FIBRATORS THAT ARE CLOSED UNDER FINITE PRODUCT

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 4 (1998) 815-818

815

RESEARCHNOTES

CODIMENSlON2

FIBRATORS THAT ARE CLOSED UNDER FINITE PRODUCT

YOUNG HO IM

Department

of Mathematics PusanNationalUniversity

Pusan609-735,KOREA MEEKWANGKANG

Department

ofMathematics DongeuiUniversity Pusan614-714,KOREA

KI MUNWOO

Department

of Mathematics PusanNationalUniversity

Pusan609-735,KOREA

(ReceivedMarch 14, 1996 andinrevised formJune 5,

1996)

ABSTRACT. Inthispaper, weshow that if

N

is aclosedmanifold withhyperhopfian fundamental group,

7r,(N)

0 for 1

< _<

n and S is asimply connected manifold, then N S satisfies the propertythat allproper, surjectivemapsfromanorientable

(n +

2)-manifold M^to a2-manifoldBfor which each

p-1 (b)

ishomotopy equivalenttoN x S"necessarilyareapproximate fibrations.

KEY WORDS ANDPHRASES: Approximate fibration, Hopfiangroup, hyperhopfian group, Hopfian manifold.

1991AMS SUBJECTCLASSICATION CODES: 57N15;55R65.

1. INTRODUCTION

In thestudy ofproper mapsbetween manifolds, the concepts of approximatefibrationsplay very important roles becauseithasniceanduseful properties justas Hurewicz fibrations. Asageneralizatio, ofHurewiczfibrations and cell-likemaps,the concept ofanapproximatefibration isinitiallyintroduced by CoramandDuvall

[I ].

A

propermapp"

M

^---,

B

betweenlocallycompactANR’siscalled

a

approximate

fibranon

^ifit

has the followingapproximatehomotopy lifting property for allspaces: givenanopencovereof

B,

an arbitraryspace

X,

andtwomapsg X^---,

M

andXxI^--,b such thatp^og F0, there exists amap G XxI Msuch that

Go

^{gandpoGise-closetoF.}

Ifa proper map

p:M B

is an approximate fibration, then the point inverses are homotopy equivalenttoeachother. Naturally,the question arises astounder whatconditionsthe converse ofthis fact holds.

Many

peoplehave been interestedin asetting which forces anyproper mapdefinedon an arbitrary manifold ofaspecifieddimension tobeanapproximatefibrationmerelyduetothefact thatall point preimages are copies of some closed manifoldN. Inthispaper,wemainlyconcentrate on such a closed n-manifold when thedimensionof

M

isparticularlyⁿ

+

^2.

Weassume allspacesarelocallycompact,metrizableANR’s,andall manifolds are finitedimensional, orientable, connected and boundaryless.

A

manifold

M

is said tobe closedif

M

iscompact, connected andboundaryless.

A

closed n-manifold

N

is calleda codimension

2fibrator

if, wheneverp

M B

is aproper mapfromanarbitrary

(n + 2)-manifold M

^{to a2-manifold}

B

such that eachpoint preimage homotopy equivalentto

N,

p

M

^---,

B

is anapproximate fibration.

Thedegree ofamap

R

N^---,

N,

whereNis aclosed manifold,isthe nonnegative integer^dsuch thattheinducedendomorphism of

H, (N; Z) Z

amountstomultiplicationbyd, uptosign Notethat

816 Y.H.IM,M.K. KANO AND K.M. WOO

adegree one map/i"N^-,Ninduceshomology isomorphisms

R. Hz(N)

^-,

Hi(N)

for all integers

_>

0. Thecontinmty setCofp"

M

^--,

B

consists of those pointsc E

B

suchthat,underanyretraction /i

p-lU

^-,

p-lc

defined over aneighborhood UC

B

of c, chasanother neighborhood

V

^{C U}

sch

that

RIp-b p-lb

^--,

p-lc

isadegreeonemapfor allb E

V.

The continuitysetCisthemaximalopen subset of

B

overwhichthe nth cohomology sheaf of themap p islocally constant. In [2], several characterizationsforanapproximate fibration were described. Asamatteroffact, theessential point whether or not a proper map is an approximate fibration depends on the fact that any retraction /i"

p-U

^--,

p-lb

restricts tohomotopyequivalences

p-lc

^--,

p-b

for all c sufficiently closeto b

[2].

Thefollowingtermsefficientlyaid to convertahomologyequivalence intoahomotopyequivalence.

A closed manifold N is called Hopfian ifevery degree one map N N which induces a aulomorphismis ahomotopy equivalence. A group

H

isHopfianifeveryepimorphism

H

His necessarilyanisomorphism,while afinitelypresented group

H

ishyperhopfianifevery endomorphism

H

^---+

H

with

(H)

normal and

H/(H)

cyclicisan automorphism. Ahyperhopfiangroupimplies aHopfianonebut the converse doesnothold.

Daverman

([3],[4])

^hasshown that each ofasimplyconnected closed manifold and anaspherical closedmanifold withhyperhopfian fundamentalgroupis acodimension2 fibrator. Theproblemwhether the classof codimension2 fibrators isclosed underfiniteproductis notyet settled. Aparticular class of allclosed surfaceswithnegative Eulercharacteristics isclosed,which is claimedby Im

([5]).

Alsowe extended thisresultto the extentthat any productofann-sphere

S"(r > 1)

^and a finiteproduct closed surfacesof genusatleast2 isacodimension 2fibrator

([6]).

The purpose ofthis paper is to extend the above results so that any finite product ofa simply connected closed manifold S" and a closed manifold N with hyperhopfian fundamentalgroup and r,

(N)

^0forI

< _<

^ris a codimension2 fibrator.

2. PRELIMINARIES

The investigation about codimension 2 fibrators comparedwith other codimensions is easierto

approachduetothefollowingresult.

LEMMA

2.1

[6].

If

G

isanuppersemicontinuousdecompositionofan

( +

2)-manifold

M

into closed n-manifolds, then thedecomposition space

B( M/G)

^{isa2-manifold}and

D B\C

^islocally finitein

B,

whereCrepresents the continuitysetof the decompositionmapp"M^--,

B

Becauseofthelocalfinitenessof

D,

we canlocalizetheproblemtothatof anopendisk

B,

proved that p is an approximate fibration over the continuity set

C,

so that p"

M,

^B is an approximate fibrationover

B

bfor someb B. Insuch a case, the homotopy^exactsequenceforp over

B\b

can be reducedto ashortexactsequenceasfollows:

0

7r2(B b)

^--,7rl

(p-1 (c))

^--,7rl

(M p-i(b))

^--,

I(B b) _ Z

^--,0,

wherec is any point of

B\b. Hence,

ifeach preimage ofphas the samehomotopy typeas

N, me

hyperhopfianproperty of

r (N)

makes the continuitysetC^thewholesetB. And so, the hyperhopfian property is very valuable on discussingcodimension2fibrators.

Westatethe established results aboutcodimension 2 fibrators.

THEOREM 2.2

[4].

All closed, Hopfian manifolds^with hyperhopfian fundamental group is a codimension 2 fibrator.

THEOREM2.3

[3]. Every

simply connected closed manifoldisa codimension2 fibrator.

Theideaof theproofis toshow thatthecontinuitysetof the decompositionmapp

M

^"+2

M/G

istheentire set

M/G.

A closed surface with negative Euler characteristic has a hyperhopfian fundamental group [4]

Therefore, from Theorem2.2and 2.3, the following corollary follows

CODIIVNSION2FIBRATORSTHATARECLOSEDUNDERFINITE PRODUCT 817

COROLLARY2.4. Everyclosed surface

F

with nonzeroEuler characteristic is a codimension 2 fibrator.

THEOREM2.5

[5]. Any

^finiteproduct N-

F1

^x

F2

^x-..x

F,,

of closed surfaces

F, (i

1,-.., ofgenusatleast2 is a codimension2fibrator.

Sinceann-sphereSr‘isasimply connected closed manifold,eachofthefimdamentalgroupand the firsthomology group ofSr‘

F1

^{x x}

^F,,,

^isisomorphic^toeach of those of

F

^{x x}

F.

^Taking

advantageof the methodintheproof of Theorem2.5andthefact thatSr‘ x

F

^x

^F,,

^{is aHopfian}

manifold,wecan obtainthenextconsequence.

THEOREM 2.6. Afinite product N S" x

F

^x

^Ft,

of n-sphere

S"(n > 1)

andclosed orientablesurfaces

Fi ⁽ⁱ

^1,-.-,

^m)

^withnegative Eulercharacteristics is a codimension 2fibrator.

3. MAIN RESULTS

PROPOSITION 3.1. Let S be a simply connected closed n-manifold and N be aclosed manifold with

r,(N)

--0 for 1

< <

n. If

R"

S N^--,S xN is a degree one map, thenso is pr^o

R

PROOF. Assume^o S

S,

wherethatpr

R

SxSNxN SisSthe first projection andxNis adegreeonemap.SBy

-

^Staking^xNa universal^{isaninclusion}coveting^map.

space

(, 0)

^of

^N, (S

^x

,

^idx

0)is

^{auniversalcoveringspace ofS xN.}

Let

"

^S--,Sx be a continuousmapfor which

(id

^x

O)o

and Sx

’

^{Sx bea}

continuousmap such that

R

^o

(id

^x

0) (id

^x

0)o

by thelitingproperty. Considerthe following commutativediagram

where q isthe projection fromS

.g _omo

_{S. Because of}

_r,(N)

_{0for 1}

_{< <}

n, we obtain that r,

()

^{0 for 1}

^{< <}

n, and then

Hi (/’)

^{0for 1}

^{< <}

^{n. Accordingtothe KtinnethTheorem,}

H,., (S

^x

) H,,(S)

^and

H,.,(S

^x

N)

isisomorphic^tothedirect sumof

,"=oH,.,_,(S)

^(R)

H,(N)

and

,"$0H,,_,_(S). Hi(N).

^{Then it} easily checked that i.oq.

(id

^x

0).

as homomorphisms from

H,,(S

^x

N)

^toitself, and by the diagram chasing,

tL(H,(S))

C

H,(S)

holds whenwe restrict/L to

H.(S) c H.(S

x

N).

Rewrite

Mr,(S

x

N)

^{in a} form {torsion-free} {torsion}. Since is an isomorphism, the restrictionof/L to

{torsion-free}

of

H,.,(S

x

N)

^{is an} isomorphism and induces aninvertible kxk matrixof the following form

K** K2 .Kit:

.K21 .K22 K2k

K K:

^K,k

Here

Kll

is the matrixcorrespondingtothemap/LIH,(S)

H,(S) H,(S)

^and

Ki

is the matrix inducedbythehomomorphism from thei-thdirectsummandtothe j-thdirectsummand oftorsion-free part

ofH.(S

^x

N).

Since

R.(Hr‘(S)) c H(S),

therestriction

Pl{torsion-free}

^of

P

^{doesn send}firstfactor

H(S)

toany direct summand except itself and thusKlj^iszerofor eachj 2,...,k.

Hence,

the isomorphism

/L[{torsion-free}

^{inducesdeCK 4-1 and}

detK +

1, so that pr

oR

oiis adegreeonemap.

COROLLARY3.2. Let

S

beasimply connected closed n-manifold andNbeaclosed aspherical m-manifold. If R: S^xN^--,SxN is a degree one map, then so is pro

R

^o S^---,

S,

where pr S N^-,Sis the firstprojectionand S^--,SxNis an inclusionmap.

818 Y.H.IM,lIK.KANGANDK.M.WOO

COROLLARY3.3. Let S bea simply connectedclosed r=manifoldand

N

bea finiteproductof closed orientable surfaces with nonpositive Euler characteristics. Then for every degree one map

R"

S^PROOF.xN

-

^S’

^Every

^x

^N,

^closedpro

^R

^{orientableoi"S -.}surface except^{Sis adegreeone}2-sphere^map. isaspherical andtheir finiteproductis a closed asphericalmanifold.

REMARK. Inthe above corollary, adegreeone map

R"

$1 x $2 $1 x $2 on aproductof closed simply connected manifolds

S

^{and $2 cannot}guarantee that thedegreeof the restriction

R[S

is one, so that the condition nonpositive Eulercharacteristic cannotbe omitted.

The following theoremisthemainresultin this section.

THEOREM3.4. LetS beaclosed simply connected manifold, andNbeaclosed manifoldwith hyperhopfian fundamentalgroupand r,

(N)

0forI

< <

r. ThenSxNis acodimension2 fibrator.

PROOF. Sincethe fundamental group ofSxNisisomorphictothe hyperhopfiangroup it suffices to show that S N is a Hopfian manifold by means of Theorem 2.2. Assume that /i"SxN^--,SxN is a degree one map. Since a degree one map between compact orientable manifoldsof the same dimension inducesaq-epimorphism

([8]), Re

^"q

(S N)

rl

(S

^x

N)

isan epimorphism^andactuallyitisaq-isomorphismbytheHopfianproperty of_rl

(N).

Toclaimthat

R

is ahomotopyequivalence, letusconsider/

7r,(S

x

N)

^--,

7r,(S

x

N)

for

>

2.

By

Proposition 3.1,thedegree of pro

R

^o S^---,Sisone and sopr.o/Loio

H, (S)

^--,

H, (S)

^{is an} isomorphism for each

>

1. Since S issimplyconnected, we canapplythe Whitehead theorem and obtain that (pro

R

^o

i)#

ri

(S)

--, ri

(S)

is a 7r,-isomorphism for

>

1. Implying the fact that

7ri(S

^x

N) _ 7r,(S)

^x

7r,(N) _ ri(S)

^{for each}

>

n,

P 7ri(S

^x

N)

^--,

r(S

^x

N)

isanisomorphism for each

>

n. Since

R

has the property

R _,

we obtainthat

R

is ahomotopyequivalence. Therefore, SxNisa Hopfian manifold.

COROLLARY

3.5. LetSbeaclosed simply connected manifold, andNbeaclosedasphefical manifold withhyperhopfian fundamental group. ThenSxNisa codimension 2 fibrator.

COROLLARY3.6. Let

,

beaclosed simply connected manifold, and

F

be aclosed surfacewith nonzero Eulercharacteristic. ThenSx

F

isacodimension 2 fibrator.

PROOF. If the Euler characteristic of

F

is negative, itsfundamental

group

ishyperhopfian [4].

Otherwise,

F

is homeomorphic to 2-sphere and then Sx

F

is simply connected. Hence, it is a codimension2 fibrator.

COROLLARY3.7. Let

S

be a closedsimplyconnected manifold, and Nbea finiteproductof closedsurfaceswithnonzero Euler characteristic. Then

’

^{xNis a}codimension2fibrator.

PROOF. If a closed surface

F

hasapositive Euler characteristic, then

F

isa2-sphere. Therefore, wecanrewrite

’

^xNas

^S"

^x

^N’,

^where

^S’

^issimply connected and

N’

isa"productof closed surfaces with negative Euler characteristic. Since

N’

is a closed asphefical manifold with hyperhopfian fundamental group[4], S^xNis a codimension 2fibrator.

ACKNOWLEDGEMENT. ^Thepresentstudies weresupportedbythe BasicScience ResearchInstitute

Program,

MinistryofEducation, 1996, ProjectNo.BSRI-96-1433.

REFERENCES

CORAM,D.andDUVALL,P.,Approximate fibrations,RockyMountainJ.Math. 7(1977),^275-288.

[2] CORAM,D.andDUVAI, P.,Approximate fibrations andamovability conditionformaps,

Pacific

^J.

Math. 72(1977),^41-56.

[3] DAVERMAN, 1LJ., Subrnanifold decompositionsthatmduce approximate fibrations, Top. Appl. 33

(1989),

^173-184.

[4] DAVERMAN, 1LJ., Hyperhopfian groups and approximatefibrations, CompositioMath. 86 (1993), 159-176.

[5]

IM,

Y.H., Decompositions into codimension two submanifolds that induce approximate fibrations, TopologyAppl.56(1994),^1-11.

[6] IM, Y.H., KANG,M.K.andWOO, K.M.,Product spaces that induce approximatefibrations, J. Korean Math.Soc.33(1996),145-154.

[7] DAVERMAN,1LJ.andWALSH, J.J.,Decompositionsintocodimensiontwomanifolds, Trans. Amer.

Math.Soc.288(1985),273-291.

[8] HEMPEL, J.,3-ManifoMs,PrincetonUniversityPress,Princeton,NJ,1976.