Product involutions with 0 or 1 dimensional fixed point sets on orientable handlebodies

VOL. 17 NO. 3 (1994) 457-462

PRODUCT INVOLUTIONS WITH

0

OR 1-DIMENSIONAL

FIXED POINT

SETS

ON ORIENTABLE

HANDLEBODIES

ROGER B. NELSON

Department

of Mathematical Sciences Ball

State

University, Muncie,

IN

47306-0590

(Received August 20, 1992 and in revised form July 19, 1993)

ABSTRACT.

Let hbeaninvolutionwitha0or1-dimensionalfixed pointseton anorientable

han-dlebody

M. We

show that obvious necessary conditions forfibering

M

as

A

x

I

sothat h r x

"

with r an involutionof

A

and r reflection about themidpoint of l also turn out to be sufficienl.

We

alsoshow that sucha "product" involution isdeterminedbyitsfixedpoint set.

KEY

WORDS AND

PHRASES.

Orientablehandlebody, involution,invariant disk,fixed point set. 1992

AMS

SUBJECT CLASSIFICATION CODES.

Primary57S17, 57S25.

1.

INTRODUCTION.

Kim and Tollefson

[1]

showedthat if

A

isacompactsurfaceand hisa

PL

involution of

A

x

I

where

h(a

x

01)=

A

x

OI

(I

[-1,

1]),

then thereexists aninvolution9of surface

a

such that his equivalentto the involutionh’ of

a

x

I

given by

h’(x,t)

(9(z),

(t))

for

(z,t)

E

a

x

I

and

(t)

or -t. Thatis, hisa

product

involution. When

A

isaclosedsurface the conditionalways

holds;

allinvolutionsarethen productinvolutions. When

A

has boundary,

A

x

I

is ahandlebody.

For

ahandlebody

M,

onenaturally asks forwhich involutionsafiberingexistssothatthe condition

holds,

or equivalently,which involutionsareproducts.

The question is answered by Nelson

[3]

for involutions of orientable handlebodies with 2-dimensional fixed point sets.

In

thatcase, obvious necessary conditionsare shown to be sufficient

as well.

In

this paper, we show that such is also the casefor orientable handlebodies carrying involutions with 0 or 1-dimensional fixed point sets.

In

this setting,weshow that clearnecessary conditions are also sufficienttoguarantee that aninvolution h ofan orientable

handlebody M

is equivalenttorxron

A

x

I,

whererisaninvolutionof

A

and

r(t)

-t.

We

alsoshow that these involutionsare

completely

determined

by

their fixedpoint sets.

In

this paperallspacesandmapsarepiecewiselinear

(PL)

inthesenseof Rourke and Sanderson

[5].

Notation generally follows

[5, 3].

An

involution h on

X

is a period two homeomorphism h"

X

X.

The orbit space of involutionh

isX,

X/

x wherex c yifand only ify x or

h(x);

theprojectionmap isdenoted byph. Thefixed point setof his

Fzx(h)

{x

c= Xlh(x

x}.

existsahomeomorphism

f

X

.’

suchthat h

f-1

oh’o

f.

Recallthat ahandlebodyof

nisaspace obtainedby choosing2ndisjoint disks in theboundary ofa3-ball andidentify,g in pairsby n

PL

homeomorphisms.

Two

handlebodies are homeomorphicif and onlyiflhey

thesamegenus andareboth orientableorboth nonorientable.

2.

TECHNICAL LEMMA.

Ifhistobeaproductinvolutionof formrx r, then certainly

Fzx(h)

iscontained in

A

x

{0}.

Accordingly,wesay that asubset

S

of

M

isA-horizontal in

M

if

M

A

x

I

and

S

canbe isotope(t

into

A

x

{0}.

Ofcourse, not all systems of pointsor arcs and loops canbe the fixed point s(’! of

someinvolution ronA. Ifasubset of

M

isisomorphic to the fixed point set ofsomeinvolutio1

A,

we say it is A-fixable.

Note

thatallinvolutionsofacampact borderedsurface

A (and

hence all A-fixable

sets)

areeasilylisted

[3].

Clearly,if

H

isequivalentto rx r, then

Fx(h)

must be both A-horizontalandA-fixable.

Constructinga

fibering

for

handlebody M

as

A

x

I,

with respectto which which involutionh isaproduct,isaninductive process.

We

decompose

M

into"smaller" h-invariant pieces whichmay

beassumed,by induction hypothesis,to carrysuchastructure. Thefiberingof

M

isreconstructed from thefiberingson thepieces. The following lemma enables us to carry out the crucial stepof

breaking

M

intoh-invariant pieces.

Fixedpointsetsofinvolutionsoncompactsurfaces have theproperty thatonecanalwaysraise

theEulercharacteristicofthesurfaceby cutting

along

an arc which either misses the fixed point. setorcrosses afixed point loopinexactlyonepoint. If hisaninvolutionon

handlebody

M,

then

Fix(h)

being both A-horizontal and A-fixableclearly implies that the genus of

M

canbelowered by cutting

along

a

properly

embeddeddiskmissing

Fix(h)

orcuttingafixedpoint loopinexactly one point. The following lemma shows that this can be accomplished equivariantly. Following Theorem 1,it will alsobeclear that thiscutting property implies that

Fix(h)

isbothA-horizontal and A-fixable.

LEMMA

1.

Let

M

be an orientable handlebody with involution h having either 0 or 1-dimensional fixed point set

Fix(h).

Suppose

also that

Fix(h)

is both

A-parallel

and A-fixable.

Then thereexistsa

properly

embedded

non-boundary-parallel

disk

D

suchthat either

D

orelse

D

is h-invariantand cutsafixed point

loop

ina

single

point.

PROOF:

A

disk

D

in

M

is in

h-general

position modulo

Fix(h)

if

D

and hD are both in

general

position withrespectto

Fix(h)

andif

D- Fix(h)

andhD-

Fix(h)

arein

general

position.

We

alwayspresume that a disk

D

is in

h-general

position and alsothat

(D- Fix(h))V

OM

and

(hD

F,x(h))

V10Marein generalposition.

We

may,asnoted

above,

assumethat thereexists properly embedded

non-boundary-parallel

disk

D

in

h-general

position.

Although D

f3hD is

formally

a

subgraph

ofthe 1-skeleton of thetriangulationof

M,

wewill

notneed to consider

D

f3hDwith this

degree

of detail.

Instead,

we

regard D

f3hD has consistingof

A-objectswhere

(a)

a

A-loop g

isa

cycle

of

edges

formingasimpleclosedcurve componentor such

We partition theA-objectsinto

(i)

loops,

(ii)

arcs withboth endpoints in

Fix(h)- OM,

(iii) arcswith neitherendpointin

Fiz(h)UOM,

(iv)

arcs withboth endpointsin

OM

and

(v)

arcswith

oneendpoint in

OM

andthe other in

Fix(h).

Note

thatarcs of type

(ii)

do not actuallyoccur.

A

A-object is called extremalon

D (or hD)

if it either bounds or cobounds with other A-objects a

disk

E

on

D

(or

hD)

theinteriorofwhich intersectsnoother A-objects.

This proof requires the

repeated

"simplification" of a disk

D

by removal of A-objects from

D

f3hD.

Although

removal ofaA-object can

always

be accomplishedwithoutaddingany pointsto

DVlhD,

thenatureofthecomponentsofDnhD maybe

changed

sothataA-loop iscreated by the removalofa

A-arc. Accordingly,

define the complexity ofadisk

D

by

c(D)

(a, l),

wherea isthe number of

A-arcs

and the number of

A-loops

in

D

flhD. The ordered pairsarelexicographically orderedsothat removal ofa

A-arc

always lowers the

complexity.

Let E

be the collectionof non-boundary-parallel disks

properly

embeddedin

M

and missing fixed pointarcs. Choose

D

E

E

such that

DfqFiz(h)

is minimal. The

hypotheses

regarding

Fi:r(h)

make this intersection contain at mostonepoint. Thereplacementsof

D

bya "simplified"

D’

E

E

will not addto this intersection.

Ifanextremal A-objecton hDisof

type

(i)

or

(iii),

the the objecteither

bounds,

orcobounds

withother

A-arcs,

adisk

E

inthe interiorofhD. Thesecases arehandledin

Kwun

and Tolefson

[2].

D

is isotopic to disk

D"

E

where

c(D")

<

c(D)

by the procedureof

[2,

Lemma

1].

Theonly remaining A-objects,oftypes

(iv)

and

(v),

aretreatedbelow.

Iftheextremal A-objecton hDisatype

(iv)

arca,then aU

/’,

fl’

anarcinhD

V10M,bounds

adisk

E’

on hD.

Also,

aU

,

anarcon

DVIOM,

boundsadisk

E

on

D,

where

E

hE’.

Suppose

U

/’

boundsadisk

E"

on

OM.

Then

E

U

E’U

E"

bounds a3-cell

C

withface

E"

in

OM

suchthat

N(C)

f3

hN(C)

,

where

N

denote asmall regular neighborhood.

Let

Gt

bean isoptoy of

M,

withsupport in

N(C),

whichmoves

E through

C’,

/across

E"

and

E

offof

E’. Let

D’

GI(D)

andthen

c(D’)

<

c(D).

If

/U/’

does not bound a disk on

OM,

then

D’

E

U

E’

is a non

boundary-parallel

disk

properly embeddedin

M

suchthat

D’

Cl

hD’

t.

To

this pointwehavedemonstrated that we canfind

D

E

such thateither

D

tqhD

I

or else

D

ClhD consistsentirely of

type

(v)

arcs.

We

need

only

show that from the lattercasewe can constructan h-invariant disk in Pcontaining exactlyonefixed point.

Suppose

DfqhDconsistsonlyoftype

(v)

arcs. Since

D

was_{originally chosen}sothat

Df3Fiz(h)

wasminimal, andsincethe above simplifications of

D

addno newpoints to

D

f’l

hD,

allofthe type

(v)

arcsemitfroma

single

fixedpoint.

D

istherefore partitioned into

"wedge shaped"

regions

E,

bounded

by type

(v)

arcsa,_ and a, andarc

fl,

on

OD.

Similarly, regions

E

on hDarebounded bya,_, a,, and

fl,

where

fl

COhD.

Definean equivalencerelationon

{E,}

by

E,

E

iff

E

hE,. Eachequivalenceclass has twomembers.

For

each

class,

replaceeither

E,

by

E

or

E

by

E.

The resultisanh-invariant disk

D

containing

exactly

onefixed point.

We

neednowonlynotethat

D

isnot

boundary-parallel.

If

itwas

boundary-paralled,

itwould coboundaninvariantballwithanh-invariantdiskon

OM.

This disk contains afixedpoint.

Bu

this is

clearly

not possible unless

D

cuts afixedpointarcinstead

3.

MAIN THEOREM.

THEOREM

1.

Let

M

beanorientable

handlebody

with involutionh having0or1-dimensional fixedpoint set

Fix(h). Suppose

alsothat

Fix(h)

is B-horizontaland B-fixable where

M

%

B

1.

Then thereexistacompact, bordered surface

A

such that

M

A

I

and h isequivalent tor r.

wherer isanonidentityinvolutionof

A

and

r(t)

-tforall E

I

[-1, 1].

PROOF: Let

D

be the disk provided

by Lemma

1. Either

D

hDor

D

3hD

.

In

either case, let

M’

M-

N(D)- N(hD)

where

N(D)

and

N(hD)

aresmall regular neighborhoods of

D

and hDchosen to coincide

(be h-invariant)

when

D

hDand tobedisjointwhen

D

3hD

.

Either

M’

is a connected

handlebody

with induced involution h’ or else

M’

M_I

t3

M1

where

M,

is invariant with induced involution h,or

h(M,)

M_,.

In

the case where

M’

isconnected,

g(M’) < g(M),

otherwise

g(M,) < g(M),

-1,1,

N(D)

’

D

[-1,

1]

andwedenote

by

D,

the

disk

corresponding

to

D

{z}

on

OM,,

OM’,

or

ON(D).

Thedisks

(hD),

aresimilarly identified. The proof now follows by inductionon thegenus

g(M). We

first handle independently the caseswhere the desiredproduct structure cannotbeassumedbyinductionhypothesis. Thisis the

casewhen

M

isa3-ball

B

with involutionh such that

Fix(h)

#

orwhen

g(M) >

andhisfree.

When

M

-

B

z,

itis well known that hisequivalenttoeitherreflection

through

the center point orrevolutionabout acentralaxis.

In

eithercase

M

B

I

and h

3

r where

fl

is reflection

through

the center point or acentralaxison

B

2.

Przytycki

[4]

classifiesallfree Z,-actionson handlebodies.

By

Theorem of that paperthere

isat mostoneorientationpreservingandoneorientationreversing freeinvolution up toequivalence on anyhandlebody

M

with g

g(M)

>

1.

Furthermore,

theseoccur onorientablehandlebodies

only when thegenus isodd.

Hence,

we needonly presentan example of eachin factored form to

completethis first step.

In

bothcases we viewthehandlebodyofgenusg as

A

I

where

A

isthe2-sphere

S

less

29

disjoint disks and h 7" r.

In

the orientationpreservingcase,r isinducedby theantipodalmap on

S

when g invariantpairsof disksareremoved toform

A.

In

the orientationreversing case,

-

is inducedbythe involutionon

S

with two fixedpoints

(i.e.

revolutionabout acentralaxis piercing thefixed

points)

whenaninvariant diskabout eachfixedpoint andg- invariant pairsofdiskare removedtoform

A.

The inductionstep consists in cutting

along

invariant

D

or

along

D

t3

hD,

that is,

deleting

N(D)

U

N(hD)

from

M,

assuming a product structure on each piece by induction hypothesis, modifying the

product

structures tobeconsistentwitheach other and then reassembling the pieces.

We

consider twocases.

Case

1: Thedisk

D

provided

by Lemma

ish-invariant.

By

inductionhypothesis,

N(D)

and

M’

or

M_I

and

M1

areeach endowed with a

product

structuresuch that

hIN(D

and h’ or h_l and

hi

areeach equal tosome r r. The difficultyisthatthe

"scars",

i.e. the images of the

D,

on these pieces, arenot necessarilyconsistentwith the

fiberings. It

isnot immediately thecasethat

D,

a

I

whereaissome arc on

OA

andthe pieceofinterestfibers as

A

I. We

show thatthe

fibering

maybesochosen

by

deforming the original fiberingto meet this condition.

Since

D

is h-invariant,we may assume that

Fix(h)

is 1-dimensional. Then each

D,

is also

pt,,M’

M". Note

that

ph,(D,)

isadiskon

OM"

containingpoint

p,,(z,)

initsinterior.

Now M’

factors as

A’

I

with h’ havingafixed point arcterminatingin z,.

Let

abean invariantarc on

OA’

containingx,. Thereis anisotopyon

O(M’*)

with support inasmall

regular

neighborhood of disk

p,,(a

I)

U

p,(D,)

moving

p,,(a

I)

onto

p,(D,)

and constant on

pt,,(x,).

Thisextends to

an isotopy

K’

of

M’*

with support in acollar

(Rourke

and Sanderson

[5,

Theorem

2.26])

of the

boundary. This isotopy may beassumedconstanton

pt,,(Fix(h’)).

Hence,

it lifts to anisotopy

Kt

of

M’

movinga

I

onto

D,

while fixing all points of

Fix(h’).

We

deform thefibering of

M’

by

K

-1 _sothatwe

mayassumethat

D,

a

I.

Thesamedeformationcanalso be carried outon

M,.

This same argument is now applied in a neighborhood of

D,

in

N(D)

with the additional requirement that if

N(D)

isfiberedas as

B

I

and

13

isaninvariantarc on

OB

containingx,, then

(plN(D))(3)

is initially isotopedintoapositioncorrespondingwiththeadjusted position of

p,(a)

on

p,,(M’).

Reassemblynowgives

M

A

I

where

A

isconstructed from thecompact bordered surfaces

B

and

A’

or

A,,

-1, 1, byidentificationalong arcs onthe boundary.

Hence

A

is alsoa compactborderedsurface.

Clearly,

h r v whererisconstructed from

a’

ora,, -1, 1,and

theinduced involutionof

B.

Case

:

DC3hD where

D

isthediskprovided

by Lemma

1.

In

thiscasewenotethat

N(D)

and

N(hD)

aredisjoint form

Fix(h).

The

argement

isbasically thesame asfor case except that

one need not take special measures on

Fix(h).

Since

p,(U(D))

ph(U(hD))

in the orbit space

M*,

the isotopy

K’

withsupport in

N(D,)

lifts to isotopy

Kt

with support in

N(D,)tO N(hD,),

-1, and commutes with h. []

4.

UNIQUENESS.

Any

productinvolution with 0-dimensionalor 1-dimensional fixed point set on an orientable

handlebody may be routinely

generated

from an involution with identical fixed point set on an orientable,

compact

bordered surface. Since classifying the involutions on compact surfaces is

an easy

matter,

all product involutions with 0-dimensionalor 1-dimensioanl fixed point sets on

orientable handlebodiescan nowbeeasily

generated.

Our

problem is that the

generated

list might contain many redundancies.

For

instance, if

M

is the orientable

handlebody

withgenus

three,

aninvolution with

two

fixedpoint arcsmay be

generated

inat least the following threeways:

(i)

T1 ron

Ax

I

where

A1

isthe

2-sphere S

less fourdisks. Involution’x isconstructed from

the involutionon

S

withasinglesimple closedcurve asfixed point setby removingtwoinvariant

disksintersectingthis fixed point

loop

andaninvariant pair of disksdisjoint from thefixed point set.

(ii) r

r on

A

I

where

A

is the torus

T

less two disks. Involution r2 is constructed

from s x

idsa

on

T

$1 x

Sx,

where sis complex conjugation, by the removal ofinvariant disks

intersectingeachof thetwo fixedpoint

loops.

(iii)

rax

ron

Aa

x

I

where

Aa

isagain the torus

T

less twodisks. Involution

ra

isconstructed

form the involutionv on

T

$1

Sx,

where

v((x,y))

(y,x),

byremovalof two invariant disks

from the singlefixedpoint

loop

ofv.

the 0-dimensionalcase.

In

either case,weseeksufficientconditionstoconcludethattwoinvoluti(,,s areequivalent. The most obvious necessary condition turns out tobesufficient.

THEOREM

2.

Let h

and9 be

product

involutionswith 0-dimensionalor 1-dimensionalfixed

point setsonanorientable

handlebody M.

If

Fix(h)

-

Fiz(9),

then h _9.

PROOF:

Thisproofisbyinduction the thegenusof

M.

Again,wehandleindependently h()se cases which cannot be assumed by induction hypotheses.

Note

that the involutions h and 9 are bothorientationpreservingwhen their fixed point sets are1-dimensionalandorientation reversing when that have 0-dimensional fixed point sets.

Hence,

the aforementionedwork of Przytycki

[4]

coversthecaseof free involutionsonhandlebodies withnonzerogenus. Theresult is wellknow for involutionsofa

3-ball.

In

general,

in ordertoshow h _{9 we}find ahomeomorphism

f

M,

M;

betweenorbit

spaces which can be lifted toa homeomorphism

f"

M

M

such that

pgf

f*Ph.

Observing

that

ph(F,z(h))

and

pg(Fiz(g))

arethebranchpoints of branched coverings,wenote that

f"

can

be lifted if

f*ph(Fiz(h))

pa(Fiz(9)).

We

nowconsider the inductionstep. Sincehand₉areboth of the formr r wecan,without

using

Lemma

1, find invariantnonseparating disks

Dn

and

Du,

intersecting

componects

of

F,z(h)

and

Fiz(9)

of thesame

type.

M[

M-

N(Dh)

and

M

M-

N(D)

have induced involutions

h’ and

9’

and

9(M)

9(M)

<

9(M).

By

induction hypothesis, thereexist homeomorphisms k’’

M

-

M

and k*"

M

M,"

such that

pu,

k’=

k*ph,. We

mayassume that

k*ph,(Dh

{i}),k*ph,(Oq))

(pg,(Dg

{i}),pg,(cq)),

i=-1,1, where

N(D_)

corresponds

to

D_

[-1,

1].

Now

(phlN(Dh))(Dh

{i})

and

(paIN(Da))(Da

{i})

arefaces ofthe 3-cells

ph(N(Dh))

and

p(N(Dg)),

respectively. The liftable isomorphism k*

M’,

M,*

induces an isomorphism

]c

between these pairs of faces.

We

notethat

]c

extendstoahomeomorphism

]c’N(Dh)*

N(D)’,

suchfixedpoint imagesaretakento fixedpoint images and

Now

let

f*

be

:(ph,(N(Dh)

0

(A

{0})))

ps(N(Da)

91

(A

x

{0})).

(k*

tO

])"

M*,

U

N(Dh)*

M7

U

N(D)*

taking

ph(Fix(h))

to

p(Fix(g)).

Then

f"

lifts to

f"

M

M,

sothat h g.

REFERENCES

1.

KIM,

P.

K.

and

TOLLEFSON,

J. L.

PL

involutionsoffibered

3-manifolds, Trans.

Amer.

Math.

Soc.

232

(1977),

221-237.

2.

KWUN,

K. W.

and

TOLLEFSON,

J.

L. Extending

a

PL

involutionoftheinteriorofacompact

manifold, Am.

J.

Math. 99

(1977),

No. 5,

995-1001.

3.

NELSON,

R. B. Some

fiber preservinginvolutions of orientable 3-dimensional handlebodies,

Houston J.

Math9

(1983),

255-269.

4.

PRZTYCKI,

JOZEF H. Free

Actionsof

Zn

onhandlebodiesand

surfaces,

Bulleti______n

d__e

L’Academie PolonaisedesSciences,Serie dessciejcesmaths, ast____r_r, e_tp_hy_. 26

(1978),

No.

7,

617-624. 5.

ROURKE,

C. P.

and

SANDERSON,

B.J.

Introductionto PiecewisLinea_______r

Topology,