L)\x[0,1]$. This is possible because $f$ and $f'$ induce the
same scheme. Notice that $F_*$ necessarily agrees with $\psi$, and
$F$ extends over $E(C)$ since $F_*$ is extended by $\psi$. After a
small perturbation, the inverse of the Pontryagin construction then
produces the ``Seifert surfaces'' $I\SV$ which exhibit that
$(L,\SV)$ is $S$-cobordant to $(L',\SV')$. Hence $p$ is injective.
To see that $H(m,q,w_i)$ is a group, note that clearly $H$ depends
only on the image of $w_i$ in $F$. Thus we can choose a
{\it reduced\/} scheme $S$ compatible with $w_i$, and apply 5.6.
\qed
\bpage
\proclaim{Corollary 6.11} If $S=(w_1,\dots,w_m)$ then there is an
action of $\Aut_{w_i}F$ on the set $C(m,q,S)$ of scheme cobordism
classes of $S$-links, with respect to which the bijection $p$ of
6.10 is equivariant.
\endproclaim
\bpage
\sub{Proof of 6.11} Given $f\in\Aut_{w_i}F$ simply define
$f^\#[(L,\SV)]$ to be $p^{-1}(f^\#(p([L,\SV])))$. It is then also
clear that the geometric description of $f^\#([L,\SV])$ in terms of
copies of the Seifert surfaces (see above 6.7) realizes this action
and hence that the geometric description of the action is
independent, up to scheme cobordism, of the choices involved. \qed
\bpage
\proclaim{Lemma 6.12} Suppose $(w_1,\dots,w_m)$ represents the
reduced scheme $S$. The isomorphism given by taking the Seifert
form, $\th_S: C(m,q,S)\lra G(m,(-1)^q)$ , is equivariant with
respect to the actions of $\Aut_{w_i}F$ defined in 6.11 and 3.4
respectively.
\endproclaim
\bpage
\sub{Proof of 6.12} By Theorem 3.6 and Corollary 5.6, we may assume
that an arbitrary scheme cobordism class $(L,\SV_L)$ takes the form
of a boundary link $(B,\SV_B)$ with Seifert form $\a$ acting on a
ribbon homology boundary link $(R,\SV,b)$ for which the loops
$\{\p\Delta_1,\dots,\p\Delta_m\}$ intersect $\SV$ in words which
reduce to $\{x_1,\dots,x_m\}$ in the free group. Of course
$\th_S(L,\SV_L)=\a$. Now consider acting on $(L,\SV_L)$ by
$f\in\Aut_{w_i}F$ such that $f(x_i)=r_i$. By 6.11, we may use the
geometric definition of $f^\#(L,\SV)$ as described above 6.7. But
changing the Seifert surface system of $L$ does not change the fact
that it is obtained as the boundary link $(B,\SV_B)$ acting on
ribbon link because $L$ itself is unchanged by $f^\#$. However now
$(B,\SV_B)$ is acting on $(R,\SV',b)$ and the loops
$\{\p\Delta_1,\dots,\p\Delta_m\}$ now intersect $\SV'$ in words which
reduce to $\{f_*(x_1),\dots,f_*(x_m)\}=\{r_1,\dots,r_m\}$ in the
free group. Thus $\th_S(f^\#(L,\SV_L))=\th(R,\SV')\op f_*(\a)$ by
3.5. Since $(R,\SV')$ is clearly still a ribbon homology boundary
link, $\th(R,\SV')=0$. Hence $\th_S(f^\#(L,\SV_L))=f_*\th_S(L,\SV)$
as desired. \qed
\bpage
We have now completed the proof of 6.3. Given any pattern $P$ and
any representative $(w_1,\dots,w_m)$ of $P$, we may combine
6.8--6.11 to show that the forgetful map from $C(m,q,S)/\Aut_{w_i}F$
to $\SP(m,q,P)$ is a bijection. Lemma~6.12 then completes the
argument. Theorem~6.4 then follows formally from 6.3 and the
functoriality of the $\G$-groups and $L$-groups. \qed
\bpage
\proclaim{Corollary 6.13} Two homology boundary links $L$, $L'$ are
homology boundary link cobordant if and only if there exist Seifert
surface systems such that $(L,\SV)$ and $(L',\SV')$ are
scheme-cobordant (see 5.1).
\endproclaim
\vskip.7cm
\subhead{\bf\S7. $\BZ_p$-Homology Boundary Links in $\BZ_p$-Homology
Spheres}\endsubhead
\mpage
Suppose $\SS$ is a closed, oriented $(2q+1)$-manifold which has the
$\BZ_p$-homology of $S^{2q+1}$ (let $J=\BZ_{(p)}$, the integers
localized at $p$). Suppose $L=\{K_1,\dots,K_m\}$ is an ordered,
oriented, embedded collection of $(2q-1)$-spheres in $\SS$ (whose
longitudes are torsion in $H_1(E(L))$ if $q=1$). Then we call
$(L,\SS)$ a {\it link in a $\BZ_{(p)}$-homology sphere\/}. If $L$
admits a system $\SV=\{V_1,\dots,V_m\}$ of ``Seifert surfaces''
where $\p V_i$ is homologous to the $i^{\text{th}}$ longitude in
$H_{2q-1}(\p E(L); \BZ_p)$ then we call $L$ a
$\BZ_p${\it -homology boundary link\/} (see \cite{H1}). We restrict
to such $L$ with $(q-1)$-connected ``Seifert surfaces'' as before and
continue to use the term simple. The Pontryagin construction
associates to $(L,\SV)$ a map $E(L)\lra\bigvee\lm_{i=1}^m S^1$ as
before and hence a free covering space $\wt X$. Then
$H_q(\wt X\ ;\ \BZ_{(p)})$ is an $A$-module $(A=\BZ_{(p)}[F])$ and
we may define on it a Blanchfield form, as in \S4, taking values in
$\La/A$ where $\La$ is the Cohn localization of
$A\overset\e\to\lra\BZ_{(p)}$. The specific analysis using the
Mayer-Vietoris sequence also holds to show that this Blanchfield
form is determined by a ``Seifert matrix''. Here, to avoid speaking
of linking numbers one can define $\th(\a_{ij},\a_{kl})$ to be the
coefficient of $\hat\a_{kl}$ for $i^+\a_{ij}$, that is the matrix
of $i^+$ with respect to the dual bases $\{\a_{ij}\}$,
$\{\hat\a_{ij}\}$ for $H_q(\SV\ ;\ \BZ_{(p)})$ and
$H_q(E(L)-\SV\ ;\ \BZ_{(p)})$. Observing that
$i^+\a-i^-\a=\pm\Sigma(\a\cd\a_{ij})\hat\a_{ij}$ where the latter
is the intersection form on $H_q(\SV\ ;\ \BZ_{(p)})$, one sees that
the matrix of $i^+$ is $\th$ and the matrix of $i^-$ is $\th\pm\SI$
where $\SI$ is the intersection matrix on $H_q(\SV\ ;\ \BZ_{(p)})$
with respect to $\{\a_{ij}\}$ (we do not stop here to get the sign
correct). Then the map $d$ is represented by
$\Delta=\G\th\pm\SI-\th$. Note that $\Delta$ is invertible when
augmented since $\SI$ is invertible. Hence the entire proof of
Theorem~4.2 goes through using the matrix
$(I-\G)(\G\th+\SI-\th)^{-1}$.
Recall that new invariants of links were introduced in \cite{CO1},
\cite{CO2} to show that not all links are concordant to boundary
links. The initial step of the definition of those invariants
entailed associated to the link $L=\{K_1,\dots,K_m\}$, a
{\it covering link\/}
$\wt L=\{\wt K_1, \wt K_{21}, \wt K_{22},\dots, \wt K_{2p},
\wt K_{31},\dots, \wt K_{m1},\dots, \wt K_{mp}\}$ consisting of the
lifts of the components of $L$ in a $p$-fold cyclic cover of
$S^{2q+1}$ branched over $K_1$ ($p$ prime). In case $L$ were a
simple homology boundary link with surface system $\SV$, $\wt L$
would be a simple $\BZ_p$-homology boundary link in the
$\BZ_{(p)}$-homology sphere $\SS$. Then we have the Blanchfield form
$B=B(L,\SV)$ in $L^\e(\BZ[F\left],\Sigma)$ and
the Blanchfield form $\wt B=B(\wt L,\wt\SV)$ in
$L^\e(\BZ_{(p)}[F'],\Sigma')$ where $F'$ is free on $1+(m-1)p$
letters. One might then define a $\BZ_p$ scheme-cobordism relation
on the set of $\BZ_p$-homology boundary links in
$\BZ_{(p)}$-homology spheres and see that the operation of forming
covering links of the type above carries scheme-cobordism classes to
$\BZ_p$ scheme cobordism classes. Therefore one expects a functorial
relationship between $B$ and $\wt B$. In fact, since every element
of $L^\e(\BZ F,\Sigma)$ is represented by a simple boundary link,
one can geometrically define a {\it transfer\/}.
\bpage
\proclaim{Proposition 7.1} If
$\phi: F\left\lra\BZ_p$ sends $x_1$ to $1$ and
$x_i$ to $0$ if $i>1$, there is a {\it transfer\/} homomorphism
$\tr: L^\e(\BZ F,\Sigma)\lra L^\e(\BZ_{(p)}F',\Sigma')$ where $F'$ is
$\ker\phi$. Moreover, for any simple homology boundary link
$(L,\SV)$, and covering link $(\wt L,\wt\SV)$ defined by $p$-fold
branched cover (branching over $K_1$),
$B(\wt L,\wt\SV)=\tr(B(L,\SV))$.
\endproclaim
\bpage
\sub{Proof} One way to show this is to note that the free $(F)$
covering space $\wt X$ of $X=E(L)$ associated to $\SV$ has
precisely the same underlying space as the free $(F')$ covering
space of $E(\wt L)$ associated to $\wt\SV$. Therefore the module on
which $B(\wt L,\wt\SV)$ is defined is merely $H_q(\wt X)\ox\BZ_{(p)}$
considered as a module over $\BZ_{(p)}F'$ via
$\phi: \BZ_{(p)}F'\hookrightarrow\BZ_{(p)}F$. The pairing itself
therefore admits a purely algebraic definition (which we shall not
give here) in terms of $B(L,\SV)$.
Another way is to define transfer using boundary links, then
establish its independence of pattern. Use 4.7 and 5.6 to replace
$(L,\SV)$, up to scheme-cobordism, by $(L',\SV')$, the action on a
ribbon homology boundary link $(R,\SW)$ with identical scheme, by a
boundary link $(L'',\SV'')$ with $B(L'',\SV'')=B(L,\SV)$. Since $L$
is scheme-cobordant to $L'$, $\wt L$ will be $\BZ_p$ scheme-cobordant
to $\wt{L'}$ and hence $B(\wt L)=B(\wt{L'})$ (neither fact have we
proved herein but appeal by analogy to the integral case). Moreover
we now argue that the covering link of ($L''$ acting on $R$) is the
same as the action of the covering link of $L''$ acting on the
covering link of $R$. This is done by observing that the punctured
$2$-disk $\Delta$ used to decompose $L'$ into two angles will lift
to a punctured $2$-disk and decompose the covering link. Upon
re-doing our additivity theorem, one calculates that
$B(\wt L)=B(\wt R)\op B(\wt{L''})$. Since $R$ is scheme-cobordant to
$0$, $B(\wt R)=0$. Finally $B(\wt{L''})=\tr(B(L''))$ by definition
of the transfer on boundary links. Thus $B(\wt L)=\tr(B(L))$ as
desired. \qed
\bpage
Proposition 6.1 was used in \cite{CO2; \S3} to calculate our
invariants associated to covering links. The invariants there were
images of $B(\wt L,\wt\SV)$ in
$L^\e(\BZ_{(p)}F'_{\text{abelian}}\ ;\ \Sigma')$, that is, ordinary
Blanchfield forms associated to the universal {\it abelian\/}
covering space of $E(\wt L)$ (in fact to successfully compute we
always reduced to a $\BZ$ covering space, which invariants
correspond to the image of $B(\wt L,\wt\SV)$ in
$W^*(\BZ_{(p)}[t, t^{-1}]\ ;\ \text{determinant }=1)$.)
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\vskip1.7cm
\line{Tim D. Cochran\hfil Kent E. Orr}
\line{Mathematics Department\hfil Mathematics Department}
\line{Rice University\ \ P.O.Box 1892\hfil Indiana
University}
\line{Houston, Texas\hfil Bloomington, Indiana}
\line{77251--1892\hfil 47405}
\end