TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 171, September
1972
BOUNDARYLINKS AND AN UNLINKINGTHEOREM^)
BY
M. A. GUTIÉRREZ
ABSTRACT.
dimensional
link
This paper gives
to be trivial.
An 777-link is an embedding
(72 + 2)-space.
isotopy
There
the standard
cobordant
trivial
link.
to the trivial
An interesting
Tzz-link is trivial
disk;
concept,
weaker
if the embedding
with boundary
The purpose
be (i) boundary,
(i)
of this
paper
triviality,
trivial;
as
[7l and [9l.
relations,
is to give
is that
any link to
if the link
to an embedding
are called
some
is
which
of dimension
for a link
on the homotopy
space,
sends
meridians
to
type
as follows:
is an epimorphism
> 4 and
(72 + l)-
manifolds.
conditions
are reflected
an
of the (72 + l)-
of m compact
of the link in the ambient
in 772generators
link:
copies
Seifert
homotopic
conditions
if, and only if, there
i. be an m-link
of boundary
to one of ttz disjoint
These
These
of the image
77,(X) onto the free group
Let
than
if it extends
An 772-link is boundary
(ii)
such
are found in [2ll,
to it is called
extends
the sphere.
(ii) trivial.
of X, the complement
into the
links
one, we say it is a slice.
an 772-link is boundary
manifolds
amongst
via our equivalence
Any link isotopic
for a higher
of the 77-sphere
relations
and definitions
to compare,
criterion
copies
equivalence
Some results
it is useful
theoretic
of ttz disjoint
are various
and cobordism.
Generally,
a homotopic
from
to generators.
\Jm S , the wedge of 772circles;
suppose
ir.(X) = rr. ( V S1)
and that
for
Then
z = 1 meridians
are sent
is a boundary
link where
i
(q - l)-connected.
In particular
As by-products,
presented
for z < q < 7 (« + l)
on to generators.
the Seifert
manifolds
for q = V2(n + l),
we obtain
some results
can be chosen
to be
X is trivial.
about
cobordism
of links
which
are
in §3.
Essentially,
this
the supervision
Received
of J. Levine,
by the editors
AMS 1969 subject
Key words
is the author's
thesis,
whose
September
help
Boundary
the author
at Brandeis
is glad
University
under
to acknowledge.
28, 1971.
classifications.
and phrases.
written
Primary 5520, 5720.
zn-link,
lower
central
series
of a group,
Seifert
manifolds.
(1)
This
paper
was
written
in Méjico
under
the auspices
of the
O. A. S. Multinational
Plan.
49I
Copyright © 1972, American MathematicalSociety
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492
M. A. GUTIÉRREZ
0. Notation.
smooth
or PL,
In this
locally
paper,
flat,
all manifolds,
categories,
As usual
5"
772S" is a disjoint
and
union
D"+1
boundary
if it extends
manifold
of Seifert
manifolds
of 272copies
and isotopies
All manifolds
, M 2. stands
belong
and the
to the
are oriented
for the disjoint
'
=1
(n + l)-disk.
union
of
In particular,
of Sn.
X: 2225"—►Sn+
to an embedding
with boundary
for £.
2mz
'
are the n-sphere
An 22z-link is an embedding
orientable
mappings
consistently.
and if M I. (i -..'-.
= 1, • • •,* 722) are manifolds,
them.
[September
^m_,
S".
(222, s > l);
V ■—►S"+
The collection
In particular,
if the
we say that
, where
is
V . is a compact,
\V .\ is called
V . ate disks,
£
a collection
we say that
<£ is
trivial.
Consider
find tubular
Let
£..:
S"—► S"
neighborhoods
, the restriction
T.
of Im(x)
X be Sn+
- U
T -, a compact
same homotopy
type
as S"+
Consider
these
arcs
arcs
and
generators
X is called
the
A and,
meridians.
when the generators
[l8l;
Gffl is by definition,
Ambient
collection
bundle
surgery.
of Seifert
ÓV¿ =Sn
of the
are mutually
x S" to a common
miS
disjoint
[20].
x Sm) and of the
basepoint;
for 22> 2, its fundamental
is denoted
Let
manifolds
G
indicates
the
by Fim)
the union
group
of
is free in 222
or by F{a
X.: mS"—* Sn+
for it.
Let
V ., we can find embeddings
where
72th member
(| G .. In particular,
, ■■■, a
]
V. = V ■ and
of the lower
link and
V. —>/ be a map with
\V \ a
/"
With the aid of the (trivial)
ip.:
the two copies
central
if F = F(tt2), F<u= 1 [lO].
be a boundary
/.:
and W. = \ix, t) £ V. x l\ fix) < t\.
V .q U V.j,
which
with boundary
are specified.
if G is any group,
1.
S
The group
Finally,
series
= L-,
manifold
- Im(£).
in X joining
called
of X. to S" ii = 1, • • •, 222); we can
W . —►Sn+
; notice
are attached
that
(0) =
normal
dW . =
by the boundary.
The
manifold
y = s"+2- U<A,(wO
is
called
5"+
Z™ = liVi0
cut
U Vn).
along
the
V ■, a compact
The composition
manifold
with
V{ = Vu C Y is called
boundary
viv
See [20].
Let <p:Skx Dn+1~k —>V. be an embedding; define 9ÍV-, <p) as W. U Dk +l
x Dn+
1
~
L
where
the handle
*
is attached
to V .
by cp; 99 (V¿, cp) is equal
Z
to V{
U xivi> 0). s = I - t.
Suppose
that for some
k > 1, the
V ■ ate zè-connected.
By Alexander
duality
HqiY; Z) =2HqiVi; Z), q < k - 1; let Ya= 5"+2 - Int Uy<aWy, then by [7, §4],
Y x is 1-connected.
If Y
is 1-connected,
then by the van Kampen
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theorem
we have
19721
BOUNDARYLINKS AND AN UNLINKING THEOREM
493
»1<*„I>
0=77j((9W .)
1
77.(Y)=0
7+1
1
r
n^y)
therefore
Y
theorem,
Y is then
. is 1-connected
and,
by induction,
so is
Y.
By the Hurewicz
(k - l)-connected.
Lemma (1). // (i) n > 2k + 1 or (ii) n - 2k or 2k - 1, n <4,
77fe(V¿)is in ker v. t, the embedding
where
cp is an embedding
Proof.
since
ß
pair
of 77, ,(Y, W .).
an embedding
This
g: D
latter
a..
(3) of [7].
v.^a.
ß'
g can be isotoped
By using
Let
= 0, ß
a tubular
of £ n,(V
is the boundary
is ¿-connected,
—> Y representing
V . is 1-connected,
g(S ) represents
to 8 (V ., cf>),
a.
We refer to the proof of Lemma
ing to a under V. = VÇ, C dY,
Since
if/.: W.—» Y U W . extends
representing
and a £
hence
which
) correspondof an element
by (1) of [6], there
is transversal
by (2) of [6] and (3) of [7l
neighborhood
of Im(g)
is
to the boundary.
so that
we get the desired
extension.
2. The Fundamental
and
H An) =0,
where
coefficients.
fundamental
i
Lemma
Proof.
groups
group.
Let 77= 77j(X); in [4l it is proven that
H (n) is the
The inclusion
qth homology
A C X induces
Hy\n) = Zm
group of 77 with trivial
a homomorphism
integral
i: F(m) —» 77 of
groups.
(2).
For n > 2, i induces
In fact,
Proposition
F(m)/F(m).
z*:
F(t72) C 77/77 .
7 induces
of F(t7z) and
In particular,
i±:
*
(3). £
77/77.,
7
an isomorphism
77. The result
the meridians
J ^
follows
on the first
now from [l8,
of it generate
is a boundary
'7 'finite,
a free
and second
Theorem
subgroup
homology
3-4].
of 77/77 .
link if, and only if, the inclusion
í¿; F(t7z) C
77/77^ is an isomorphism.
Proof.
that
onto
If X. is a boundary
»# is an isomorphism.
\/m S
by projection
Let
link,
77/77 = F(t7z) via i„ as in [l6l.
\Jm Sl be the wedge of t?2 circles,
to get a diagram
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Suppose
now
we can map A
494
M. A. GUTIÉRREZ
[September
X
a -E-►v
which
by [3, p. 194] can be completed
fundamental
s1
if and only if the corresponding
diagram
of
groups
77
\
p*_^
F(?72)
can be completed,
So there
PL,
that
is,
is a map
locally
flat)
? Firn)
if i^ is an isomorphism.
q: X —►\Jm S
map.
Choose
which can be approximated
points
x . in the
z'th circle
by a smooth
(resp.
of \/m S ; the manifolds
V'i = q~
' lix i.) can be deformed to V i. C Sn +2 with dV ii■= L ..
Let
X and X
F-isotopy
Sn+
be two m-links
of dimension
on the z'th component
- U. ,.L.
if there
and an orientation
72. X
is a torus
preserving
arises
V = D
from it
x S
homeomorphism
by a simple
contained
/: S"+
in
—>5" +
such that /(L .) = L* for / / i and either
(i)
L z. is the core of V (i.e.
v
v
is an isomorphism,
(ii)
L 2' is the core of fiv)
'
Suppose
(i) is the case;
Theorem
(4) (Smythe).
inducing
an isomorphism
For a proof
Corollary
L z. = (0) x S") and H72(LA
= Hn (Sn)
-»//
z
i
ifiV))
n '
or
and H n ÍL i.) =
let
H n iS")i —>Hn (V) is an isomorphism.
r
27 be the group
The group
27/77, =
of X and
n is a retract
p/p2
ani
p that
of p under
preserving
of X.
a map
xp: p —> 27
meridians.
see [17].
(5).
Every
link F-isotopic
to a boundary
In fact,
if X
is a boundary,
i: Fim) =
link is itself
a boundary
link.
Proof.
meridians.
On the other
and the generators
Let
hand
by [l8],
are meridians;
X be a boundary
link,
xp: p/pu
hence
ÍV.!
H 1. = 27,(V.)
I
1
H 1. —>G
the obvious
„
tion recall
cover
v.,:
it
for the cover
of this
wedge
of circles.
copy of y with boundary
IntV'.,(izz)
the
of Seifert
homomorphisms.
r
manifolds
group
We can
to 27 . To motivate
ts
the cover
Let in fact,
"7
X should
Yiw),
2m_AV -Aw) U Vixiw));
to Int V -Awa .). Suppose
and so ipi: F ^
V ■ with fundamental
X of X associated
\Jm S C X as the meridians;
tr/n^
a collection
of X cut along
and
=
are
tt/ttu
X is boundary.
Y be the complement
construction
p/pM and the generators
G is presented
contain
tot w £ F{a.,
X is obtained
for it.
G.
Let
Call
find an explicit
r
its constructhe universal
■• •, a
], be a
by identifying
by (y x, ■• -, y a; R x, ■■■, R a )>
then
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19721
BOUNDARY LINKS AND AN UNLINKING THEOREM
Proposition
(2) tt=
(6).
(w, ya:
G ^> = rríydí/)),
Proof.
We have
the following
RT(ya),
wy^w'^y^,
presentations:
»,«•> = *<--*))
. Here
*<*>: 7/. _
we Firn).
(1) follows
the groups
group
495
G^w'
amalgamations
from Neuwirth's
ate in the vertices
are performed
from considering
theorem
[l3l.
of the universal
along
the edges
We can simply
cover
of such
assume
of \/m S
cover.
that
and that the
Assertion
(2) follows
the group extension
1 —> 77 —> 77-► E(t72) —» 1
which splits
3.
since
F(7T2)is free [ll,
Cobordism.
A link X. is split
by (72 + l)-spheres
that
they
S C X.
are cobordant
Im(K) meets
cobordant
Given
if there
IVl.
if the
L . can be separated
XQ and Jl,,
exists
link is called
K: mS" x I —►Sn+
x I where
split-cobordant.
72> 2, ÍV î a collection
of Seifert
for it.
Theorem
(7).
Every
boundary
link of dimension
Proof. As in III.6 of [4l, we can add handles
dimension
taken
other
72, we say
and K|ttz5" x {t\ = £( for ' = 0, 1. A link
it be an 772-boundary link of dimension
manifolds
from each
772-links of dimension
an embedding
d(S"+2 x /) tranversally
to a splitted
Let
Chapter
below
the middle
to be disjoint.
embeddings
and,
The result
by general
n > 2 is split-cobordant.
to the V"+
position
is a collection
[l9l,
in D"+3 up to one
these
of manifolds
handles
W"+
can be
in D"+
and
j .: V ■x I —*W . satisfying
(i) W. O Sn+2 = V. and jfx, t) =Y2(t + l)x in D"+i.
(ii) dW.= IC U j.(dV. x /) U V¡, where V\ O V¡ - 0, VJ n f{(dV{x /) = dV[
= j.(dVi x 0).
(iii)
V- is connected
up to the middle
dimension.
(In particular
for 72 even,
V. is a disk.)
(iv)
W. is obtained
from j.(V ■x ¡) by adding
We use now the engulfing
argument
of Lemma
handles
of index
< }/2n.
4 of [9l for W y in the notation
of [2, Theorem 2], X = V[ and V (of [2]) = D"+3 with cuts along the W\ (l < i < m).
The hypothesis
of the engulfing
theorem
is verified
as in [9] so we can find a
ball B*+3 in V with Bj D W, = V[. Now we repeat the argument for V = D" +3
with cuts
along
Vf, U B ,, W2, ••-,"/
we get
B ,,-•-,
0
1
D"+
- ß contains
4.
retic
Unlinking
criterion
and to the engulfing
process.
By induction
B 772 with B Z. f~lW2. = V'.
Then B ,1 # • ■■ ü B 772 is a ball B and
Z
a cobordism
of X to a link split
spheres
in codimension
for determining
whether
two.
by the "spheres
The following
an 772-link of dimension
or not.
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dB y
is a homotopy
72 > 4 is trivial
theo-
496
M. A. GUTIÉRREZ
Theorem
(8).
X is homotopy
is generated
sult
X be an m-link
equivalent
[l5]
for 272= 1, 72= 3.
setting,
results
of the cover
of X cut along
[7] for the case
> 7.
to this
Cappell
772= 1, 72> 4 and by
work,
Lee [5] proved
[l] in his thesis
is to be taken
with integral
X of X made in §2:
the Seifert
n Ax) = F(m)
the re-
obtained,
in a
to ours.
all homology
the construction
copies
link, where
X; if
trivial.
Simultaneous
similar
n > 4 with complement
of the trivial
X is itself
has been found by Levine
In this paragraph
call
then
for 222> 2 and odd dimensions
more general
of dimension
to the complement
by the meridians,
This result
Shaneson
Let
[September
manifolds.
These
coefficients.
X is obtained
copies
Re-
by pasting
are called
Yiw), w £
Fim), and have boundary 2ÍV .Aw) U V .Aw)).
Lemma
(9).
Let
be 1-connected
Proof.
X be a boundary
if, and only if,
In fact, if the manifolds,
Proposition
(6), assertion
X is in this
case
link;
the Seifert
(2).
the universal
VÇ are simply connected,
it n = F{a x, ••-,
cover
F(m)w=
since
the map Int V . C X factors
By a result
27.(int V .) = 27.(V.)
that
of Serre [l4],
/': D
map above;
The technique
resent
a1..
1 The
by general
V,,
••-,
an innermost
component
of /
without
altering
by induction.
Lemma
and (i) v.
V ,, • • •, V,,
(10).
Suppose
• 27,(V.) —» z7,(y)
simply
of the
manifolds
If 2 < k,
, 1-connected.
is a monomorphism
about
the
Suppose
that,
a £ rr X(V^) repa = 0; if 2 > k +
(1) to eliminate
of Lemma
• •■
V ■ such
f1. are disjoint.
and let
on V ■ as in Lemma
V,
V .)
say by a2,
to the
connected.
V .).
77.(int
of the remark
a A
(D ) O (\J
we can make
the Seifert
Vj
the map
X of
X by construction.
regular
because
choose
] the cover
generated,
the images
us to make
Vk ate 1-connected;
1, a £ ker v. „ and we can do surgery
follows
[l9]
to
27= F(t22) by
a
Then,
though
is finitely
fl. exist
A
position
of [7, §5] allows
by induction,
1.
—» X (l < 2 < 272, 1 < j < r) be transverse
f2.(S
) C Int V.2 represents
1
r
inclusion
can be chosen
by the meridians.
Conversely,
—» 27= F is zero because
a2 . Let
manifolds
n = 27.(X) z's free generated
a.
The result
(9) are
So,
now
(k — l)-connected
for t = 0, 1, all
i, (ii) 27,(lnt V¿)
—»77,(X) z's zero for all i; then 27,(VC) = 0 for all i.
Proof. Let
a £ 77^VÇ ); by (ii) there is a map /: Dfe+1 —»X such that f(Sk) C
Int V .
and represents
inverse
image
D + ; let
/').
We may assume
component
If of D *
As in Lemma
in [7].
that
/ is ¿-regular
, V . is a not necessarily
M be an innermost
nected submanifold
(some
a.
by / of Um
of it, i.e.
such
that
with dW = M and suppose
(4) of [7],
/|zVf extends
By a sequence
of such
to all
connected
V . so that
ze-manifold
there
exists
a con-
f\M maps M to V .
to W and we can eliminate
manner
described
/-1(U
V¿) = Sk so that vt t^a = 0 for some t and, by (i),
modifications
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the
in
we will have
a = 0.
M in the
19721
BOUNDARY LINKS AND AN UNLINKING THEOREM
We can now prove
the following
Proposition
Let
complement
(11).
n>2k+l
and x. an m-link of dimension
n^^nffls1),
includes
the assumption
meridians.)
nected
Then
Seifert
Proof.
that
the fundamental
Jl is a boundary
manifolds
i<k.
group
group,
Following
link and we can find a collection
an application
of Lemma
ate not monomorphisms.
so by Lemma
of X is generated
by the
of k-con-
for it.
(9) suppose,
Lemma (10), that the V . are all (k - l)-connected
—►itÁY)
n > 4 whose
X verifies
(Uk)
(U.
497
By [l4l,
of
(k > 2), and that Vit*'. ni/A/A)
ker V-,t
(1) it can be eliminated
in the notation
is a finitely
by surgery;
generated
by Lemma
abelian
(10) the
V. ate
now connected.
We must kill ttAV •) when n = 2k or 2¿ - 1 under assumption
2k - 1, ker V.
is generated
is free abelian;
by Lemma
therefore,
(10), the
by primitive
by Lemma
elements
(1) v. * can be made
V ■ can be exchanged
^zi*|T.
by the Mayer-Vietoris
theorem,
where
©
integral
is the integral
Laurent
group
polynomials
T..
and then,
With the notation
By Lemma (5) of
of §2 and
= rr.X = 77,X = 0 is presented
0 = Uk*!* -* Z'W®
= H AV )
monomorphic
.(V .) has torsion
monomorphic.
H,X
ttAv)
for 2^-disks.
For 72= 2k, notice that 77,(V .) = H,
[7], we can make the
(LIA). If n =
[12] because
[8] and
by
0ra -^ "kW S6,* -* "k* = 0
ring of Fizz y • • ■, a
in m noncommuting
1 viewed
variables
as the ring of
t,,
•--,
t
and where
d is given by the formula
d(a ® 1) = v.n*a
IU
From the sequence
Lemma
® /.111- i/.,*a
® 1
for a e //,R. (V I ).
, d is an isomorphism.
(12).
Under
the present
hypothesis,
ker v.
is generated
by primitive
elements.
Proof.
itive
Assume
element.
z = 1 and let
Now,
pu.
,a
= 0 so that
Represent
i/j
ß
to p': D + —►X; ß'
extends
(w £ F(m)),
by an embedding
contained
D + . Consider
manifold
,a'
a £ ker v. „.
in H Ay)
iy
a
(U
/3'-1(U
V Aw))
V-,(l));
a = pa
£ Hl(Y)
—» Y(l) C X.
can be assumed
in X, so ß'~
the components
W and hence
ß: S
Then
and
is a torsion
Since
to be transversal
® © , W establishes
together
a homology
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is a primelement.
X is contractible,
is a ¿-dimensional
they,
a
to the
V Aw)
manifold
with
in
S , bound
a
498
M. A. GUTIÉRREZ
[September
m
(3)
"1(/®
1 = 22 {uioa, • h - vaa\ • 1}
z= l
where
o".(resp.
Since
each
loop
d is an isomorphism,
<f. represent
M2 by an arc
y,y2
y,
Now consider
nonconnected
those
that
each
fj^a)
= 0.
Then
that
In fact,
if cri =*Çj +
M . of D + , we can join
arc
it bounds
y2 in /3 (W).
a 2-disk;
Let
y £ T.
torsion
M,
The resulting
by a Whitney
process
that represents
can be eliminated
by 2k + 1 disks
generalization
reasoning:
in particular
v. y = 0 for some
In such a way we can assume
ker vi
the above
elements,
y £ H ,(V Aw)) = H A]/ ■) be represented
and
y = 0 and the intersection
V ■ can be exchanged
proof is a direct
Notice
all represent
components.
the
o. = o. £ T ..
of D + —VI and repeat
V . (w))
of [7, p. 13].
Thus,
V-Al)).
component
component.
is a monomorphism,
(resp.
(\J
are innermost
by the method
submanifolds
Y, hence
V„(l)
submanifold.
and by another
in
with
/3'" Kv i0(l)) = Mj # M"2.
of ß'~
by an innermost
that
by a connected
in V ._(l)
is nullhomotopic
the components
of ß'(W)
from (3) we conclude
we can alter ß' so that
the
the intersection
oi (a A can be represented
zf2 and the
and
o\ ) represents
by surgery
proof which
Since
v. \T.
y can be eliminated
ß'(D
and the theorem
of Levine's
s.
) C Y(l)
and
and so
by Lemma
is proven.
(10),
The present
is remarkably
simple
and
geometrical.
Theorem
(Lee's
homotopyzJ
morphic
(8) is equivalent
to the following
form of the Unlinking
equivalent
'
to K
As a last
Theorem).
statement:
Let
M"
, « > 3, be a closed
to K nm =SlxS"#---#S1xSn(m
in the category
remark,
that the fundamental
out it both results
both
s
'
then
manifold,
M is iso-
PL.
in Proposition
group be generated
are false.
times);
(3) and in Theorem
by meridians
(8), the condition
is essential.
In fact,
with-
See [22].
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