fundamental groups having the whole information of spaces

l

of thege,es are

spacesspace

al grouproposem theheal

Topology and its Applications 146–147 (2005) 317–328

www.elsevier.com/locate/topo

Fundamental groups havingthe whole information of spaces

Greg Connera, Katsuya Edab,∗

a Department of Mathematics, Brigham Young University, USAb School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan

Received 6 November 2002; received in revised form 26 May 2003

Abstract

We introduce a new construction of spaces from groups using homomorphic imagesfundamental group of the Hawaiian earringH. According to this construction the Menger sponSierpinski gasket, Sierpinski carpet and the direct product of their countable many copirecovered from their fundamental groups. 2004 Published by Elsevier B.V.

MSC:55Q20; 55Q70; 57M05; 57M07; 20F34

Keywords:Fundamental group; Hawaiian earring

1. Introduction and definitions

Fundamental groups are defined for arbitrary spaces and are isomorphic amongof the same homotopy type. Therefore we usually think the fundamental group of ahas less information than that of the space itself. Is there a space whose fundamenthas the same information as that of the space itself? In the present paper we pa new construction of a space from a group and show that this construction frofundamental groups of certain spaces produces the original spaces, which implies that tfundamental groups of such spaces have the same information as that of the topologicspaces themselves. The roots of this investigation are in the following [2, Theorem 1.3]:

* Corresponding author.E-mail addresses:[email protected] (G. Conner), [email protected] (K. Eda).

0166-8641/$ – see front matter 2004 Published by Elsevier B.V.
doi:10.1016/j.topol.2003.05.005

318 G. Conner, K. Eda / Topology and its Applications 146–147 (2005) 317–328

LetX andY be one-dimensional, locally path-connected, path-connected, metric spacesps of

lg., them theirle

arring

lows:

tinuum.

of

of

n-

r

y,

which are not semi-locally simply connected at any point. If the fundamental grouX andY are isomorphic, thenX andY are homeomorphic.

As our construction of spaces is based on this fact, the topologiesof one dimensionafractals which are Peano continua and have non-trivial fundamental groups, e.Sierpinski gasket, the Sierpinski carpet and the Menger sponge, are recovered frofundamental groups according to this construction. We shall see that finite or countabproducts of such spaces are also recovered from their fundamental groups.

In the remaining part of this section we state basic definitions. The Hawaiian eis the plane continuumH = ⋃∞

n=1{(x, y): (x + 1/n)2 + y2 = 1/n2} ando denotes theorigin (0,0). Each simple closed curve of the Hawaiian earring is parametrized as folen(t) = ((cos(2πt) − 1)/n,sin(2πt)/n) for 1 � n < ω,0 � t � 1. (Here,en refers tothe nth earring, that is thenth simple closed curve.) The fundamental groupπ1(H) isisomorphic to the freeσ -product××n<ωZn, where eachZn is a copy of the integer groupZ.We refer the reader to [3] for the notion of infinitary words and a freeσ -product××σ

i∈IGi .A locally path-connected, connected, compact metric space is called as a Peano con

2. Construction of spaces and n-slender groups

The fundamental group of the Hawaiian earringπ1(H) plays a principal role in thisconstruction. The groupπ1(H) is presented algebraically as××n<ωZn (abbreviated by××ωZ) [6,9,3,1]. Letδn denote a generator ofZn which corresponds to a windingen tothenth circle of the Hawaiian earring.

Let HG be the set of all subgroups of a groupG which are homomorphic images××ωZ. We remark that finitely generated subgroups are homomorphic images of××ωZ, butinfinitely generated free groups are not homomorphic images of××ωZ. Moreover, any freeproduct∗i∈IGi with infinitely many non-trivial factors is not a homomorphic image××ωZ by [3, Corollary 2.2]. (See Theorem 4.1 for a related result.)

The notation “F � X” means “F is a finite subset ofX”. A finite subsetF of HG said tobecompatible, if there existsH ∈HG such that

⋃F ⊆ H . ForH ∈ HG andF � HG H is

compatible withF , if {H } ∪ F is compatible. We say a subfamilyC of HG is compatible,if any finite subset ofC is compatible. LetXG be the set of all maximal compatible noempty subfamilies ofHG which contain an uncountable subgroup.

Since××ωZ � F ∗××ωZ for a finitely generated free groupsF , x ∈ XG is closed undeconjugacy, that is,u−1Hu ∈ x for H ∈ x andu ∈ G. In addition,x ∈ XG is downward-closed, that is,H ∈ x andH ′ � H imply H ′ ∈ x for H ′ ∈HG.

Since every compatible family extends to a maximal compatible family,XG is non-empty if and only ifHG contains an uncountable subgroup.XG is a one-point set or emptif G is a homomorphic image of××ωZ.

For subgroupsH,H ′ of G, H � H ′ if there existsF � G such thatH � 〈H ′ ∪ F 〉holds. IfH is finitely generated, thenH � {e} holds.

The first proposition shows the root of the above definition.

G. Conner, K. Eda / Topology and its Applications 146–147 (2005) 317–328 319

Proposition 2.1. Let X be a path-connected, one-dimensional metric space andG be

ve a

that

blyerfact

m

d,ies as(

the fundamental groupπ1(X,x0). Then each uncountableH ∈ HG belongs to a uniquex ∈ XG.

Proof. Let h :××ωZ → H be a surjective homomorphism. By [2, Theorem 1.1] we haunique pointx∗ ∈ X, a pathp from x∗ to x0 and a continuous mapf : (H, o) → (X,x∗)such thath = ϕp · f∗ whereϕp is a base-point-change isomorphism fromπ(X,x∗) toπ(X,x0) according top.

First we show that for uncountableH,K ∈HG, {H,K} is compatible if and only if thepointx∗ determined byH andy∗ determined byK are the same. To see this, supposex∗ = y∗. Then we have continuous mapsf : (H, o) → (X,x∗) andg : (H, o) → (X,x∗)and pathsp and q from x∗ to x0 so that H = Im(ϕp · f∗) and K = Im(ϕq · g∗).Define f ′ : (H, o) → (X,x∗) so thatf ′ · e2n = f · en and f ′ · e2n+1 = g · en. Then〈[p−q]〉 ∗ Im(ϕp · f ′∗) ∈ HG contains bothH and K, which shows thatH and K arecompatible. Conversely suppose thatH andK are compatible. Then we haveH ′ ∈ HG

such thatH,K � H ′. Let g′ : (H, o) → (X, z∗) be a continuous map andp′ be a pathfrom z∗ to x0 such that Im(ϕp′ ·g′∗) = H ′. Suppose thatx∗ is not equal toz∗. Theng′ mapsalmost all the circles ofH in a neighborhood ofz∗. On the other hand there are uncountamany essential loops in a neighborhood ofx∗, which is a contradiction. We refer the readto the proof of [2, Lemma 4.2] for a precise proof to deduce the contradiction. Thisimplies that an uncountableH ∈HG belongs to a uniquex ∈ XG. �

We introduce a topology on the setXG as follows.ForY ⊆ XG, Y is closed if each pointx with the following property belongs toY ; there

exists a sequence(xn: n < ω) of elements ofY satisfying the following condition (∗).(∗) For given uncountableHn ∈ xn (n < ω) there existH ′

n ∈ xn satisfying the following:

• Hn � H ′n;

• for arbitraryan ∈ H ′n (n < ω) there existsh :××n<ωZn → G such thath(δn) = an for

everyn < ω and Im(h) ∈ x.

In generalXG may not satisfy any separation axiom, but we write limn→∞ xn = x when(∗) holds. IfG is the fundamental groupπ1(X,x0) for a one-dimensional Peano continuuX, then (∗) is equivalent to each of the following simpler statements, i.e.,

(T1) for given uncountableHn ∈ xn (n < ω), there existe �= an ∈ Hn andgn ∈ G such{g−1

n angn: n < ω} ⊆ H for someH ∈ x;(T2) there exist uncountableHn ∈ xn (n < ω) such that for arbitraryan ∈ Hn (n < ω)

there existsH ∈ x such that{an: n < ω} ⊆ H .

Each of these is equivalent to the condition (∗), because an uncountableH ∈ HG

determines a point ofX for a one-dimensional, locally path-connected, path-connectemetric space. Therefore we need not use the notion of maximal compatible familwell. However, it seems to be necessary to use maximal compatible families and∗) tointroduce a reasonable topology.


Theorem 2.2. LetA be an Abelian group. ThenXA is an empty or one-point space.

e

t

ables of.2].

e

ts

e,

Proof. Let B � A be a homomorphic image of××ωZ. ThenB is a homomorphic imagof the abelianization of××ωZ, where the abelianization of××ωZ is isomorphic toZω ⊕ C

for some algebraically compact groupC [5]. Here,Zω � Zω ⊕ Z

ω andC � C ⊕ C forthisC. ThereforeHA itself is a compatible family and henceXA is an empty or one-poinspace. �

As our construction of spacesXG shows, our concern is concentrated to uncountsubgroups of a given groupG. For our purpose we are interested in a certain clasgroups. We recalln-slender groups from [3, Section 3] together with [3, Proposition 3

Definition 2.3. A groupS is n-slenderif, for each homomorphismh :××n<ωZn → S, theset{n < ω: h(δn) �= e} is finite.

The classS consists of all the groupsG such that, for any non-trivial elementg ∈ G,there exist ann-slender groupS and a homomorphismh :G → S with h(g) �= e.

Since a finite direct product ofn-slender groups is againn-slender [3, Theorem 3.6], whave,

Proposition 2.4. The following conditions are equivalent for a groupG:

(1) G belongs toS;(2) G is a subgroup of a direct product of n-slender groups;(3) G is a subgroup of an inverse limit of n-slender groups.

Lemma 2.5. Let G ∈ S andh andh′ be homomorphisms from××n<ωZn to G. If h(δn) =h′(δn) for everyn < ω, thenh = h′.

Proof. Let S be an n-slender group andg :G → S a homomorphism. There exism < ω such thatg · h(××n�mZn) = g · h′(××n�mZn) = {e}. Henceg · h = g · h′ by theassumption. �Lemma 2.6. LetG ∈ S be a homomorphic image of××n<ωZn. ThenG is finitely generatedor uncountable.

Proof. Let h :××n<ωZn → G be a surjective homomorphism. Suppose thath(δn) = e forn > m. Since there existsh0 :××n<ωZn → G such thath0(δn) = h(δn) for everyn < ω

and Im(h0) = 〈h0(δ0), . . . , h0(δm)〉, G is finitely generated by Lemma 2.5. Otherwiswe choose a strictly increasing sequencein (n < ω) and homomorphismsgn from G ton-slender groupsSn by induction so thatgnh(δin ) �= e andgkh(δin ) = e for k < n. For

f :ω → {0,1}, defineWf = δf (0)i0

δf (1)i1

· · · δf (n)in

· · · , whereδ0ik

is an empty word. Thenh(Wf ) and h(Wf ′) are distinct forf �= f ′, which implies thatG has the continuumcardinality. �


Proposition 2.7. Let Si (i ∈ I) be n-slender groups. ThenXG is an empty or one point

e

-[3,

s from],

-

cted

,adopt

r-map isntin-

entalse ofducts.

space in the following cases:

(1) G is the inverse limit of n-slender groups. In particular,G = lim←−(∗i∈F Si : F � I);

(2) G =××σi∈I Si .

Proof. We first prove the conclusion for the case (1). LetG = lim←−(Tj ,pij : i, j ∈ J ) and

pj : G → Tj be the projection. Leth0, h1 :××n<ωZn → G be homomorphisms. Definh :∗n<ωZn → G by h(δ2n) = h0(δn) andh(δ2n+1) = h1(δn). SinceTj is n-slender,pj · hextends on××n<ωZn uniquely and consequentlyh extends on××n<ωZn uniquely by theuniversal property of the inverse limit. Now we have Im(h0) ∪ Im(h1) ⊆ Im(h), whichimplies that each two elements ofHG are compatible and henceXG is an empty or onepoint space. Since the class ofn-slender groups is closed under taking free productsTheorem 3.6], the second statement of (1) is a special case of the first one.

To show the conclusion for the second case, we recall standard homomorphism[4]. For G =××σ

i∈I Si , let h0, h1 :××n<ωZn → G be homomorphisms. By [4, Theorem 2.3there existu0, u1 ∈ G and standard homomorphismsh0, h1 :××n<ωZn → G such thath0 = u−1

0 h0u0 andh1 = u−11 h1u1 hold.

Then, according to [3, Proposition 1.9] and [4, Lemma 2.4],h :××n<ωZn → G isuniquely determined by the equationsh(δ2n) = h0(δn) andh(δ2n+1) = h1(δn). Now wehaveu0 Im(h0)u

−10 ∪ u1 Im(h1)u

−11 ⊆ Im(h), which implies thatXG is an empty or one

point space. �Remark 2.8.

(1) In the definition ofXG we use maximal compatible families and do not use direfamilies. ForH,H ′ ∈ x ∈ XG there may not existK ∈ x such thatH ∪ H ′ ⊆ K,there existsK ′ ∈ HG such thatH ∪ H ′ ⊆ K ′. If we restrictx ∈ XG to be directedit is unclear whether a directed family extends a maximal one. Therefore wemaximal compatible families.

(2) Our construction of topological spaces from groups is not categorical. Homomophisms may not induce continuous maps. Moreover, even when a continuousinduced, the injectivity of a homomorphism does not imply that of the induced couous map [2, Remark 6.9 (2)].

3. Direct products

In this section we are concerned with direct products of groups. Since the fundamgroup of a direct product of path-connected spaces is the direct product of thocomponents, one may expect that our construction may commute with direct proWe show this is indeed the case under certain conditions.


Lemma 3.1. Let H = ∏m<ω Hm where eachHm is a homomorphic image of××n<ωZn.

fed

o

le

ThenH is also a homomorphic image of××n<ωZn.

Proof. It suffices to show that∏

m<ω××n<ωZ(m,n) is an homomorphic image o××(m,n)∈ω×ωZ(m,n), whereZ(m,n)’s are copies ofZ. To define a homomorphism we nea notion of infinitary words and other ones from [3, p. 244].W(Z(m,n): m,n < ω) is theset of words, that is, infinitary words, whose letters are inZ(m,n)’s. For a wordW and asubsetI of ω × ω, WI is a word obtained fromW by deleting letters not belonging t⋃

(m,n)∈I Zm,n. Defineh :××(m,n)∈ω×ωZ(m,n) → ∏m<ω××n<ωZ(m,n) by: h(W)(m) = WIm ,

whereIm = {(m,n): n < ω}. Thenh is a surjective homomorphism.�In the sequel of this section, whenG = ∏

m<ω Gm, let pm denote the projection fromG to Gm for eachm.

Lemma 3.2. Let G = ∏m<ω Gm and x ∈ XG. Then

∏m<ω pm(H ′) ∈ x for H ′ ∈ x and

consequentlyx = {H ∈ HG: H �∏

m<ω pm(H ′) for H ′ ∈ x} holds.

Proof. Sincepm(H) ∈ HGm for H ∈ x and each{pm(H): H ∈ x} is compatible,{H ∈HG: H �

∏m<ω pm(H ′) for H ′ ∈ x} is compatible and containsx by Lemma 3.1. We

have the conclusion by the maximality.�Lemma 3.3. LetH0 � K0 andH1 � K1. ThenH0 × H1 � K0 × K1 holds.

Proof. There existF0 � G0 andF1 � G1 such thatH0 � 〈K0 ∪ F0〉 andH1 � 〈K1 ∪ F1〉.Now H0 × H1 � 〈K0 × K1 ∪ F0 × {e} ∪ {e} × F1〉 and we have the conclusion.�Lemma 3.4. Let G = G0 × G1 and x ∈ XG. If H × {e} ∈ x and {e} × K ∈ x, thenH × K ∈ x.

Proof. Let F � x. Since the family{X0 × X1: X0 ∈ HG0, X1 ∈ HG1} is cofinal inHG,we can findH0 × H1 andK0 × K1 with Hi,Ki ∈ HGi so thatH × {e} ∪ ⋃

F ⊆ H0 × H1and{e} × K ∪ ⋃

F ⊆ K0 × K1. Since⋃

F ⊆ (H0 ∩ K0) × (H1 ∩ K1), (H × K) ∪ ⋃F ⊆

H0 × K1 ∈HG. We haveH × K ∈ x by the maximality ofx. �Lemma 3.5. Let Gm (m < ω) be groups inS and definep∗

m(x) = {pm(H): H ∈ x}. IfXGm is non-empty, thenp∗

m(x) belongs toXGm .

Proof. Let im :Gm → G be the canonical embedding, that is,im is the homomorphismsatisfyingpmim(g) = g and pkim(g) = e for k �= m and g ∈ G. To show thatp∗

m(x)

contains an uncountable group by contradiction, we suppose the negation. Thenpm(H)

is finitely generated by Lemma 2.6. SinceXGm is non-empty, we have an uncountabK ∈ HGm . Since any finitely generated subgroups ofGm are compatible withK, im(K)

is compatible with anyF � x, which impliesim(K) ∈ x by the maximality ofx. HenceK = pmim(K) belongs top∗

m(x), which is a contradiction.


Sincep∗m(x) is compatible, now it suffices to show thatp∗

m(x) is maximal. LetH ′ ∈

.

ly

ce

se

eegee next

HGm be compatible with anyF � p∗m(x). Sinceim(H ′) is compatible with anyF � x by

Lemma 3.2,im(H ′) belongs tox and consequentlyH ′ belongs top∗m(x). �

Theorem 3.6. LetGm (m < ω) be groups inS, G = ∏m<ω Gm andI = {m < ω: XGm �=

∅}. Then the one of the following holds:

(1) XG � ∏m∈I XGm , if I is non-empty;

(2) XG is a one-point space, ifI is empty and the set{m < ω: Gm �= {e}} is infinite;(3) XG is empty, otherwise.

Proof. We only prove the first case. We use the notionim in the proof of Lemma 3.5Define ϕ :XG → ∏

m∈I XGm by: ϕ(x)(m) = p∗m(x). Thenϕ is surjective and injective

by Lemmas 3.2, 3.4 and 3.5. It now suffices to show thatϕ is continuous and inversecontinuous. To see that eachp∗

m is continuous, let limn→∞ xn = x in XG and choose anuncountableKn ∈ p∗

m(xn). Then im(Kn) ∈ xn by Lemma 3.2. There exists a sequen(Hn: n < ω) with im(Kn) � Hn ∈ xn which witnesses limn→∞ xn = x and so we haveKn � pm(Hn). Hencepm(Hn) is uncountable and belongs top∗

m(xn). For bn ∈ pm(Hn),choosean ∈ Hn so that pm(an) = bn. Then there existsh :××n<ωZn → G such thath(δn) = an (n < ω). Nowpmh(δn) = bn (n < ω) and Im(pmh) ∈ p∗

m(x), which implies thecontinuity ofp∗

m. Hence,ϕ is continuous. To see thatϕ is inversely continuous, suppothat limn→∞ p∗

m(xn) = p∗m(x) for everym ∈ I . LetHn ∈ xn be uncountable for eachn < ω.

Sincepm(Hn) ∈ p∗m(xn) we have a sequence(Kmn: n < ω) with Kmn ∈ p∗

m(xn) whichwitnesses limn→∞ p∗

m(xn) = p∗m(x) for eachm ∈ I . Then

∏

k�n,k∈I

Knk ×∏

k�n,k /∈I

{e} ×∏

k>n

pk(Hn) ∈ xn and

Hn �∏

k<ω

pk(Hk) �∏

k�n,k∈I

Knk ×∏

k�n,k /∈I

{e} ×∏

k>n

pk(Hn)

hold by Lemmas 3.2 and 3.3. Letan ∈ ∏k�n,k∈I Knk × ∏

k�n,k /∈I {e} × ∏k>n pk(Hn) for

n < ω. Fix m. Sincepm(an) ∈ Kmn or pm(an) = e for n � m, there existshm :××n<ωZn →Gm such thathm(δn) = pm(an) (n < ω) and Im(hm) ∈ p∗

m(x). Defineh :××n<ωZn → G by:pmh = hm (m < ω). Hence, we haveh(δn) = an (n < ω) and Im(h) = ∏

m<ω Im(hm) ∈ x

by Lemma 3.2. �

4. Free products

We call a groupG quasi-atomic, if for each homomorphismh :G → A ∗ B there existsa finitely generated subgroupA′ of A or B ′ of B such that Im(h) is contained inA′ ∗ B orA ∗ B ′.

By definition, finitely generated groups and Abelian groups are quasi-atomic, but frproducts of infinitely generated groups arenot quasi-atomic. Every homomorphic imaof a quasi-atomic group is also quasi-atomic. This section is devoted to prove ththeorem.


Theorem 4.1. LetGi be a finitely generated groups for eachi ∈ I . Then××σ Gi is quasi-

of

tion 2].s

s

t

i∈I

atomic.

For a wordW ∈ W(Gn: n ∈ ω) andn ∈ ω, ln(W) is the number of appearanceselements ofGn in W . Forx ∈ ∗j∈J Hj , l(x) denotes the length of the reduced word forx.Theorem 4.1 strengthens [3, Corollary 2.5] and the proof uses the content of [3, SecFirst we recall [3, Lemma 2.3]. We state it ina more precise form which follows from itproof.

Lemma 4.2 [3, Lemma 2.3].LetHj (j ∈ J ) be groups. Letm + n + 2 � k for m,n, k ∈ N

andu,xi, z ∈ ∗j∈J Hj (1 � i � M). If l(u) � m,u = x1zk · · · xMzk and l(xi) � n for all

1 � i � M, then one of the following holds:

(1) z is a conjugate to an element of someHj ;(2) z = x−1f xy−1gy for somef ∈ Hj and g ∈ Hj ′ with f 2 = g2 = e, and some

x, y ∈ ∗j∈J Hj such thatxi = zpx−1f x or xi = y−1gyzp for somei andp.

Lemma 4.3 [3, Corollary 2.5].Leth :××n∈ωGn → ∗j∈J Hj be a homomorphism for groupGn(n ∈ ω) andHj(j ∈ J ). If everyGi is finitely generated, then there existsF � J suchthath(××n∈ωGn) � ∗j∈F Hj .

Definition 4.4. A subsetC1 of A ∗ B is the set of all conjugates to elements ofA ∪ B, i.e.,C1 = {x−1ux: u ∈ A ∪ B, x ∈ A ∗ B}. Also letC2 = {xy: x, y ∈ C1}.

The proof of the next lemma is straightforward and is omitted.

Lemma 4.5. The reduced wordW for an element ofC2(⊆ A ∗ B) is one of the followingforms:

(1) empty;(2) V −1uV whereu ∈ A ∪ B and V is a reduced word, that is,W ∈ C1 as an elemen

of A ∗ B;(3) V −1

2 V −10 u0V0V

−11 u1V1V2 whereu0, u1 ∈ A ∪ B andV0,V1,V2 are reduced words;

(4) V −12 v0V

−10 u0V0v1V

−11 u1V1v2V2 whereu0, u1, v0, v1, v2 ∈ A ∪ B andV0,V1,V2 are

reduced words andv0v1v2 = e.

Lemma 4.6. Let A′ and B ′ be subgroups ofA and B, respectively. Ifa0 ∈ A \ A′,b0 ∈ B \ B ′ anda1 ∈ A \ 〈A′ ∪ {a0}〉, then

(1) u−1a0uv−1b0vu−1a1u /∈ C2 for u,v ∈ A′ ∗ B ′;(2) wu−1a0uv−1b0vu−1a1u does not belong to

{(x−1f xy−1gy

)px−1f x, y−1gy

(x−1f xy−1gy

)p:

f,g ∈ A ∪ B, x, y ∈ A ∗ B, p � 0}

for u,v,w ∈ A′ ∗ B ′.


Proof. Let U andV be reduced words foru andv, respectively. If the left most letter

ya.

t

n bye

d

o

eand

a′ of U belongs toA′, thena′−1a0a

′ ∈ A \ A′ anda′−1a1a

′ belongs toA \ 〈A′ ∪ {a0}〉.Hence, we may assume thatU−1a0U,V −1b0V andU−1a1U are reduced. LetW be thereduced word ofV U−1. WhenW is empty or the left most letter ofW−1 does not be-long toA′, the reduced word ofU−1a0W

−1b0Wa1U is of the formU−1a0X−1b1Xa1U

for someb1 ∈ B \ B ′ and some wordX. When the left most letter ofW−1 belongs toA′, the reduced word ofU−1a0W

−1b0Wa1U is of the formU−1a2X−1b1Xa3U for some

a2 ∈ A \ A′, b1 ∈ B \ B ′, a3 ∈ A \ 〈A′ ∪ {a2}〉 and some wordX. In the both cases, theare not of the form indicated in Lemma 4.5 and the conclusion (1) holds by that lemm

To show (2) by contradiction, suppose there existf,g ∈ A ∪ B,x, y ∈ A ∗ B andp � 0such thatwu−1a0uv−1b0vu−1a1u = (x−1f xy−1gy)px−1f x or y−1gy(x−1f xy−1gy)p.Consider the reduced form ofu−1a0uv−1b0vu−1a1u shown in the proof of (1). Thearrangement ofa0, b0, a1 implies thatp � 1. Sincea1 or its conjugate inA appears jusone time in the reduced word forwu−1a0uv−1b0vu−1a1u, we get a contradiction.�Proof of Theorem 4.1. First we prove the conclusion for a homomorphismh :××n<ωZn →A ∗ B. By Kurosh’s Theorem [8, Section 34] or [7, Chapter 17], the image ofh is a freeproduct of copies ofZ and conjugate groups of subgroups ofA or B. By Lemma 4.3the number of components of this free product is finite. To prove the conclusiocontradiction, suppose that there exists noA′ nor B ′ required in the conclusion. Sincthe component isomorphic to a free group is finitely generated, there existu,v ∈ A∗B andinfinitely generated subgroupsA∗ andB∗ such thatu−1A∗u andv−1B∗v are free factorsof Im(h). By our assumption neitherA∗ nor B∗ is contained in any finitely generatesubgroupA′ of A nor B ′ of B. Note that one can findu,v,A∗ andB∗ with the aboveproperty forh(××n�mZn) as well. We choosekm < km+1 andxm ∈××n�mZn by inductionas follows:

We choosex1 so that h(x1) �= e and k1 = 2. In the m-step, let km = m +max{l(h(xi · · ·xm−1)): 0 � i � m − 1} + km−1. Applying the above argument th(××n�mZn), we obtainu−1A∗u and v−1B∗v. Take finitely generated subgroupsA′ ofA andB ′ of B such thatu ∈ A′, v ∈ B ′ andh(xi) ∈ A′ ∗ B ′ for every 0� i � m − 1. Byapplying Lemma 4.6 toA∗ \ A′ andB∗ \ B ′, we obtainxm ∈××n�mZn such that

(1) h(xm) /∈ C2;(2) h(xi) · · ·h(xm) does not belong to

{(x−1f xy−1gy

)px−1f x, y−1gy

(x−1f xy−1gy

)p:

f,g ∈ A ∪ B, x, y ∈ A ∗ B, p � 0}

for any 1� i � m − 1.

Let Wm be the reduced word forxm for each 1� m < ω. We use the following notion in thproof of [3, Lemma 2.4]. LetSeq be the set of all finite sequences of natural numbersdenote the length ofs ∈ Seq by lh(s). An elements ∈ Seq is denoted by〈s1, . . . , sn〉 wheresk ∈ N (1 � k � n). For s, t ∈ Seq, s ≺ t if s(i) < t(i) for the minimali with s(i) �= t (i)

or t extendss.


Let V = {(s,p): s ∈ Seq, 1 � s(i) � ki for 1 � i < ω, p ∈ Wi} with the lexico-

ing

nt

as

y

graphical ordering andV (s) = Wlh(s)(p). Then V is a word inW(Zn: n < ω). LetVm = V ∩ {(s,p): lh(s) � m, si = 1 for 1� i � m} andVm = V � Vm. Finally, chooseN so thatN � l(h(V )). Thenh(V ) is equal to the element expressed in the followfigure:

h(x1) h(x2) · · · h(xN−1)h(VN)kN

. . .

h(xN−1)h(VN)kN

h(x2) · · · h(xN−1)h(VN)kN

. . .

h(xN−1)h(VN)kN

.... . .

h(xN−1)h(VN)kN

h(x1) h(x2) · · · h(xN−1)h(VN)kN

. . .

h(xN−1)h(VN)kN

h(x2) · · · h(xN−1)h(VN)kN

. . .

h(xN−1)h(VN)kN

.... . .

h(xN−1)h(VN)kN .

Applying Lemmas 4.2 and 4.6 we concludeh(VN) ∈ C1 and by the same argumeh(VN+1) ∈ C1, which impliesh(VN+1)

−kN+1 ∈ C1. Since xN = VNV−kN+1N+1 , we have

h(xN) ∈ C2, which contradicts the construction.Next we treat with the caseh :××σ

i∈IZi → A ∗ B. We modify the above constructionfollows: We choose countableJm ⊆ I and finiteSm ⊆ I in addition toxm andkm < km+1by induction so thatJm ∩ Sm−1 = ∅ and xm ∈ ××i∈JmZi and

⋃m<ω Jm = ⋃

m<ω Sm.Then we apply the above procedure taking infinitely generated subgroupsA∗ andB∗ for××i∈I\Sm−1Zi and we can defineV as before and get a contradiction.

Since××σi∈IFi is isomorphic to××σ

j∈J Zj for finitely generated free groupsFi ,××σi∈IGi is

a homomorphic image of××σj∈J Zj and the conclusion holds.�

Corollary 4.7. LetA andB are groups inS. ThenXA∗B is the topological sum ofXA andXB .

Proof. Let G be the free productA ∗ B andH ∈ HG be an uncountable subgroup. BTheorem 4.1 we first assume that there existsF0 � A such thatH � 〈F0〉 ∗ B. As in theproof of Theorem 4.1,H is isomorphic to a finite free product of copies ofZ and conjugate


groups of subgroups ofA or B. Every letters inA and subgroups ofA which appears in

ren

hat

up

,of

etric

s

etric

this presentation is in〈F0〉 and there exists an uncountable subgroupH0 of B such thatH is a free product of a conjugate subgroup ofH0 and a finitely generated group. HeH0 ∈ HB . We haveF1 � B such thatH � 〈H0 ∪ F0 ∪ F1〉. We have a similar conclusiowhenA is replaced byB.

By the preceding fact we easily see that the only one of{H ∈ x: H ∈ HA} ∈ XA and{H ∈ x: H ∈ HB} ∈ XB holds for x ∈ XG. On the other hand we also easily see t{〈H ∪ F 〉: F � A ∪ B, H ∈ x} ∈ XG for eachx ∈ XA ∪ XB . Let YA = {x ∈ XG: {H ∈x: H ∈HA} ∈ XA} andYB = {x ∈ XG: {H ∈ x: H ∈HB} ∈ XB}. ThenXG is the disjointunion ofYA andYB andYA andYB are homeomorphic toXA andXB , respectively.

To show thatYA is closed inXG, suppose that limn→∞ xn = x for xn’s in YA. TakeuncountableHn ∈ xn. We have a sequence(Kn: n < ω) with Hn � Kn ∈ xn whichwitnesses limn→∞ xn = x. SinceKn is uncountable,Kn contains a conjugate subgroto an uncountable subgroup ofA. We choosean ∈ Kn so thatan �= e and a conjugate toan element ofA. We remarkpA(an) �= e andpB(an) = e. Then we haveh :××n<ωZn → G

such thath(δn) = an (n < ω). SincepB(an) = e for everyn < ω, we havepBh(x) = e

for everyx ∈××n<ωZn by Lemma 2.5. By the proof of Lemma 2.6 Im(h) is uncountablewhich implies Im(h) contains an subgroup conjugate to an uncountable subgroupA.Thereforex belongs toYA. We obtain the corresponding statements also forYB . �

5. Recovering a space

A point x ∈ X is called awild point, if X is not locally semi-simply connected atx.Xw denotes the subspace consisting of all wild points ofX. A spaceX is calledwild, ifX = Xw holds.

Theorem 5.1. LetX be a locally path-connected, path-connected, one-dimensional mspace andG be the fundamental groupπ1(X). Then XG is homeomorphic toXw .Consequently, in addition ifX is wild, XG is homeomorphic toX itself.

Proof. As we stated in the proof of Proposition 2.1, each uncountableH ∈ HG determinesa unique pointx∗ ∈ X and also determines a unique pointx ∈ XG. MoreoverH isconjugate to the image of a homomorphism induced by a continuous mapf : (H, o) →(X,x∗), where o is the unique wild point ofH. Hence x∗ is a wild point. Thiscorrespondencex �→ x∗ is a bijection, sinceH andH ′ are compatible only if the pointof X uniquely determined byH andH ′ are the same for uncountableH,H ′ ∈HG. To seethe continuity, letY ∗ = {y∗: y ∈ Y } for Y ⊆ XG. It suffices to show thatY ∗ is closed ifand only ifY is closed. This easily follows from the fact that forxn ∈ XG (∗) holds if andonly if limn→∞ x∗

n = x∗ by [2, Theorem 1.1]. �Together with Theorem 3.6 we have,

Corollary 5.2. LetXn be locally path-connected, path-connected, one-dimensional mspaces such thatXw

n �= ∅ andG be the fundamental groupπ1(∏

n<ω Xn) � ∏n<ω π1(Xn).


ThenXG is homeomorphic to∏

n<ω Xwn . Consequently, in addition ifXn = Xw

n for

and

llows.

(T2)

n the4.7

pl.,

3–

00)

956)

6)

everyn < ω, XG is homeomorphic to∏

n<ω Xn.

Remark 5.3. Let eachXn be a copy of one-dimensional wild Peano continuumY = ∏

n<ν Xn, whereν � ω. ThenXπ1(Y ) � Y by Corollary 5.2, but the topology ofXπ1(Y )

does not coincide with the one defined by the condition (T2). This can be seen as foTakeHn ∈ yn to be subgroups ofπ1(X0) × {e}, wheree is the identity ofπ1(

∏1�n<ν Xn).

Then the satisfaction of (T2) is the information only on the first co-ordinate and sodoes not give the topology ofY .

Concerning (T1) we have no examples which show the difference betweetopologies (∗) and (T1). Since we feel some difficulty to get Theorem 3.6 and Corollarybased on the definition (T1), we adopt (∗).

References

[1] J.W. Cannon, G.R. Conner, The combinatorial structure of the Hawaiian earring group, Topology Appl. 106(2000) 225–271.

[2] K. Eda, The fundamental groups of one-dimensional spaces and spatial homomorphisms, Topology Apsubmitted for publication.

[3] K. Eda, Freeσ -products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263.[4] K. Eda, Freeσ -products and fundamental groups of subspaces of the plane, Topology Appl. 84 (1998) 28

306.[5] K. Eda, K. Kawamura, The singular homology ofthe Hawaiian earring, J. London Math. Soc. 62 (20

305–310.[6] H.B. Griffiths, Infinite products of semigroups and local connectivity, Proc. London Math. Soc. 6 (1

455–485.[7] M. Hall Jr, The Theory of Groups, Macmillan, New York, 1959.[8] A.G. Kurosh, The Theory of Groups, vol. II, Chelsea, New York, 1960.[9] J.W. Morgan, I.A. Morrison, A van Kampen theorem for weak joins, Proc. London Math. Soc. 53 (198

562–576.

fundamental groups having the whole information of spaces

Documents