On the fundamental group of $mathbb R^3$ modulo the Case-Chamberlin continuum

ON THE FUND AMENT AL GR OUP OF R 3 MODULO THE CASE-CHAMBERLIN CONTINUUM KA TSUY A EDA, UMED H. KARIMOV, AND DU ˇ SAN REPOV ˇ S Abstract. It has b een kno wn f or a long time that the fundamental group of the quotien t of R 3 b y the Case-Chamberlin con tinuum is non trivial. In the presen t paper w e pro ve that this group i s in fact, uncoun table. 1. Introduction In the 1960’s , during the ea rly days of the decompo sition theo ry , the quo tien t space X 3 of the Euclidean 3-space R 3 by the classica l Cas e-Chamberlin co n tinuum C (see [3]) was one of the mo s t int ere sting examples. One of the most imp ortant questions was whether X 3 is simply connected. It was settled – in the negative – by Armentrout [1] and Shrikhande [10]. How ever, it rema ined an o pen problem until present day to determine how big is the fundamental gr o up of X 3 . In this pap er we give the solution for this pro blem – na mely , we show that the fundamental group π 1 ( R 3 /C ) is uncountable . Consider the Case-Cha m b erlin inv erse se q uence P (see [3], [5, p.628]): P 0 f 0 ← − P 1 f 1 ← − P 2 f 2 ← − · · · where P 0 = { p 0 } is a singleton, P i is a b ouquet of tw o circ le s S 1 a i W S 1 b i , and p i is the base po in t o f the b ouquet S 1 a i W S 1 b i , for every i > 0 . Fix an o rientation on ea c h of the cir cles of the b ouquet. Let f i : S 1 a i +1 _ S 1 b i +1 → S 1 a i _ S 1 b i be a piecewise linear mapping which maps the base p oint p i +1 to the base p oint p i and maps the na tur al generato rs a i +1 and b i +1 of π 1 ( S 1 a i +1 W S 1 b i +1 ) to the co mm u- tators [ a i , b i ] and [ a 2 i , b 2 i ] of π 1 ( S 1 a i W S 1 b i ), resp ectively . The Case-Chamberlin contin uum C is then defined as the inverse limit lim ← P of the Case-Chamber lin inv ers e seq uence P (se e [3]). Obviously , C is a 1-dimensio nal contin uum and therefore it is embeddable in R 3 (see [4]). It is well-known that the homotopy types o f the quotient s pace R 3 /f ( C ) ar e the same for all e mbedding s f of C in to R 3 (see [2]). The main result of our pap er is the following theorem: Theorem 1.1. L et C b e t he Case-Chamb erlin c ontinuum emb e dde d i n R 3 . Then the fun damental gr oup π 1 ( R 3 /C ) of the quotient sp ac e R 3 /C is un c ountable. Date : May 29, 2007. 2000 Mathematics Subje ct Classific ation. Pr imary: 54F15, 55Q52, 57M05; Secondary: 54B15, 54F35, 54G15. Key wor ds and ph r ases. Case-Chamberlin con tinuum, quotien t space, fundamen tal group, low er cen tral ser ies, we ight, commu tator. 1 2 KA TSUY A EDA, UMED H. KARIMOV, AND DU ˇ SAN REPOV ˇ S 2. Preliminaries Let G b e a gr oup. B y the c omm utator of the elements a an b of G we mean the element [ a, b ] = a − 1 b − 1 ab o f G . Let G n be the low er central series which is defined inductively (see [9]): G 1 = G, G n +1 = [ G n , G ] , where [ G n , G ] is the g roup genera ted b y the set { [ a, b ] : a ∈ G n , b ∈ G } . Obviously , G n ⊇ G n +1 , for every n . By the w eight w ( g ) of an element g ∈ G we mean the maximal num ber n such that g ∈ G n if s uc h a num ber exis ts , and ∞ otherwise. So the weigh t of any element of a per fect group is equal to ∞ . W e shall need the following r esult from [8, Ch. I, Pro positio n 10.2]: Prop osition 2.1. : F or any fr e e gr oup F t he lower c entr al series F n has trivial interse ction, i.e. T ∞ n =1 F n = { e } . That is, in a n y free gro up the weigh t of an element x is finite if and only if x 6 = e . Let C ( f 0 , f 1 , f 2 , . . . ) be the infinite mapping cylinder o f P (see e.g. [7, 11]) and let e P b e its natura l compactification by the Case-Chamber lin con tinuum C . L e t P ∗ be the quotient space of e P by C . Obviously , P ∗ is homeo morphic to the o ne-po int compactification of a n infinite 2-dimensional po lyhedron C ( f 0 , f 1 , f 2 , . . . ). Let C ( f k , f k +1 , f k +2 , . . . ) be the mapping cy linder of the inv ers e sequence: P k f k ← − P k +1 f k +1 ← − P k +2 f k +2 ← − · · · . W e shall denote the corr espo nding one–p oint co mpa ctification by C ( f k , f k +1 , f k +2 , . . . ) ∗ . W e sha ll c o nsider C ( f k , f k +1 , f k +2 , . . . ) ∗ as a subspac e of P ∗ and we sha ll denote the compactification p oint by p ∗ . W e consider P i , for i ≥ 0, as a subspace o f C ( f 0 , f 1 , . . . ) and we consider C ( f k , f k +1 , f k +2 , . . . ), for k ≥ 0, as a subspa ce of e P . Obviously , P 1 is a strong deformation r etract of C ( f 1 , f 2 , . . . ). W e have the following ho momorphism ϕ i +1 = ( f 1 · · · f i ) ♯ : π 1 ( P i +1 ) → π 1 ( P 1 ) which is a monomo rphism, s ince it is the co mpositio n of mono morphisms ( f i ) ♯ : π 1 ( P i +1 ) → π 1 ( P i ) . Note that fo r a fixed i , the ele ments [ a i , b i ] and [ a 2 i , b 2 i ] are fre e generator s o f a subgroup ( f i ) ♯ ( π 1 ( P i +1 )) o f π 1 ( P i ) (se e Exerc is e 12 o n p.119 of [9]). Since ϕ i is a monomorphism, we can consider the group π 1 ( P i ) as a s ubg roup of π 1 ( P 1 ) = F , where F is a free gro up on tw o genera tors a 1 and b 1 . In par ticular, by ident ification, we hav e a 2 = [ a 1 , b 1 ] , a 3 = [ a 2 , b 2 ] = [[ a 1 , b 1 ] , [ a 2 1 , b 2 1 ]] , etc. Since a i 6 = e , the weigh t w ( a i ) is a finite num be r (cf. Prop osition 2 .1 a bov e). It follows b y definition of a i that w ( a i ) ≥ i , for every i . THE FUNDAMENT AL GROUP π 1 ( R 3 /C ) IS U NCOUNT ABLE 3 Cho ose an incr easing s equence o f natural num b ers { n i } as follows: Let n 0 = 1 and n 1 = 2. If n k is already defined, then let n k +1 be an y natural num ber suc h that n k +1 > w ( a n k ) for k ≥ 1. Then we hav e a n k / ∈ F n k +1 . Let I i be the unit segment w hich connects the points p i +1 and p i and whic h corres p onds to the ma pping cylinder of the ma pping f i | { p i +1 } of the one-p oint set { p i +1 } to the one- p oint set { p i } , for i ≥ 0 . T o define a certain kind of lo ops we need a new notion. F or t wo paths f , g : I → X satisfying f (1) = g (0), let f g : I → X b e the path defined by: f g ( s ) =  f (2 s ) if 0 ≤ s ≤ 1 / 2 , g (2 s − 1) if 1 / 2 ≤ s ≤ 1 . W e also let f ( s ) = f (1 − s ) for 0 ≤ s ≤ 1 . Two paths are simply said to be homotopic , if they a r e homotopic rela tive to the end p oints. A lo op in X is a path f : I → X , satisfying f (0) = f (1). F or a sequence of units a nd z e ros ε = ( ε 1 , ε 2 , ε 3 , · · · ) , ε i ∈ { 0 , 1 } define a pa th g ε : I → P ∗ so that the following prop erties hold: (1) g ε (0) = p 1 and g ε (1) = p ∗ , (2) g ε maps [(2 k − 2) / (2 k − 1) , (2 k − 1) / 2 k ] homeomorphically on to S n k − 1 i = n k − 1 I i starting fro m p n k − 1 to p n k for k ≥ 1, and (3) g ε maps [(2 k − 1) / 2 k , 2 k / (2 k +1)] onto S 1 a n k as a winding in the p ositive direction, if ε k = 1, and g ε maps [(2 k − 1 ) / 2 k , 2 k / (2 k + 1)] to the p o in t set { p n k } constantly otherwise, for k ≥ 1. Let h : I → P ∗ be a path fro m p ∗ to p 1 which maps I homeomor phica lly o n to S ∞ i =1 I i ∪ { p ∗ } . Finally , let f ε = g ε h . Then f ε is a lo op with base p oin t p 1 corre- sp onding to a ε = a n 1 ε 1 a n 2 ε 2 a n 3 ε 3 · · · . 3. Proof of Theorem 1.1 F or our pro of of Theorem 1 .1 we shall need the following tw o lemmata: Lemma 3.1. L et C b e t he Case-Chamb erli n c ontinuum emb e dde d in R 3 . Then the quotient sp ac e R 3 /C is homotopy e quivalent t o the 2-dimensional c omp actum P ∗ . Pr o o f. The pro of is completely analogo us to the pro of of the first asser tion of The- orem 1.1 of [6] and therefore we shall o mit it.  Lemma 3. 2. L et p 0 , p 1 , p ∗ b e distinct p oints in a Hausdorff sp ac e X and let f b e a lo op with b ase p oint p 1 such that f − 1 ( { p 0 } ) is empty and f − 1 ( { p ∗ } ) is a singleton. If f is nul l-homotopi c, then ther e exists a lo op f ′ in X \ { p 0 , p ∗ } such that f and f ′ ar e homotopic in X \ { p 0 } . Pr o o f. Since f is null-homotopic, we hav e a homo top y F : I × I → X from f to the constant ma pping to p 1 , i.e. F ( s, 0) = f ( s ) , F ( s, 1) = F (0 , t ) = F (1 , t ) = p 1 for s , t ∈ I . 4 KA TSUY A EDA, UMED H. KARIMOV, AND DU ˇ SAN REPOV ˇ S Let { s 0 } be the sing leton f − 1 ( { p ∗ } ). Let M b e the connectedness comp onent of F − 1 ( { p ∗ } ) co n taining ( s 0 , 0 ), and O the connectedness component of I × I \ M containing I × { 1 } . Define G : I × I → X b y: G ( s, t ) =  F ( s, t ) if ( s, t ) ∈ O , p ∗ otherwise. Then G is also a ho motopy from f to the constant mapping to p 1 and G − 1 ( { p 0 } ) is co n tained in O . Consider G − 1 ( { p ∗ , p 0 } ) ∩ O and I × I \ O . By definition of M , G − 1 ( { p ∗ , p 0 } ) ∩ O is compact and disjoint from ( I × I \ O ) ∪ I × { 0 } . Using a polyg onal neighborho o d of ( I × I \ O ) ∪ I × { 0 } whos e closure is disjoint from G − 1 ( { p ∗ , p 0 } ) ∩ O , w e get a piecewise linear injective path g : I → I × I such that Im( G ◦ g ) ⊆ X \ { p 0 , p ∗ } , g (0) ∈ { 0 } × I , and g (1) ∈ { 1 } × I and Im( g ) divides I × I in to tw o comp onent s, one of which contains G − 1 ( { p 0 } ) and the other contains M ∪ I × { 0 } . W e now see that G ◦ g is the desir ed lo op f ′ .  Pr o o f of Theor em 1 .1. By Lemma 3.1, it clearly s uffices to consider π 1 ( P ∗ ) in- stead o f π 1 ( R 3 /C ). Supp ose therefor e , that the gr oup π 1 ( P ∗ ) were at mos t co un t- able. W e ca n assume that p 1 ” is the base p oint of the space P ∗ and all of its subspaces considered below. Since the set of all sequences of units and zeros is uncountable, there would then exist an uncountable s e t E , such that for every ε, ε ′ from E , the lo ops f ε and f ε ′ with the base p oint p 1 would be homotopy equiv alen t. Fix a lo o p f ε 0 ( ε 0 ∈ E ). Then every lo op f ε f ε 0 is null-homotopic for every ε ∈ E . Since { s : g ε g ε 0 ( s ) = p ∗ } is a singleton, we can apply Lemma 3.2 to g ε g ε 0 . Since f ε f ε 0 is homotopic to g ε g ε 0 in P ∗ \ { p 0 } , we co nclude that f ε f ε 0 is homotopic to a lo op f ′ ε in P ∗ \ { p 0 , p ∗ } , where the homotopy is in P ∗ \ P 0 . Since E is unco un table and P ∗ \ { p 0 , p ∗ } is homotopy eq uiv alent to the b ouquet of tw o circle s S 1 a 1 W S 1 b 1 , that is, π 1 ( P ∗ \ { p 0 , p ∗ } ) is countable, there exis t distinct ε and ε ′ in E such that f ′ ε is homotopic to f ′ ε ′ in P ∗ \ { p 0 , p ∗ } and hence in P ∗ \ P 0 . It follows that f ε f ε 0 is homotopic to f ε ′ f ε 0 and hence f ε is homotopic to f ε ′ in P ∗ \ P 0 . Let k b e the minimal num b er such that ε k 6 = ε ′ k , say ε k = 1 and ε ′ k = 0. Let Y k be the quotient space o f P ∗ \ P 0 by the the clo sed subspace C ( f k +1 , f k +2 , f k +3 , . . . ) ∗ . Consider the pr o jection q : π 1 ( P ∗ \ P 0 ) → π 1 ( Y n k +1 ) and let [ f ε ] and [ f ε ′ ] be the homoto p y classes co n taining f ε and f ε ′ resp ectively . Since a n k +1 , b n k +1 ∈ F n k +1 , F / F n k +1 is a quo tien t group of π 1 ( Y n k +1 ). Then, q ([ f ε ]) = q ( a ε 1 n 1 ) · · · q ( a ε k − 1 n k − 1 ) q ( a n k ) and q ([ f ε ′ ]) = q ( a ε 1 n 1 ) · · · q ( a ε k − 1 n k − 1 ). Since a n k / ∈ F n k +1 , it follows that q ( a n k ) is non-trivial and hence f ε is not ho motopic to f ε ′ in P ∗ \ P 0 . This contradiction sho ws that o ur initial assumption was false a nd therefore π 1 ( P ∗ ) ∼ = π 1 ( R 3 /C ) is indeed a n uncountable gr oup, as asserted.  Question 3. 3. L et C b e t he Case-Chamb erli n c ontinuum emb e dde d in R 3 . Is the first s ingular homolo gy gr oup with inte ger c o efficients H 1 ( R 3 /C ; Z ) of the quotient sp ac e R 3 /C also unc ountable? THE FUNDAMENT AL GROUP π 1 ( R 3 /C ) IS U NCOUNT ABLE 5 4. Acknow ledgements W e w ere supp orted in part b y the J apanese-Slov enian research grant BI–JP/0 3 – 04/2, the Slov enian Resear c h Agency resea rch program No . J1 –6128– 0101–0 4 a nd the Grant-in-Aid for Scientific resear c h (C) of Japan No. 16 540125 . 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Sch upp, Combinato ri al Group Theory , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-V erlag, Berli n 1977. [9] W. Magnu s, A. K arras and D. Solitar, Combinato ri al Group Theory , Do ver, New Y ork, 1976. [10] N. Shrikhande, Homotopy pr op e rt ies of de c omp osition sp ac es , F u nd. Math. 11 6 (1983), 119– 124. [11] L. C. Sieb enmann, Chapman ’s classific ation of shap es. A pr o of using c ol lapsing , Manuscripta Math. 16 (1975), 373–384. School of S cience and Engineering, W aseda University, Tokyo 169-855 5, Jap an E-mail addr ess : eda@logic.i nfo.waseda.ac.jp Institute of Ma thema tics, A cademy of Sciences of T ajikist an, Ul. Ainy 299 A , Dushanb e 734063, T ajikist a n E-mail addr ess : umed-karimo v@mail.ru Institute of M a thema tics, Physics and Mechanics, and F acul ty of Educa tion, Uni- versity of Ljubljan a, P.O.Box 2 964, Ljubljana 10 01, Slovenia E-mail addr ess : dusan.repov s@guest.arnes.si