Internat. J. Math. & Math. Sci.
(1990) 39-44
VOL. 13 NO.
39
A DECOMPOSITION INTO HOMEOMORPHIC HANDLEBODIES
WITH NATURALLY EQUIVALENT INVOLUTIONS
ROGER B. NELSON
Department of Mathematical Sciences
Ball State University
Munci e, IN 47306
(Pceived September 12, 1988)
Downing [6]
ABSTRACT.
extended the well-known result that any closed 3-manifold X
el(X-H) is homeomorphic to H in the case where X is
We show that if X is a compact 3-manifold
contains a handlebody H such that
a compact 3-manifold with nonvoid boundary.
with involution h having 2-dimensional fixed point set, then X contains an h-invariant
handlebody H such that the
i,,duced
involutions
on
H
and
el(X-If)
are
naturally
equivalent.
KEY WORDS AND PHARSES.
3-manifold, involution, handlebody, equivalent involutions.
1980 AMS SUBJECT CLASSIFICATION CODES.
Primary 57Q15, 57S17, 57s25.
I NTRODUCT ION.
In this paper all spaces and maps are piecewise linear (PL) in the sense of
Rourke
and
if].
Sanderson
subspaces are PL.
Bing [2].
spaces can be triangulated and all maps and
Thus all
This is no restriction as every 3-manifold has a tr[angrulation by
[3], [4], and Rourke and Sanderson [I].
Notation follows Nelson
Let X be a closed 3-manifold with triangulation T.
and H
If S is the l-skeleton of T
S, then it is a classical result, see Seifert and
el(X-H) are homeomorphic handlebodies. Downing
2
is a regular neighborhood of
Threlfa]l
[5,p.219], that H
and H
[6] extended this result, showlug that every 3-manifold with nonvoid boundary also
It is the purpose of
contains a handlebody which is homeo,orph[c to its complement.
this
paper
to
show
that
symmetries are respected.
these
res,Its
urther
can
These snmetries
ale
extended
so
that
certain
those realized as involutions of X
with orientable 2-dimensional fixed po.nt sets.
Recall
that
Involdtlons h :X
i
/
i
homeomorphism f:X
involut[on h:X
an
involution
Xi,
i
l,
X such
X
ix
;s
a
h,meomorphsm
are equivalent,
that h
2
X is F(h)
of
,Y[h(x)
o h
x}.
2
denoted h
o f.
The
on
h
X
2
Fixed
of
period
two.
if there exists a
point
We shall prove the following:
set
of
R.B. NELSON
0
THEOREM I.
Let X be an orlentable 3-manlfold and h:X
orlentable
fixed
invarlant handlebodies H
and H
dimensional,
Let f:H
THEOREM 2.
point
F.
set
Then
hlH
in X such that
thereby establlshlng the equivalence
assumed to be the identity on
H
H[
hlH
Io
H
h
such that
hlH
By Nelson [4; ThE. C] the involutions
hlH 2
and
(i) their fixed point sets are homeomorphic and
exist
complementary,
hlH
f-I
of Theorem I.
2
H2
H
2
there
hlH2.
2
be the homeomorphlsm
/
X an involution with 2-
o
(hiM 2)
flHl
Then
h-
o f
may be
are equivalent If and only If
(If) elther both fixed point sets
In Nelson [3; lemma 2.1] it was shown that a 2-
separate or both fall to separate.
dlmensional fixed point set of an Involut[on on an orlentable handlebody separates the
handlebody if and only if the l-dlmens[onal fixed point set of the involution induced
Snce H
on the boundary separates the boundary.
and H
can be chosen so that H N F
need only show that H
share a common boundary, we
2
H2N
ence,
F.
Theorem
follows
immediately from the following lemma:
LEMMA I.
Let X be
orlentable
dimensional,
invariant handlebodies
a
copact
fixed
H!
and H
point
3-manlfold
set
F.
1en
in K such that H
X an
and h:X
OF
exlst
there
H
Involutlon wlth 2-
complementary,
2
2
The 3-manlfold X may be either orlentable or nonorlentable in this lemma.
be assned to be orlentable in Theorem
in order
h-
F.
invoke Nelson
to
X mast
[4; ThE. C].
Section 2 contains the proof of thls lemma in the case when X is a closed manifold.
The proof is modified in section 3 to cover the case when X has nonvold boundary.
proof of Theorem 2 is provided In section 4.
of proofs which appear elsewhere.
The
Both sections 3 and 4 are modifications
These sections refer heavily to the proofs which
they modify.
2.
)[
IS CLOSED.
Our approach is to pick an initial handlebody H
until H
F
+
showlng H
of F
H
2
Fi=
F
that will be repeatedly modified
F.I
of F.
This identity is established by
I for each component
/
+
0 F and H 0
FI, where F i and F are "h omeomorphic halves"
I
F.-- 0
i
i
i
We note first that any component F
The Initial H
surface.
i
of F is a closed, orlentable
Any such surface admits an involution gl with a separating fixed point set
+
This separates
and F
Fiinto F I
I
+
consisting of at most two simple closed curves.
where
gl (FI)
Let
X
Fi
be the orbit space of h.
manifold with
F.z
is
B
F.
gl-invarlant.
triangulation of
of T* to X.
Triangulate
Since
X can
BX
B
Since F is 2-dimenslonal,
is collared,
Rouke and Sanderson [I"
be extended to a trfangulatlon T* of
Then T is an h-lnvarlant triangulation of palr
triangulated identically to F
i
is a bordered 3-
so that the triangulation of component
for each i.
X
Cot.
2.26], the
Let T be the llft
+
(X,F) such that F i
is
41
DECOMPOSITION INTO HOMEOMORPHIC HANDLEBODIES
(n)
Denote by T
the nth derived subdivision of T and by S
edges and vertices of T
H2-- (X-HI).
and
triangulates F, H
(n)
H
and H
10
F
2
lying on the l-skeleton S of T.
Procedures:
each F
The
i
I
handlebody
of
result
each
T
i
(2))
T
Since
minus k disks, where k
to
I
either
by
procedure
3-handle
a
plus
20
H
of
is
index
that
one
modified
the
the
and
new
plus a 3-handle of index one. Hence,
2
left with homeomorphl c, complementary
is isomorphic to H
complementary handlebody
after
N(S (2),
F consists of k disks.
i
i
Two procedures are used to repeatedly modify T near
modification
isomorphic
is
HI=
handlebodles.
orientable surface F
is the connected,
i
is the number of 2-simplices in the triangulation of F
Modification
Let
orlentable
homeomorphlc
are
(n) the collection of all
we
modification
again
are
handlebodies.
We say such a pair is near F if
i
The 3-simpllces of T are in h-invarlant pairs.
F
the two 3-simplices of the pair intersect in
The procedures below correspond to
i
We will not need to
the cases where 3-simplices intersect in an edge or 2-face in F
i
employ modifications exploiting s implices intersecting F in a vertex.
i
Procedure =:
The net result of this procedure is to remove one edge from TO F
F.I
Intuitlvely, H 0
contains "less" of F
than H O F
i
i
i
Then e is the common edge of two 3Let e be any edge of T lying in F i
Let e and e 2 be the
s and s 0
such that h(s
slmpllces, s and s
s2-- {e}
2
2
(2) to
(I)
Modify S
edges of T
connecting the vertices of e with the barycenter of s
(2)
(2)
lying on e and adding
all vertices and edges of T
by deleting from S
get
I)
to
(2)
S(2)all
H
by
H2--
T(2)lying
vertices and edges of
l-- NI(2) T(2)) I
Procedure
B:
a
3-handle
of
(clUe2)U h(elUe 2).
in a point.
As a result,
ala2a 3 be any 2-simplex of T in F.
ala2a3a4 and ala2a3a 5 such that
HI0
Then
h(a
3-slmplices
is a double tetrahedron we shall retriangulate.
slmpllces
ala2a4a 5
a
a2a3a4a 5
la3a4a 5
F
i
contains a disk component not
4)
ala2a 3
a
is the common face of
The union of these slmpllces
5
The new triangulation consists of the
and their faces.
this new triangulation is h-invariant.
It is easily checked that
Let T be the triangulation of X obtained by
replacing the former simplices of the double tetrahedron by the new ones.
I-- IN((2)’ (2))I" I
Hl,
the edge
Employing
a4a5.
the
near each F
i
2 clIX-H2)
Modifications:
<
is homeomorphlc to H
is also homeomorphlc to H
The
and
Substitute
plus a 3-handle whose core is
plus a 3-handle of index one.
8modifications are repeated independently
Let f be the number of 2-simpllces and p the number of vertices in the
We assume that the original T was chosen so
(and also of
triangulation of
that p
one
i
Let
for
index
The net result of this procedure is to add an h-lnvarlant edge to T
which intersects F
in H 0 F
plus
We replace
plus a 3-handle of index one, the cocore of which
cI(X-H I) is homeomorphic to
lies in a disk bounded by
to H
homeomorph[c
is
h(elU e2).
(elDe2)
on
F
F).
2f for each fixed point component F
i
42
R.B. NELSON
CASE I.
>
p
f.
First apply
p-f
this out we ralst have that p-f -< f.
(F i
3f
both II
+
lfi
when
F and
to F (or F
S
F.=
i
2.
Next,
F consist
H2
of
In order to carry
ala2a3eF
disk
p
x(F i)
(F
apply,
0 and easily checked in the
procedure a to
and
components
one
in F
edge
every
Then
homeomorphic
component
minus p disks.
CASE 2.
>
f
p.
F consist of
B f-p
First apply procedure
procedure a to remove all edges of T from
H20
where
But this is equivalent to the statement t1at
e which is immediate when
case
remaining
time,;
F.
+
+
F.
times to sl.nplices in
is that both H
The res,lt
disk components and one component ho,.eomorphic to F
+
Then use
IO
F and
(or F
less f
disks.
X IS NONVOID.
3.
.
In this section we make the adjustments necessary in the above proof for the
case X
*
Notice that only tle initial choice of H
F.
First, it was noted that
collection of disks.
longer
is
will be a closed or[entable
Such a surface also admits an involution
o
fixed point set consisting
it
that
true
reg,lar
a
that Downing’s construction of the handlebody H
B.,
I,
i
closed
a
is
this is
surface less a
with a separating
the
of
neighborhood
3-manifold
B.
and
klB
and
accordingly
q
l-skeleton
of
Rather, we must show
can be carried out equivariantly.
X’= X
n be the boundary components of X and let
where M. is the handlebody with M
X’
gi
,
*
simple closed curves and arcs.
triangulation of X will have a homeo,norphic complement in X.
Let
If K
is a ,nlosed, orientable surface.
floweret, F
no longer necessarily true.
No
was depeudent upon X having
The modification procedures do not depend upon assumptions about X.
empty boundary.
mapping B
has
i
classical
a
Uk(UMi)
homeomorphically to 8M
decomposition
i
into
H{). By Downing [6; Lemma I], each
X’ onto
where
j
1,2, are specially
Nij=
Ni
(M.
and N are respectively, just regular neighborhoods of
positioned in H and H
2.
the l-dimensional sets denoted Y and Y in Downlng’s proof.)
It is then shown
-(2N
that X
X’
and that
H
and
are homeomorphic
H{
homeomorphic handlebodies H
and
Ni H,
is isotoped by G in
Mi
H!
Ui Mi
cl((X’
handlebodies with H
2
The
that
f(H
M.
f(N.)
2)
cl(xH
for all i.
f(Hl)).
and
H
We
do
dimensional
fixed
Bi
Let M
h’
point
i
A
X’ such
yields a homeomorphism f:X’
homeomorphic handlebodies such that
that f(H
how.ever,
I)
and f(H
2)
are
h-
may, however, be assumed h-invar[ant as in section 2.
of an involution on
into M
know,
yet
not
involt[on
constructed to commute with h’, then
to
-Ni2
2
-UNi)
i
(i) h can be extended to an
To extend h
H2
HI).
moving each
Mi to N
f(H;) and f(H)_ are
isotopy G of X’
invariant.
to
cl(X
we
need
set)
Mi.
on
’
f(H1)
:K’
and f(H
only know
M.
is
But if
X’ and (ii) if the isotopy G can be
2)
that
will be h-invariant as needed.
any
equivalent
Then attaching
M[
(free or with l-
involution
by
to X by
kl--klB
I where A is a 2-spher minus disks.
kl
i
to
the
restriction
allows h to be extended
If h is a free involution
43
DECOMPOSITION INTO HOMEOMORPHIC HANDLEBODIES
of
BMi,
then
him
given (m,t)
is equivalent to
i_
(s(m),t) where
involution on A derived from the antipodal map on $
2
a is the free
by removing invariant pairs of
disks.
If
has a l-dimensional fixed point set,
hlBM i
In this case
closed curves.
him
i
with arcs as fixed point components.
is equivalent
it
will
only
consist
of simple
where s is an involution of A
to
If the fixed poit set separates, than s may be
2
by renoving invar[ant disks from
derived from the reflection in a great circle of S
the fixed point circle and invariant pairs of disks disjoint from the circle of fixed
points.
Suppose
him i
curves.
Let M.
m: S x S
S x S
loop.
has a nonseparating fixed point set consisting of k simple closed
E
I where A is S
given
himi
Again,
by
x S
(s2, 1)
Y(sl s2)
here
l,e Involution
minus disks,
s
has
one
is derived from y by
fixed
nonseparating
point
removing k invariant disks from
the nonseparatlng fixed point and an appropriate number of invar[ant pairs of disks
disjoint
rom
the fixed point set.
Let A x [-I,I] be embedded in X’ by j.such that
locally defined product structure.
[-1,3]
extended to an embedding, also called j, of A x
h’lJ(A
j(A x {I})
h’lj(A
x [-I,]) preserves the
Using an invar[ant collar C on B
3])
B.i and so that
x [-I
Tollefson [7; Thin. B],
where :
Bix
(b,t)
Since J(A x {I}) is
B. ,j(A x [I 3]) is an invariant subspace of
h’IC
((h’IBi)(b),t).
into
X’, such
fiber
is
[-1,3]
+ B
C with
h’lj(A
in X,
can be
that
By Kim and
preserving.
x [-1,3] is given by
i
an invarlant subspace of
x [I 3]) fiber preserving.
Replacing T x [-1,3] in the proof of Downing [6; lemma I] by A x [-1,3], the isotopy
G
t
with support in
J(A x [-1,3]), taking Mi
to N
i
may be assumed to commute with h’.
That is, we may assume that G
preserves the fiber structure on J(A x [-1,3]).
t
In the case where the boundary component B is not h-invariant, such care in
i
choosing the isotopy G is unnecessary. Any isotopy G
with support in a small
neighborhood of M.
i
4.
t
will suffice as long as h’G
t
t
is applied near
Mj
hM
i
PROOF OF THEOREM 2.
In this section we show that the equivalence
That is, we show that the homeomorphism
assumed to be the identity on H
I-
of Nelson [4, Theorem B] and Nelson
f:Hl+
H
hl}{i~ hlH 2
is in some sense natural.
H establishing th[s equivalence may be
2
8H
2. This
[3, Theorem 4.1].
we do by modifying the proofs
The proof of the equivalence in the above referenced papers is by induction on
the common genus of the handlebodles H
(hlHi)-invarlant, nonseparating,
D. lowers the genera of
along the
H’. Hi
the
N(Di )’
induction
i
and H
properly
Nelson [4; Lemma 2.1] provides
2
embedded disks D c H. such that cutting
i
F(hlHi)
i
I, 2. Let
I, 2, be the handlebodies resulting from cuttlng along D
i
hypothesis,
equivalences be established by
the fixed point sets
hlH hlH
fl :HI
and
a nd
hlN(D I) hlN(D2).
f2:N(D + N(D2). By the
also
Let
By
these
homogeneity of
44
R.B. NELSON
the manifolds
(Hi)*
fm(HN N(D2)).
N(DI)*
and
equivalence
The
hlHl1 hlH2
reconstructed from the homeomorphism
fl
established
is
f2
and
lHl
so that
In order to modify the above argument so that f
DlC H
need only to be able to choose
and
this proof consists in showing that D
Consider the disk
[4 ;Theorem B].
Nelson
p(D.)1
Then
arcs,
if
D.i
is
where
and
a
disk
aic
p(F(h
Hi)
can
if
B relative
2
H
embedded
be
in
icP(Hi).
2
isotoped
onto
p(H 2).
We
to
DI-
P(Di)
D2.
find
which
is
we
The remainder of
i
U
,
to
such D
p(F(hlH2))
and D
2
as provided by
space
orbit
the
be isotoped onto D
relative
i
can
2
2
2
can be so chosen.
H
D can
H
is the identity on H N H
F(hlH i)
and
onto
D2c
and D
N(DI)
f:H[+
by
2
properly embedded in H i and cutting
Let p denote the projection onto the
properly
only
isotoped
(H N
I, 2, one ny assume that
i
by
un[on
relar, lve
and
2
the
H
of
two
to
can
F
be
following
procedure.
as a fixed loop on H
and regard D
Fix D
D2cH
find
If
and
2.
2
2
2
respectively, then we are done.
H
--cI(H
2
on H
N(D2))
2
2’ .,Note
lower
of
Employ Nelson [4; Theorem B] to
p(F(hlH2))onto and
can be isotoped relative to
If not, cut H
along D 2 to get a handlebody
2
D may
genus.
still
be
that with this cut the bordered surfaces
regarded
p(FNH)
and
as
a
fixed
p(H)
loop
which make
and
up (H ’) have lower genera than p(F N
respectively. Now apply Nelson
2
Theorem
B] again to H taking additional care that D misses the "s cars" of
[4,
2
2
previous cuts (so that D may be regarded as properly embedded n H ). This can be
2
2
done because of the homogeneity of
Repetition o this procedure must
p(H2)
)
(H)*.
eventually yeild the desired disks D
with both
p(H;)
and p(F
and D
2
since the procedure eventually terminates
H 2) being disks.
REFERENCES
I.
ROURKE,
2.
BING,
C.P. and
SANDERSON,
B .J., Introduction
Piecewlse-Linear
to
Topology,
Springer-Verlag, Berlin, 1972.
R.H., An Alternative Proof that 3-Manifolds can be Triangulated. Ann. of
Math 69(1959), No. 2, 37-65.
3.
NELSON,
R.B., Some
Fiber
Preserving
Involutlons
of
Orieatable
3-dimenslonal
Handlebodles, Houston J. of Math. 9(1983), No. 2, 255-269.
4.
5.
6.
7.
NELSON, R.B., A Unique Decomposition of involutions of Handlebodies, Proc. Amer.
Math. Sot., 93(1985), No. 2, 358-362.
SEIFERT, H. and THRELFALL, W., Lehrbuch der ToP01oies Chelsea Publishing
Company, New York, 1947.
DOWNING, J.S., Decomposing Compact 3-manlfolds into Homeomorphic Handlebodies,
Proc. Amer. Math. Soc. 24(1970), 241-24.
KIM, P.K. and TOLLEFSON, J.L., PL Involutions of fibered 3-manifolds, Trans.
Amer. Math. Soc. 232 (1977), 221-237.