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\topmatter
\title Homology Cobordism and Generalizations of Milnor's Invariants
\endtitle
\author Tim D. Cochran* and Kent E. Orr**\endauthor
\endtopmatter
\document
\subhead{\bf\S0. Introduction}\endsubhead
Two ordered links are said to be {\it concordant} if there are
disjointly embedded, locally flat cylinders in $S^3 \times I$, transversely
intersecting the boundary in the respective links. Concordance is a classical
equivalence relation on links dating to work of Fox and Milnor, and playing a
fundamental role in manifold theory, in singularity theory and the study of
characteristic classes.
Concordant links have homology cobordant complements, implying the groups of
these link complements have isomorphic nilpotent quotients. Nilpotent link
group quotients were exploited in Milnor's seminal work [M1], [M2]. This
was formalized and related to concordance questions in important work of
Stalling's [St], and Stalling's ideas were vastly extended in
Bousfield [B].
Milnor, using ideas of Magnus, defined his
$\bar\mu$-invariants [M2]. These are a numerical measure of the failure of
distinct classical link groups to have isomorphic lower central series
quotients,
and in particular, of the link complements to be homology cobordant.
During the past decade, a fairly complete understanding of Milnor's invariants
has emerged. (See, for instance, [C1], [O], [HM]) As Milnor
knew, his invariants obstructed the link longitudes lying in the intersection
of the lower central series of the fundamental group of the link exterior.
Recent work suggests slice links may have longitudes
lying in the intersection of the transfinite lower central series of the group
closure (See [L2], [CO2]). Unfortunately, it is still unknown if links with
vanishing Milnor invariants can have a nontrivial transfinite lower central
series.
Potentially, this doorway opens to previously unknown slicing obstructions.
This paper attempts to lay a groundwork for tapping this resource.
For example, it is unknown if a link with vanishing Milnor invariants is
concordant to a fusion of a boundary link. A fusion of a boundary link is a
geometrically constructed link obtained by appropriately piecing together the
components of a boundary link, and is a class of links more general than
boundary links but with vanishing Milnor invariants [C1], [C2]. J. P.
Levine defines an invariant which vanishes if and only if a given
link is concordant to a sublink of an homology boundary link [L2]. The
existence or non-existence of transfinite Milnor invariants would either
distinguish Levine's invariants from Milnor's invariants, or give an important
geometric interpretation of the vanishing of Milnor's invariants.
The idea of this paper is simple. We attempt to reach beyond the usual lower
central series into the transfinite lower central series of a link
group by investigating the lower central series of finite index subgroups of
link groups. This we achieve by defining Milnor invariants for links obtained
as finite, often iterated, covers of the original link. We call these {\it
covering links}, following [CO1]. (See that paper for examples of how
concordance invariants of covering links detect links not
concordant to boundary links.) To define these new invariants, we need a
Milnor-Stallings type theory describing lower central series quotients of the
groups of these covering links. This is already available via Stallings
theorem and the observation that homology cobordisms become $\bzp$
homology cobordisms after passing to $p$-fold covers.
We give mod p and $\bzp$ version of these localized Milnor
invariants, but sadly, the paper concludes with a result limiting the value of
these mod p obstructions. The $\bzp$ slicing obstructions merit
further investigation.
\subhead{\bf\S1}\endsubhead
\proclaim{Theorem 1} Suppose $f: G\to\pi$ is a homomorphism of
groups which induces an isomorphism on $H_1(\ ;\bz)$ and an
epimorphism on $H_2(\ ;\bz)$ for a prime $p$. Suppose
$\phi:\pi\to\BZ\to\bz$ is an epimorphism, $\tl\pi=\kl\phi$,
$\wt G=\kl(\phi\circ f)$. Then the induced homomorphism $\tl f: \wt
G\to\tl\pi$ is $2$-connected with $\bz$ coefficients. If $G$, $\pi$
and $H_2(\tl\pi;\BZ)$ are finitely generated (for example if $G$ is
finitely generated and $\pi$ is finitely presented), then $\tl f$ is
$2$-connected with $\bzp$ (and thus with $\BQ$) coefficients as well.
\endproclaim
\pf{Theorem 1} The proof is very similar to the proof, given in [CG;
Lemma~4.2],
that a $p$-fold branched cyclic cover of a slice knot is a $\bzp$-homology
sphere. Let $\psi$ denote the epimorphism
$\pi\tha\BZ$, $\i\pi$ its kernel and $\i G$ the kernel of
$\psi\circ f$. Let $f: X\to Y$ be a map of Eilenberg-Maclane spaces
inducing $f$ on $\pi_1$, and $M$ its mapping cylinder. Let $\wt X$, $\i
X$, $\wt Y$, $\i Y$, $\wt M$, $\i M$ denote the induced covering
spaces. The short exact sequences of chain complexes $0\lra
C_*\mx\overset t-1\to\lra C_*\mx\overset q\to\lra C_*(M,X)\lra 0$
and $0\lra C_*\mx\overset t^p-1\to\lra C_*\mx\overset q'\to\lra
C_*(\wt M,\wt X)\lra 0$ induce long exact sequences in homology
with $\bz$ coefficients. Considering the first of these,
$$
\lra H_2(M,X)\overset\p_*\to\lra H_1\mx\overset t_*-1\to\lra H_1\mx
\overset q_*\to\lra H_1(M,X)
$$
we see that the hypothesis that $f$ is $2$-connected implies the
outer-most terms are zero and hence that $t_*-1$ is an isomorphism
on $H_0\mx$, $H_1\mx$ and an epimorphism on $H_2\mx$. It follows
that $(t_*-1)^p$ and hence $t^p_*-1$ share these properties. Now
considering the second long exact homology sequence:
$$
\lra H_2(\wt M,\wt X)\overset\p'_*\to\lra H_1\mx\overset
t^p_*-1\to\lra H_1\mx\overset q'_*\to\lra H_1(\wt M,\wt X)\lra
$$
we see, conversely, that $H_2(\wt M,\wt X;\bz)\cong H_1(\wt M,\wt
X;\bz)\cong0$. It follows immediately that $\tl f$ is $2$-connected
with $\bz$ coefficients, as claimed.
If $G$ and $\pi$ are finitely generated then so are $\wt G$,
$\wt\pi$, $H_1(\wt G;\BZ)$, $H_1(\wt\pi;\BZ)$ and $H_1(\wt\pi,\wt
G;\BZ)$. If $H_2(\wt\pi;\BZ)$ is finitely generated it then follows
that $H_2(\wt\pi,\wt G;\BZ)$ is also. Under these conditions the
vanishing of $H_i(\wt\pi,\wt G;\bz)$ for $i\le2$ implies the same
for $\bzp$ and $\BQ$ coefficients. \qed
\rem If $\pi$ is finitely generated then $\tl f$ will induce an
isomorphism on $H_1$ with $\bz$ coefficients without {\it any\/}
assumption about $f_*$ on $H_2(G)\to H_2(\pi)$. For, following the
above proof, we see that we still get that $t^p_*-1$ is onto. But a
surjective endomorphism of the finitely generated vector space
$H_1(\wt\pi,\wt G;\bz)$ is necessarily an isomorphism.
\rem If $G$ is finitely generated and $\pi$ is finitely presented, Theorem
$1$ is
true without the condition that the homomorphism $\phi:\pi\to\bz$ factors
through $\BZ$.
\ex The second part of the theorem is false if $G$ is not finitely
generated. Let $G$ be the semi-direct product of $\BQ$ with $\BZ$
where $t\in\BZ$ acts by negation. Let $\pi$ be $\BZ$, the
abelianization of $G$, and let $f$ be the abelianization and $\phi$
be the epimorphism to $\BZ_2$. Certainly $f$ is $2$-connected. But
$\wt\pi\cong2\BZ$ and $\wt G\cong\BQ\x2\BZ$ so $\tl f$ is not a
monomorphism on $H_1$ with $\BZ_{(2)}$ coefficients though $\tl f$
is $2$-connected with $\BZ_2$ coefficients.
\vskip.8cm
\subhead{\bf\S2. New Concordance Invariants for Links in
$\bz$-Homology 3-spheres}\endsubhead
\bpage
In this section we define families of link concordance invariants.
In brief, our new invariants are $\bzp$ versions of Milnor's
invariants applied to certain ``covering links'' of the original
link.
Let $R$ be the finite ring $\BZ_p$ or a subring of the rationals. Recall the
$R$ lower central series of $G$ is defined by
$$G_1^R = G$$
$$G_{n+1}^R = \ker{\left\{G^R_n \rightarrow R \otimes \left( G^R_n
/[G,G^R_n] \right)\right\} }.$$
The above homomorphism is $1 \otimes q$ where $q$ is the quotient
homomorphism.
Let $\wt L=\{K_1,\dots,K_m\}$ denote an ordered, oriented link of
circles embedded in $\Sigma$, an oriented $\bz$-homology sphere ($p$
prime). Milnor's $\ov\mu$-invariants can be extended, in a
straightforward fashion, to links in $\bz$-homology spheres using
the $R$-lower central series of $G=\pi_1(\Sigma\backslash L)$ where
$R = \BZ_p, \BZ$ or $\BQ$.
A sketch of these extensions
follows. Choose a meridional map $\phi: F\\lra G$
which, by Alexander Duality, induces an isomorphism on $H_1(\ ;
R)$. Let $t_n(G) = G/G^R_n$. $R$ is implicit and determined
by context.
\proclaim\nofrills{Lemma 2.1 {\rm(see [M3] for the case
$R=\BZ$)}}: The meridional homomorphism
$$\phi_n:t_n(F)\lra t_n(G)$$
is surjective and its kernel is the
normal closure of $\{[x_i,w_{i,n}]\mid1\le i\le m\}$ where
$\phi_n(w_{i,n})$ represents an $i^\supth$ longitude of $\wt L$.
Thus $t_n(G)$ has a presentation
$$\.$$
\endproclaim
\pf{2.1} Choose an embedding $g$ of a wedge $W$ of $m$ circles into
$\Sigma$ whose image is $\wt L$ together with a set of base paths
compatible with $\phi$. The map $F\lra
P\equiv\pi_1(\Sigma\backslash W)$ is $2$-connected with $R$
coefficients and thus induces isomorphisms $t_n(F)\lra t_n(P)$
([St], see also [BK; Lemma~3.7]). Moreover, $\Sigma\backslash L$ is
obtained from $\Sigma\backslash W$ by adding $m$ $2$-cells (and one
$3$-cell) along the commutators $[x_i,\ell_i]$ where $\ell_i$ is a
parallel of $K_i$ on $\p(\Sigma\backslash W)$. Hence $t_n(G)$ is
the desired quotient of $t_n(P)$ (if $R\sbq\BQ$ use the fact that
$\ox R$ is exact). \qed
\bpage
There are also $\bmod p$, $\bzp$, and $\BQ$ versions of the Magnus
embedding. Let $\SP$ be the group of units in ring of power series
in the non-commuting variables $\{t_i\mid1\le i\le m\}$ with
coefficients in $R$. Let $h$ be the Magnus homomorphism
$F\overset M\to\lra\SP$ given by $h(x_i)=1+t_i$. This is an
embedding [S, \S6]. This induces a map $F/F_n\overset
M\to\lra\SP/\sim$ modulo terms of total degree at least $n+1$.
Therefore $M$ induces an embedding $M:t_n(F)\to\SP/\sim$. Hence,
given a multi-index $I=i_1i_2\dots i_{n-1}j$ of length $n$, we can
define $\mu(I)$ for $\wt L$ to be the coefficient of
$t_{i_1}t_{i_2}\dots t_{i_{n-1}}$ in $M(w_{j,n})$ where
$\phi_n(w_{j,n})=\ell_j$ in $t_n(G)$. If the longitudes of $\wt L$
each lie in $G_{n-1}$ then by 2.1, $w_{j,n}$ is unique in $t_n(F)$
and hence $\ov\mu(I)$ is a unique element of $R$.
\proclaim{Corollary 2.2} The longitudes of $\wt L$ lie in
$G^R_{n-1}$ if and only if $\phi_n: t_n(F)\to t_n(G)$ is an
isomorphism.
\endproclaim
\pf{2.2} Suppose $\ell_i\in G^R_{n-1}$ so $\ell_i=0$ in
$t_{n-1}(G)$. By induction we may assume $\phi_{n-1}$ is an
isomorphism, so $w_{i,n-1}=0$ in $t_{n-1}(F)$. If $R=\bz$ it
follows that $w_{i,n}\in F^{\bz}_{n-1}$ so the relations
$[x_i,w_{i,n}]$ are redundant. If $R\sbq\BQ$ it follows that
$w_{i,n}=\Sigma a_j\ox r_j$ where $a^{m_j}_j\in F^{\BZ}_{n-1}$ for
some $m_j\neq0$ and consequently $[x_i,w_{i,n}]=0$ in $t_n(F)$.
Conversely now suppose $\phi_n$ is an isomorphism implying that
$[x_i,w_{i,n}]=0$ in $t_n(F)$. Recall that, by definition,
$[\ell_i]\in G/G^{\bz}_2$ contains $x_i$ with zero exponent sum, and
hence the same is true of $[w_{i,n}]\in F/F^{\bz}_2$. Now we
require a result that, under this condition,
$[x_i,w_{i,n}]\in F^{\bz}_n$ implies $w_{i,n}\in F^{\bz}_{n-1}$.
The proof of Lemma~5 of [Levine; Link Concordance and Algebraic
Closure, p\.~242] for the case $R=\BZ$ works also for the cases
$R=\bz$ and $R\sbq\BQ$ using the fact that $n^\supth$ order
commutators may be detected in all cases by the ``Magnus''
embedding. It follows then that $\ell_i\in G^{\bz}_{n-1}$ if
$R=\bz$ and if $R\sbq\BQ$, that $\ell_i$ raised to a power lies in
$G_{n-1}$. In each case one concludes $\ell_i\in G^R_{n-1}$. \qed
\bpage
Therefore $R$-valued $(R=\bz,\bzp,\BQ)$ Milnor's $\ov\mu$-invariants
can be defined, generalizing Milnor, for any link in a
$\bz$-homology 3-sphere. (The first
non-vanishing $\BZ_{(p)}$ and $\BQ$
invariants are equivalent invariants.) Moreover these can be
seen to be unchanged by a ``$\bz$ concordance''. By a
$p$-concordance between
$\wt L_0$ and $\wt L_1$ in $\Sigma_0$ and $\Sigma_1$ respectively,
we mean that there is an oriented $\bz$ homology cobordism $W$
between $\Sigma_0$ and $\Sigma_1$ and a proper flat embedding $C$
of $m$ annuli such that $(\p W, C\cap\p W)=(\Sigma_0,\wt
L_0)\amalg(-\Sigma_1,\wt L_1)$ (as in [CO1, p\.~522]). Then
it is well known that the exterior $W\backslash C$ of the
$p$-concordance is an $R$-homology cobordism between the link
exteriors and that such an homology cobordism induces isomorphisms
on the quotients $t_n(G)$ [Stallings]. Therefore the
$\ov\mu$-invariants of $\wt L_0$ and $\wt L_1$ coincide when the
links are $p$-concordant.
Now, given any link $L=\{J_0,J_1,\dots,J_k\}$ in $S^3$ or indeed in
any $\bz$-homology 3-sphere, which has all linking numbers zero
$\bmod p$, we shall describe how to associate various ``covering
links $\wt L$'' to $L$ in such a way that varying $L$ in its
$p$-concordance class only varies $\wt L$ in its
$p$-concordance class. Our proposed ``new'' $\tl\mu$-invariants then
are simply the ``$R$-valued $\ov\mu$-invariants'' of the ``covering
link'' $\wt L$.
\defi{2.3} Let $\tl\mu_R(L)$ denote one of the $R$-valued Milnor
invariants of a covering link $\wt L$ associated to $L$ as
described in this chapter.
\vskip.8cm
Following [CO; \S1] exactly, given $L$, we construct the $p$-fold
cyclic cover of $S^3$ branched over $J_0$ to yield $\Sigma$ and let
$\wt L$ be the inverse images of the components. In [CO; \S1] we
usually excluded the branch set but here we usually include it. The
covering link is canonically oriented but {\it not canonically
ordered\/}. To do so entails specifying the ``first'' component for
each set of $p$-lifts of a component of $L$. The first such choice
will not actually affect invariants since $\bz$ acts as an
equivalence of links. Therefore there are actually $k-1$ arbitrary
choices. Hence if $k>2$, the $\tl\mu(L)$ suffer from greater
ambiguity than Milnor's invariants and perhaps one should view
$\tl\mu(L)$ as a set. It does not seem fruitful to invent a notation
for this at this time. Note that the property that all $\tl\mu$-invariants
vanish is unambiguous with respect to ordering of components. One extremely
important feature is that this process can be iterated to yield more
complicated
covering links
$\wt{\wt L}$. Note that a composite of $p$-fold branched covers may
not even result in a regular covering. Thus we have defined a vast
tree of potentially new and useful link invariants.
\proclaim{Proposition 2.4} If $L$ is a link in a $\bz$-homology
sphere then the set of $\tl\mu$-invariants for $L$ vanishes if and
only if the longitudes of the appropriate covering links of $L$ lie
in each term of the corresponding $R$-lower central series.
\endproclaim
\pf{2.4} Use 2.2 and the fact that the Magnus maps are embeddings.
\qed
\proclaim{Proposition 2.5} If $L$ and $L'$ are $p$-concordant then
their $\tl\mu$-invariants (corresponding to iterated $p$-fold
covers) agree.
\endproclaim
\pf{2.5} See the discussion preceeding 2.3.
\vskip.8cm
\subhead{\bf\S3. What the $\tl\mu$-invariants may detect}\endsubhead
\bpage
In this section we to assess what these may be measuring above and beyond
Milnor's $\ov\mu$-invariants. Although there are
philosophical arguments that these invariants should detect more
than Milnor's invariants alone, we show that all currently-known
links with vanishing $\ov\mu$-invariants also have zero value for
these generalized invariants.
A discussion is in order concerning why one might hope that these
invariants carry {\it more\/} information than the ordinary Milnor's
$\ov\mu$-invariants of $L$. Recall that K.~Murasugi has shown that
the information contained in Milnor's $\ov\mu$-invariants of $L$ is
captured by {\it ordinary linking numbers\/} between certain
components of certain covering links $\wt L$ associated to $L$ by
certain {\it nilpotent\/} covers of the exterior of $L$. Applying
this philosophy to $\tl\mu(L)$ we might expect that $\tl\mu(L)$ is
captured by ordinary linking numbers between components of covering
links obtained from $L$ by doing a (sequence of) $p$-fold branched
covers, {\it then\/} a nilpotent cover. Since the composition of
these covering spaces will not, in general, be a nilpotent cover,
nor even a regular cover, there is no reason to expect that our
$\tl\mu(L)$ are zero if Milnor's $\ov\mu(L)$ are zero.
Nevertheless, the only known class of links in $S^3$ which have
vanishing Milnor's $\ov\mu$-invariants are {\it fusions of
boundary links\/} (perhaps slightly more general is the class of
{\it sublinks of homology boundary links\/}, but no examples of
such are known except fusions of boundary links [Cochran-Levine])
and these also have vanishing $\tl\mu(L)$ as we shall prove below.
The discussion will also serve to focus on precisely what the
$\tl\mu(L)$ may be detecting over and above the set of $\ov\mu(L)$.
\proclaim{Proposition 3.1} Suppose $L$ is a link and $G$ its group.
Suppose there exists a homomorphism $\psi:G\to P$ such that
\roster
\item"i)" $\psi$ induces an isomorphism on $H_1(\ ;\bz)$
\item"ii)" $H_2(P;\bz)=0$
\endroster
Then the $\bz$-valued $\tl\mu$-invariants vanish for $L$. If, in
addition,
\roster
\item"iii)" $P$ is finitely presented
\endroster
then the $\bzp$ and $\BQ$-valued $\tl\mu$-invariants vanish for $L$.
\endproclaim
\pf{3.1} Let $\phi:F\\lra G$ be a meridional map.
Then $\psi$ and $\psi\circ\phi$ are $2$-connected with $\bz$
coefficients so, by Theorem~1.1, $\tl\psi:\wt G\to\wt P$ and
$\wt{\psi\circ\phi}:\wt F\to\wt P$ are $2$-connected with $\bz$
coefficients. Hence $\tl\phi:\wt F\to\wt G$
induces isomorphisms $\tl\phi_n:t_n(\wt F)\to
t_n(\wt G)$ for all $n$. Note that $\tl\phi$ is a meridional map for
the covering link $\wt L$ so $\tl\phi_n$ may be used to compute
$\tl\mu(L)$. If $\tl\ell_j$ is a longitude of a component of $\wt L$
then $\tl\ell_j\in\wt G^R_{n-1}$, by 2.2. Hence by 2.4, all
``first-stage'' $\tl\mu$ vanish for $L$. This argument may then be
iterated since the $2$-connectivity of $\wt{\psi\circ\phi}$ implies
$H_2(\wt P;\bz)=0$. \qed.
\proclaim{Corollary 3.2} If any of the following holds for the link
$L$ in $S^3$ then all $\tl\mu$ invariants vanish for $L$
\roster
\item"i)" $L$ is concordant to a sublink of an homology boundary
link; (in particular if $L$ is a fusion of a boundary link).
\item"ii)" $L$ is concordant to an $E$-link in the sense of
{\rm[Cochran LCIHT; p\.~643]}.
\item"iii)" $L$ is an $\wh F$-link in the sense of {\rm[Levine;
LCACII, p\.~580]}.
\item"iv)" The longitudes of $L$ lie in a subgroup $N$ of $G$ which
is the normal closure in $G$ of a finite number of elements, and
such that $N=[N,G]$.
\item"v)" The link group $G$ has the same algebraic closure as the
free group on a set of meridians (i.e., $\wh F\cong\wh G$ in the
sense of {\rm[Levine; LCACII]}).
\endroster
If any of the following holds for the link $L$ in $S^3$ then all of
the $\bz$-valued $\tl\mu$-invariants vanish for $L$.
\roster
\item"vi)" the longitudes of $L$ lie in every term of the {\it
transfinite\/} lower central series of the link group;
\item"vii)" the link group has the same Bousfield-Kan
$\bz$-completion as the free group with the isomorphism induced
by a meridian map {\rm[B]}. (In fact conditions {\rm i} and {\rm ii}
of {\rm3.1} are equivalent to this condition {\rm vii}).
\endroster
\endproclaim
\pf{3.2} There are implications
i$\Rightarrow$ii$\Rightarrow$iii$\Rightarrow$iv$\Rightarrow$v by
(respectively) [Cochran, Theorem~2.9], [Levine, II, proof of
Proposition~10], [Levine, II, definition on page~580]. By
Proposition~6 of [Levine II], $\wh F$ is a direct limit of
finitely-presented groups $P_n$ with $H_2(P_n;\BZ)=0$ and
$H_1(P_n;\BZ)\cong H_1(\wh F;\BZ)\cong\BZ^m$ so the map
$G\lra\wh G\cong\wh F$ factors through a map $G\lra P$ satisfying
the hypotheses of 3.1~iii. Hence a link satisfying v) has vanishing
$\tl\mu$-invariants.
There is an implication vi$\Rightarrow$vii derived as follows. Let
$G_\d$ be the intersection of the (integral) transfinite lower
central series of $G$. By Stallings' exact sequence [Stallings,
p\.~170], $H_2(G)\lra H_2(G/G_\d)$ is surjective. Since $H_2(G)$ is
generated by the tori on $\p(S^3\backslash L)$, if $\ell_i\in G_\d$
these classes become spherical and hence vanish in $H_2(G/G_\d)$
implying that the latter group is zero. Thus the composition of a
meridional map $F\lra G$ with the projection $G\lra G/G_\d$ is
$2$-connected with any coefficients. It follows that this
composition induces an isomorphism $EF\lra E(G/G_\d)$ between the
Bousfield $\bz$-homology localizations [B, \S1]. Since
$G\lra G/G_\d$ is also $2$-connected, it induces an isomorphism
$EG\lra E(G/G_\d)$ so $E(F)\approxeq E(G)$ as claimed. A similar
argument shows that i) and ii) of 3.1 imply vii) of 3.2. Conversely
if $F\overset\phi\to\lra G$ induces an isomorphism
$\hat\phi:EF\lra EG$ then the composition of $G\lra EG$ with
$(\hat\phi)^{-1}$ satisfies 3.1 i) and ii) since
$H_2(EF;\bz)\cong H_2(F;\bz)\cong0$. \qed
\bpage
It is not known whether or not the vanishing of Milnor's
$\ov\mu$-invariants for $L\sbq S^3$ implies any of conditions
i)--vii), since the only links which have been shown to have zero
Milnor's invariants are fusions of boundary links. Since the
vanishing of Milnor's invariants is equivalent to the longitudes
lying in $G_\om=\bigcap^\infty_{n=1}G_n$, we see that the $\tl\mu$
may obstruct the a priori stronger condition iv) above. However it
is known that the vanishing of Milnor's invariants implies that $G$
has the same $\bz$-{\it completion\/} as the free group and this
leads to the following disappointing result that the $\bz$-valued
$\tl\mu$-invariants are not very useful.
\proclaim{Proposition 3.3} If $L$ is a link in $S^3$ for which
Milnor's $\ov\mu$-invariants vanish, then the $\bz$-valued
$\tl\mu$-invariants also
vanish for $L$. (We are not claiming these hypotheses imply {\rm vi)} or {\rm
vii)} of {\rm3.2}).
\endproclaim
\pf{3.3} If the Milnor's $\ov\mu$-invariants vanish for $L$ then it
is well known that the meridian map $F\overset\phi\to\lra G$ induces
an isomorphism between the nilpotent completions (the
$\BZ$-completions in the sense of [BK; \S12]). Recall that the
$R$-completion of $G$, which we denote $\ov G_R$, is
$\varprojlim t^R_n(G)$. Since $\ov F_\BZ\cong\ov G_\BZ$, $\ov
F_{\bz}\cong\ov G_{\bz}$ induced by $\phi$. If $1\lra\wt G\overset i\to\lra
G\lra\bz\lra1$ induces a ``one stage'' covering link $\wt L$ of $L$ then
$\tl\phi:\wt F\lra\wt G$ is a meridian map for $\wt L$. In the exact sequence
$1\lra\wt F\overset j\to\lra F\lra\bz\lra1$, the generator $t$ of $\bz$ acts on
$H_1(\wt F;\bz)\cong\BZ^{1+mp}$ nilpotently. This is true since $t^p_*=\id$
so $(t_*-\id)^p=0$. Since $H_1(\wt F;\bz)\cong H_1(\wt G;\bz)$ the
same holds for $1\lra\wt G\overset i\to\lra G\lra\bz\lra1$.
Therefore, by [BK; p\.~63--64], these induce short exact sequences
between the $\bz$-completions $1\lra\ov{\wt F}\overset\bar
j\to\lra\ov F\lra\bz\lra1$, $1\lra\ov{\wt G}\overset\bar
i\to\lra\ov G\lra\bz\lra1$. Consequently a diagram chase shows that
$\tl\phi$ induces an isomorphism $\ov{\wt F}\lra\ov{\wt G}$. Then
by 2.2 and 2.4, the $\bz$-valued $\tl\mu$-invariants of $L$
(corresponding to $\wt L$) vanish. This argument may then be
iterated to show that all $\bz$-valued $\tl\mu$-invariants vanish.
\qed
Perhaps the $\bzp$-valued obstructions will detect links with vanishing Milnor
invariants.
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