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 anorientablehan-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 sufficientas 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 onlyiflheythesamegenus 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(tinto

### A

x### {0}.

Ofcourse, not all systems of pointsor arcs and loops canbe the fixed point s(’! ofsomeinvolution 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 whichmaybeassumed,by induction hypothesis,to carrysuchastructure. Thefiberingof

### M

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

### 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,

wewillnotneed to consider

### D

f3hDwith this### degree

of detail.### Instead,

we### regard D

f3hD has consistingofA-objectswhere

### (a)

a### A-loop g

isa### cycle

of### edges

formingasimpleclosedcurve componentor suchWe 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)

arcswithoneendpoint 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 adisk

### 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,

orcoboundswithother

### 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

diskproperly 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.

Ifitwas

### boundary-paralled,

itwould coboundaninvariantballwithanh-invariantdiskon### OM.

This disk contains afixedpoint.### Bu

this is### clearly

not possible unless### D

cuts afixedpointarcinstead3.

### 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,

thedisk

### 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 thecasewhen

### 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 paperthereisat mostoneorientationpreservingandoneorientationreversing freeinvolution up toequivalence on anyhandlebody

### M

with g### g(M)

### >

1.### Furthermore,

theseoccur onorientablehandlebodiesonly when thegenus isodd.

### Hence,

we needonly presentan example of eachin factored form tocompletethis 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 toan isotopy

_{K’}

of### M’*

with support in acollar### (Rourke

and Sanderson### [5,

Theorem### 2.26])

of theboundary. 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_{so}

_{that}

_{we}

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,andtheinduced 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 thatone 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 orientablehandlebody 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 isan easy

### matter,

all product involutions with 0-dimensionalor 1-dimensioanl fixed point sets onorientable 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 fromthe involutionon

### S

withasinglesimple closedcurve asfixed point setby removingtwoinvariantdisksintersectingthis 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 constructedfrom s x

_{idsa}

on ### T

$1 x### Sx,

where sis complex conjugation, by the removal ofinvariant disksintersectingeachof thetwo fixedpoint

### loops.

### (iii)

### rax

ron_{Aa}

x### I

where_{Aa}

isagain the torus### T

less twodisks. Involution### ra

isconstructedform the involutionv on

### T

$1### Sx,

where### v((x,y))

### (y,x),

byremovalof two invariant disksfrom 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-dimensionalfixedpoint 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;

betweenorbitspaces which can be lifted toa homeomorphism

_{f"}

### M

### M

such that### pgf

### f*Ph.

Observingthat

### 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. Sincehand_{9}areboth 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 involutionsh’ 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.