A NEW LOOK AT MEANS ON TOPOLOGICAL SPACES

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 4 (1997) 617-620

617

A NEW LOOK AT MEANS ON TOPOLOGICAL SPACES

PETERHILTON

Department

of Mathematics University of CentralFlorida Orlando,

FL

32816-1364

USA

and

Department

ofMathematical Sciences

SUNY,

Binghamton Binghamton,NY13902-6000USA

(Received June 5,

1997)

ABSTRACT. We use methods ofalgebraic topology to study when a connected topological space admits ann-meanmap.

KEY WORDS

AND

PI-IRASES: Means,

topological spaces,comeans 1991

AMS SUBJECT CLASSIFICATION CODES:

55PXX 1.

INTRODUCTION

Carath6odoryandAumann

(see [1 ],[2])

wereamongthepioneers whofirstconsidered thequestion of whatpath-connected regions

X

in^]R or

C

could support ann-mean,thatis,amap_#

X A"

satisfying

(i)

z/

^1"

X X,

where/is the diagonalmap

& X X

and

(ii)

cr

^#"

^{X X,}

^{where crE}

^S,,

^{the symmetric group on n} letters, acting on

X

by permuting components Oneof their main concerns wastofindoutif the existence of such ann-mean, n

>

2, impliedthat

X

wassimplyconnected

In

1954,

Beno

Eckmann

[4]

attackedthe questionwiththe tools of algebraic

topology He

supposed

X

to beapolyhedronandonly requiredconditions

(i),

(ii)aboveuptohomotopy

One

ofhisprincipal conclusionswasthat if

X

iscompactand admits a(homotopy)^n-meanfor

a,

^{lln, then}

^X

^iscontractible

In

1962,Eckmann,together^withTudor

Ganea

andtheauthor, returnedtothestudy ofn-means in a moregeneral setting

(see [5]).

Thus the n-mean defined in

[4]

was amorphism in the category

T

^of

based connectedCW-complexesand basedhomotopyclassesofbasedmaps

In

thisgeneralityone was ableto exploit the ideaof mean-preserving functors. Thus if

C,

79 arecategorieswith products and

F C

79 is aproduct-preserving

functor,

then

F#

is an n-mean in79for anyn-mean in

C

Moreover, onecould alsoexaminethedual questaon of theexistenceofn-comeans

Itturnsoutthat the concept of P-local objects and P-locahzatzon, where

P

s afamdy ofprimes,and the results relatedtotheseconcepts inthecategories

Th

^{andA/’,thecategoryof}ndpotent groups(see [6]),enable one tommplify many argumentsm

[5]

andtoextend the results of that paper

2.

MEANS

IN

THE CATEGORY

OF

GROUPS

Let be the category ofgroups

Let

nbe aninteger,n

>

2, and let

P

be thefamdyofprimes/9such that

pn We

then

prove

THEOREM

2.1. Thegroup

G

admits an n-mean#in

:

(7is commutative andP-local

In

that case,ifwewrite

G

additively,/isgiven by

618 P HILTON

(2 1)

PROOF. Note

first that if

G

iscommutative, then

G

isP-local ifand onlyif

G

admits unique divisionbyn Itis then plain that

(2 1)

defines an n-mean on

G

Conversely, let# bean n-meanon

G For

9,h E

G (at

ths stage, we write

G

multiplicatively),set

(9,

^e,..-,

e) ,, (h,

e,...,

e)

⁽⁵ Then, bycondition(ii),

(e,

g,

., e) (e,

e,

., g)

", sothat, bycondition(i),

Similarly, h 6 But

/(9,

^h,-..,

e) 76, #(h,

9,’"",

e) 57,

and

#(9,

^h,.-.,

e) #(h,

9,’"^",e) Thus_{7 commutes}with

5,

sothat_{9 commutes}withhand

G

is commutative

To

show that

G

sP-local itremainstoshow thatn^th rootsareuniquein

G. But,

again using properties

(i)

and(ii), ^weconclude that

#(9 ’,

e,-..,

e) #(g,

9,’"",

9)

g, so thatg is determinedby₉ Thus

G

is commutative and

P-

localand, writing additively,wehave

(.qx

if2,

",fin)

^,=1

(9,

0,

-,0) -1

^{,=1 gt}

^--(,ql

^{n1 -1-g2}"1-"’"

- ^gn).

COROLLARY

2.2. Let

G

be agroup and letnl

>_

2, _n2

>_

2 be ,ntegers Then

G

admits an n n2-meanifandonlyif

G

admitsann -meanand an_n2-mean

3. MEANS

IN

TIlE

CATEGORY T

Let X

beaconnected CW-complexwithbasepoint. We prove,with n,

P

as in Section2,

THEOREM

3.1.

Suppose X

admits an n-mean _#"

X’ X

in

Th

^Then

^X

^{is a P-local}

commutative

H-space

PROOF.

We regardthe ^thhomotopygrouprr, asdefiningaproduct-preservingfunctor from

Th

^to

Then/.

^7r,#

(Tr, X)

rr,

X

is an n-mean in Itfollows that7r,

X

s commutauve(ths^sonly sgmficant for

1)

and P-local and that_#.has theform

(2

1).

Let

i1"

^X--,

X

be the obxaous embedding. Then

(il),

^IS the endomorptusm ₉

!9

^{of the}

commutative P-localgroup7r,

X

Itfollows that

(il).

^isanautomorphism for all i, so

that/z

s aself- homotopy-eqmvalence of

X

Letp

X X

behomotopy reverseto

#il.

Let

Zl ^{X X}

be the obwous embedding and letrn p/zz,2

X

X. Thentseasytosee thatmISa commutauve

H-structure

onX

Weconclude that

X

saP-localcommutativeH-space

IZI

FromTheorem2.1wededuce, more easilythan in

[5],

TIOREM

3.2. Ifacompact, connected

polyhedron X

admitsann-meanforsomen

>

2,then

X

iscontractible.

PROOF.

Since the homotopy groups of

X

areP-local, soare thehomologygroups

H,X, >

(see [6]). Now

Browder has shown

[3]

that acompact, connected polyhedron

X

whichis an

H-space

satisfies Poincar6duality. Thus, if

X

is not contractible, there exists apositivedimension

N

which contains the universalclass giving riseto theduality isomorphism

H,(X) Hv-’(X). In

particular,

HvX Z,

but this isabsurd,since

Z

isnotdivisiblebyn I"1

REMARK

1. Wehavenotinvoked commutativity ofthe

H-structure

inthisargument Ifwedo so, we may apply^atheoremofHubbuck showingthat

X

wouldbeequivalenttoaproductof circles, which isalsoimpossible foranon-contractible P-localspace

ANEWLOOKATMEANSON TOPOLOGICALSPACES 619

REMARK

2. Theorem 3.2 is delicate Then-solenoid is compact and admits an n-mean but isnot apolyhedron TheEilenberg-MacLanespace

K(Q, m)

is apolyhedronand admitsann-meanforevery n, butis notcompact

We havenotproved--and doubt the truth of--theconverseofTheorem 31

However,

onemay readilyprove

THEOREM

3.3. If

X

isaP-local, connected, commutative,associative

H-space,

then

X

admitsa unique homomorphic n-mean. Further, ifthe connected

H-space (X, m)

^{admits a}homomorphic n-

mean,then

(X, m)

iscommutative and associative.

The casen 2 admits averyneatand precise statement.

If/" ^{X X}

is a 2-meanon

X,

we define p as in theproof of Theorem3.1 ashomotopyinverseto#il,andrn p#isacommutative

H-

structureontheP-localspace

X,

where

P

isthefamily of odd primes Conversely,ifm

X X

^isa commutative

H-structure

ontheP-localspace

X,

wedefine7-tobehomotopyinverse to

rnA X X (notice

that

rnA

induces doubling onthe homotopy

groups

of

X

and istherefore a

self-homotopy-

equivalence).

Then/

^7-mis a 2-mean on

X.

THEOREM

3.4. The

function/

p#setsupaone-onecorrespondencebetween 2-means on the P-local connectedCW-complex

X

andcommutative

H-structures

on

X

PROOF. Ifm p/, then

z/

p# p, so _% defined above, is homotopyinverseto p and

7-rn # If_7-#=/,then7-

uil

^{so, again,p is}homotopyinverseto"randp/.t m Thus the function m 7-rnis inversetothe function

4.

THE DUAL STORY

Whereas theproductin afamiliar category(like Th,

)

takesafamiharform essenlaally independent of the category, theformof thecoproduct dependsvery muchonthe categorym queslaon Thethreecategories whmhwillcomeintoqueslaonhereareTh,

,

^andAb, the category of abehan groups

Let Cbeacategoryadrmttmgfinitecoproducts,wewillwrite

C v D

for thecoproductof

C

and

D

nC and

Cn

for the coproduct ofncopies of

C

inC. Obwously, the symmetricgroup

S,,,

acts on

C,.,,

wewillwrite

V C, C

for the codiagonal,whichisthe morphism that coincides wth the identaty on each copy ofCm

C,

Thenan n-comeanon

C

s amorphsm forallaE

S, We

prove

THEOREM

4.1.

In

PROOF. Let

G

be a non-trivial group and letg E

G,

g e If_#

G G,

is ann-comean,ⁿ

>

2, then

itfollows from(i)that#g

-

^e

^{Now G,}

^isthefree productofncopies of

G,

soanon-trivial element of

G,

s umquelyexpressible as

h,h,...h,,

where

G(,)

^{tsthe thcopy of}

^G

ⁱⁿ

^G,,

^h,

G(,),

^h,

:f:

^{e, and}iq iq+l, q 1,2,..-,k-1 Suchanelement is obviouslymovedunder anypermutation awinchmoves il, so that

condition

(d)

^sviolated. I-I

THEOREM4.2. In .Ab,the abelian group

A

admits ann-comean,n

>

2, if and onlyifit admitsan n-mean

In

that_case/

A A,

isgiven by

#(a)= (a ^a, ,a). ⁽⁴¹⁾

PROOF.

Wenotefirst

that,

in.Ab,

C

^y

D C D,

so that

A, A

If

A

admits ann-mean, then, by Theorem 2.1,itisclear that

(4.1)

is ann-comean.

Suppose

converselythat#

A A,,

isann- comean

It

is thenplain from

(ii)

that

p(a) (c,

a,.-.,

a)

for some^c

A

such that, by (i),^{nc a.} It remainstoshow thatdivisionbynisuniquein

A. But

,(,) (,,, r,...,,) (,, ,,..-,,),

sothataisdeterminedbyna

620 P HILTON

REMARK. Notethat the situationsfor means and comeans areverydifferent

Means

in conclde wath meansⁱⁿ4b,onthe otherhand,therearenonon-tnwal comeansm but there arenon-tnvaalcomeansn 4b,and,moreover, the objectsm A,badmttangn-comeanscoincidewththose admittingn-means

Wenowstudy n-comeans m

Th.

Using the samenotalaonas inTheorem3.1,weprove

TItEORENI

4.3. Suppose X s a connected CW-complex admlttang an n-comean _#X---,

Xn

ⁿ

Th, n

>

9 Then

X

s asimply connected P-local commutative

H’-space

PROOF. Now X,

isjustabouquet ofncopies of

X

Since_7rl

Th

^scoproduct-preservmg,

7q#s an n-comean onthefundamental group 7qX, sothat,by Theorem41,

X

ssmplyconnected Nowthe homology groups

H, >

1, arecoproduct-preservng functors

Th

^A,b,sothat,by Theorems2 and42, thehomologygroups

HX

^arethe P-local. Since X ssmply connected, ths mphes thatX s P-local Finallyweadoptahne of reasomng entirely analogoustothatmtheproofof Theorem3 toconclude that

X

admits acommutative

HI-structure m-X X2 (Notice

^that, since

X

is simply connected, a map

f ^{X X}

^inducinghomology isomorphismsis a

homotopy

equivalence.)

Noticethattherearestraightforwardand validdualsof Theorems3.3 and3 4

On

the otherhand, Theorem 3.2 doesnotdualize.

For

example,the

Moore

space

M(Z/2, m),

rn

>

2, characterizedas the unique simply connected homotopy type with ^H9_

Z/2, ^H,

0,

>

3, is a compact

(m + 1)-

dimensionalpolyhedronwhich admits an n-comeanfor every oddn

REFERENCES [1] AUMANN,

G,Math.Annalen 111(1935),713-730

[2] AUMANN, G,

Math.Annalen 119

(1943),

^210-215

[3 BROWDER, W, On

torsion in

H-spaces,

Ann.Math.

(2)

⁷⁴(1961),^24-51

[4] ECKMANN, B., Raume

mitMittelbildung,

Comm.

Math. Helv.28

(1954),

^329-340

[5] ECKMANN, B., GANEA, T

and

HILTON, T.,

Generalized means, Studtes ^mMathematical Analysis^andRelated Topics, Stanford University

Press (1962),

^82-92.

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and

ROITBERG, J.,

Locahzation

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