Recognizing indecomposable subcontinua of surfaces from their complements

RECOGNIZIN G INDECOMPOSABLE SUBCONTINUA OF SURF A CES FR OM THEIR CO MPLEMENTS CLINTON P . C URR Y Abstra ct. W e pro v e t w o theorems whic h allo w one to recognize inde- composable sub contin ua of closed su rfaces without b oundary . If X is a sub contin uum of a closed surface S , w e call the comp onents of S \ X t h e c om pl ementary domains of X . W e prove that a conti nuum X is either indecomp osable or the un ion of tw o indecomp osable contin ua whenever it has a sequence ( U n ) ∞ n =1 of distinct complementary domains suc h that lim n →∞ ∂ U n = X . W e define a slightly stronger condition on the com- plementary domains of X , called the double-p ass c ondition , whic h w e conjecture is equ ival ent to indecomp osability . W e p rov e that this is so for contin ua whic h are not the b oundary of one of their complemen tary domains. 1. Int roduction F o r us, a c ontinuum is a compact connected metric sp ace. A close d sur- fac e is a compact and connected, but not necessarily orienta ble, 2-manifold without b oun dary . W e are in terested in conditions which imply that a sub- con tin uum of a closed su rface is top ologically complicated. F urther, we w ould lik e for these conditions to rely u p on ho w the con tin uum is em b ed- ded in its am b ien t space, rather than to use in ternal c haracteristics of the con tin uum. W e use inde c omp osability as a cr iterion for top ological complexit y . A con tin uum X is de c omp osable if it can b e written as the un ion of a pair of prop er sub con tin ua A and B . The p air ( A, B ) is then called a de c omp osition of X . A contin uum which is n ot decomp osable is called inde c omp osable . Non-degenerate indecomp osable conti nua (i.e., ind ecomp osable con tinua consisting of more than a p oin t) are certainly quite complicated and ha v e a ric h in ternal s tructure. F or example, a n on-degenerate con tin uum X is indecomp osable if and only if, f or all p ∈ X , the c omp osant of p C p = [ { prop er su b cont inua of X con taining p } 2000 Mathematics Subj e ct Classific ation. Primary 54F15; S econd ary 57N05. Key wor ds and phr ases. I ndecomp osable contin uum, complementary domain, closed surface, doub le-pass condition. The auth or was supp orted in part by N SF-DMS-0353825. 1 2 C. P . CURR Y is a dense set of fi rst categ ory [5], in whic h case there are uncountably many disjoin t comp osant s. In con trast, d ecomp osable con tin ua hav e either one or three comp osants. The r ic h stru cture of indecomp osable contin u a provides man y interesting in ternal metho ds o f recognizing indecomposability . W e are instead int er- ested in recognizing indecomp osabilit y b ased on the cont inuum’s in teraction with the space in whic h it lies. This app roac h is in th e spir it of some classical w ork of Kurato wski [7], Ru tt [11], and Burgess [2 ]. In Section 2, w e r ecall s ome earlier theorems which the current w ork ex- tends. In S ection 3, w e extend a theorem of C . E. Bur gess ab out planar con- tin ua. In Section 4 we extend the charact erization of planar indecomp osable con tin ua in [3] to certain su b cont inua of closed surf aces – those su b cont inua whic h are not th e b oundary of an y of their complemen tary domains. Finally , in Section 5 we state questions and conjectures. I would lik e to thank my advisor, Dr. John Ma y er, for many helpful discussions and tireless pro ofreadin g of p r eliminary d r afts. I would also lik e to thank the referee for careful attent ion that improv ed the qu alit y of this pap er. 2. Prior Work and Notions Let S b e a clo sed surface, and X ⊂ S b e a con tin uum. A comp on ent of S \ X is called a c ompl ementary dom ain of X . If X equals the b ound ary of one of its complement ary domains, then X is called unshielde d ; otherwise, it is shielde d . In order to shorten s ome statemen ts, we sa y that a contin u um X is 2- inde c omp osable if X is decomp osable bu t cannot b e written as the essentia l union of three p rop er su b contin u a. By a th eorem of Burgess [2, Theorem 1], a 2-indecomp osable cont inuum is in some sense u niquely decomp osable. Sp ecifically , there exists a decomp osition ( A, B ) of X where A and B are indecomp osable con tin ua. F urther, if ( C, D ) is any other decomp osition, then C and D eac h con tain exactly one of A and B . The fi rst and most famous result in the vei n of this w ork is by Kurato wski. It deals exclusiv ely with con tinua which are un shielded. Theorem 2.1 ([7]) . L et X b e a planar c ontinuum. If X is the c ommon b oundary of thr e e of its c ompl ementary dom ains, then X is either inde c om- p osable or 2-inde c omp osable. C. E. Burgess generalized this and other related theorems. T h e follo wing is a corolla ry to the main result in [2]. It is an improv ement o ver Theorem 2.1 in that it can detect ind ecomp osabilit y in shielded con tin ua. Theorem 2.2 ([2]) . L et X b e a planar c ontinuum, and let ( U n ) ∞ n =1 b e a se quenc e of distinct c omp lementary domains of X . If X = lim n →∞ ∂ U n , then X is either inde c omp osable or 2-inde c omp osable. INDECOMPOSABLE CON TINUA IN SURF A CES 3 Our Theorem 3.7 extends this theorem. It states th at the ab o v e holds not only for cont inua in the plane, but for con tin ua in all closed s u rfaces. T o state the next theorem, w e need some terminology from [3 ]. W e sligh tly extend the d efi nitions to apply to an arbitrary closed surface, whereas the w ork fr om [3] applies only to S 2 . Definition 2.3 (generalized crosscut) . Let S b e a closed surface and U a connected op en sub set with non-degenerate b ound ary . A gene r alize d cr oss- cut of U is a homeomorphic copy A ⊂ U of the op en inte rv al (0 , 1) suc h that A \ A ⊂ ∂ U . Generalized crosscuts are similar to crosscu ts from p rime end theory , ex- cept that th e closure of a cr osscu t is by definition a compact arc. T he closure of a generalized crosscut, ho we v er, need not b e lo cally connected. Definition 2.4 (shadow) . Let A b e a generalized crosscut of U . A comp o- nen t V of U \ A is called a cr osscut neighb orho o d of A . The shadow of A corresp ondin g to V is the set V ∩ ∂ U . Observe that a generalized crosscut has either one or t w o shadows. W e can no w state the double-p ass c ondition . Definition 2.5 (double-pass condition) . Let X ⊂ S b e a con tin uum, and let ( U n ) ∞ n =1 b e a sequence of complementary domains of X . W e sa y that ( U n ) ∞ n =1 satisfies the double-p ass c onditio n wh en , for any choic e of general- ized crosscuts K n of U n , there exists a sequence of shadows S n of K n so that lim n →∞ S n = X . The main result of [3] is the f ollo wing theorem. Theorem 2.6 ([3]) . A c ontinuum X ⊂ S 2 is inde c omp osable if and only if it has a se quenc e ( U n ) ∞ n =1 of c omplementary domains satisfying the double-p ass c ondition. W e extend th is theorem in S ection 4 . Sp ecifically , we pro v e th at a shielde d con tin uum in a surf ace is indecomp osable if and only if it has a sequ ence of complemen tary d omains satisfying the d ouble-pass condition. 3. An Extens ion of a Theorem of Burg ess In this section, w e extend T heorem 2.2 for planar con tin ua to con tin ua in arbitrary closed su rfaces. Namely , we pr o v e that a con tin uum in a su r face whic h is the limit of b ound aries of distinct complemen tary domains is either indecomp osable or 2-indecomp osable. 3.1. Graphs and C at’s Cra dle s. The follo wing lemma giv es us a con v e- nien t wa y to gain in formation ab out the genus of a particular surface. Definition 3.1. The c omplete bip artite gr aph K m,n is th e graph with m + n v ertices, { a 1 , . . . , a m , b 1 , . . . , b n } , and mn edges joining eve ry v ertex in { a 1 , . . . , a m } to ev ery vertex in { b 1 , . . . , a n } . 4 C. P . CURR Y Lemma 3.2. If S is a close d surfac e c ontaining an emb e dding of the gr aph K 3 , 4 g − 1 , then the genus of S i s at le ast g . Pr o of. According to [4], the minimal gen us g of an y surface in which th e graph K m,n can b e embedd ed is l ( m − 2)( n − 2) 4 m . (Here, ⌈ x ⌉ denotes the small- est in teger n suc h that x ≤ n .) Substituting m = 3 and n = 4 g − 1 giv es that the m inimal gen us for a surface to con tain a cop y of K 3 , 4 g − 1 is equal to  g − 3 4  = g .  F or the remainder of this section, let S denote a particular closed su rface of gen us g . Belo w we w ill use th e follo wing conv enient notation: F or an arc A , let In t( A ) d enote the op en su b-arc of A joining the endp oint s of A . Definition 3.3 (cat’s cradle) . Let d 1 , d 3 ∈ S b e distinct p oin ts. A collection of arcs { A α } α ∈ J is called a c at’s cr ad le (b etwe en d 1 and d 3 ) if (1) d 1 and d 3 are the endp oin ts of eac h A α and (2) for distinct α, β ∈ J , Int( A α ) ∩ In t( A β ) = ∅ . In addition, if eac h A α in tersects a set D 2 disjoin t from { d 1 , d 3 } , then { A α } α ∈ J is a cat’s cradle thr ough D 2 . W e n o w prov e a general fact ab out a cat’s cradle through a closed disk in a sur face. O n the face of it, there are many w a ys in which the arcs of a cat’s cradle could b e arranged. It is certainly p ossible for the arcs to int ersect the disk D 2 in parallel c hords (in a wa y that one might call line arly or der e d , defined precisely b elo w). On the other hand, the arcs could b e arranged in suc h a w a y that no c hord in the intersectio n divides the disk b et we en any other tw o c hords in th e in tersecti on. The follo wing lemma sho ws that this second p ossibilit y d o es not o ccur often in a surface of finite gen us. Lemma 3.4. L e t d 1 , d 3 ∈ S , and let D 2 ⊂ S b e a close d disk. Supp ose ( A n ) 4 g +3 n =1 is a c at’s cr ad le b etwe en d 1 and d 3 thr ough D 2 , wher e g is the genus of S . Then no c omp onent of D 2 \ S 4 g +3 n =1 A n c an have closur e which me ets every element of ( A n ) 4 g +3 n =1 . Pr o of. Supp ose some comp onent V of D 2 \ S 4 g +3 n =1 A n has closure w hic h meets ev ery arc A n . Designate a p oin t d 2 ∈ V , and c ho ose arcs ( A ′ n ) 4 g +3 n =1 satisfying the follo wing for all different in dices m and n : (1) A ′ n joins d 2 to a p oint of A n , with no p rop er sub -arc of A ′ n doing so; (2) A ′ n ∩ A ′ m = { d 2 } ; and (3) Int( A ′ n ) ⊂ V for eac h n . The graph S 4 g +3 n =1 ( A n ∪ A ′ n ) is th en homeomorphic to t he graph K 3 , 4 g +3 , b eing the complete b ip artite graph b et w een the sets { d 1 , d 2 , d 3 } and S 4 g +3 n =1 ( A n ∩ A ′ n ). Lemma 3.2 indicates that the gen us of S is at least g + 1, contradicti ng the assump tion that it is g .  Lemma 3.6 will b e used in the p r o of of Theorem 3.7 to bu ild interesting graphs in S . T o state it, w e n eed a d efi nition. INDECOMPOSABLE CON TINUA IN SURF A CES 5 Definition 3.5 (linearly ordered) . Let ( A ′ α ) α ∈ J b e an ordered sequence of pairwise disjoin t compact arcs in a closed disk D 2 , irreducible with resp ect to intersect ing ∂ D 2 t wice. W e sa y that ( A ′ α ) α ∈ J is line arly or der e d if, for α < β < γ in J , A ′ α is separated from A ′ γ in D 2 b y A ′ β . If J is finite, we call a comp onent of D 2 \ S n ∈ J A ′ n an end c omp onent if its closure meets only one elemen t of ( A ′ n ) n ∈ J . In what follo ws a chor d of a d isk D 2 is an arc in D 2 whic h inte rsects ∂ D 2 exactly at its end p oints. Lemma 3.6. L e t ( A n ) ∞ n =1 b e a c at’s cr ad le fr om d 1 to d 3 thr ough a close d disk D 2 , and supp ose that A n ∩ ∂ D 2 is finite for e ach n . Th en ther e is a subse qu e nc e ( A n i ) ∞ i =1 and a c ol le ction of sub-ar cs ( A ′ n i ) ∞ i =1 , A ′ n i ⊂ A n i ∩ D 2 , so that ( A ′ n i ) ∞ i =1 is line arly or der e d in D 2 . Pr o of. First, using Lemma 3.4, notice that only finitely man y comp onen ts of D 2 ∩ S ∞ n =1 A n do not meet D 2 in its in terior. Hence, by passing to a subsequence, w e ma y assume that D 2 \ A i is not connected for an y i ∈ N . In p articular, A i ∩ D 2 con tains a non-degenerate c hord for every i ∈ N and therefore separates D 2 . There are t w o cases. Case 1. One case is that, for an y elemen t A α , th er e is an elemen t A β suc h that n o elemen t A γ separates A α from A β in D 2 . W e will build by induction an increasing sequence ( n k ) ∞ k =1 and a corresp onding sequence of c hords ( A ′ n k ) ∞ k =1 with A ′ n k ⊂ A n k ∩ D 2 suc h that, for all k ≥ 1, the follo wing t w o conditions are met: (1) Th e c hords A ′ n 1 , . . . , A ′ n k are linearly ordered; and (2) some end comp onent of D 2 \ S k i =1 A ′ n i b ound ed b y A ′ n k in tersects infinitely many elemen ts of ( A n ) n>n k . (Notice th at there is a unique suc h end comp onent if k 6 = 1.) F or the base case of our in duction, set n 1 = 1, and let A ′ n 1 ⊂ A n 1 b e an y non-degenerate c hord in D 2 . Th er e are then tw o comp onen ts to D 2 \ A ′ n 1 , eac h an end comp onent of D 2 \ A ′ n 1 . Infin itely many elemen ts of the collect ion ( A n ) n ≥ n 1 m ust intersect one of them. W e ha v e therefore satisfied the requirements for k = 1. No w supp ose that c hords A ′ n 1 , . . . , A ′ n k ha v e b een found whic h satisfy conditions 1 and 2. Consider the collection { A N 1 , . . . , A N m } of arcs whic h are n ot separated f rom A ′ n k b y an y other A n . (Th er e ma y only b e fi nitely man y by Lemma 3.4.) Let V denote the end comp onen t of D 2 \ S k i =1 A ′ n i meeting A ′ n k Because eac h A N i in tersects ∂ D 2 in fi nitely man y p oin ts, we see that S m i =1 A N i divides V into finitely man y comp onents. By c hoice of N 1 , . . . , N m , no m emb er of ( A n ) n> max( N 1 ,...,N m ) in tersects the comp onent of V \ S m i =1 A N i whic h meets A ′ n k . T herefore, infinitely many member s must in tersect an- other comp onent W of V \ S m i =1 A N i . Let A ′ n k +1 b e a minimal s ub-arc of 6 C. P . CURR Y S m i =1 A N i whic h separates W f r om A ′ k . Then W is contai ned in the end comp onent of D 2 \ S k +1 i =1 A ′ n i meeting A ′ n k +1 . That end comp onen t meets infinitely man y mem b ers of ( A n ) n>n k +1 , so we h a v e extended our linearly ordered collectio n to k + 1 elemen ts. By induction, w e ha v e a s equ ence of c hords ( A ′ n i ) ∞ i =1 . S ince eac h fi nite subsequence ( A n i ) k i =1 is lin early ordered, ( A n i ) ∞ i =1 is lin early ordered. Case 2. T here exists A α suc h that any A β is separated from A α in D 2 b y some A γ . Cho ose n 1 > α , and let A ′ n 1 ⊂ A n 1 b e an y non-degenerate c hord. Find a sequence ( A n i ) ∞ i =1 suc h that A n i is separated from A ′ α b y A n i +1 . Recall that D 2 , as a tw o-dimensional disk, is uni c oher ent , which implies that an y closed set w hic h separates t w o p oints of D 2 also has a comp onen t whic h do es so. Therefore, a chord A ′ n i +1 ⊂ A n i +1 also separates A n i from A ′ α . Th is pro cess inductiv ely yields the ordered sequen ce ( A n i ) ∞ i =2 , whic h is evident ly linearly ordered.  3.2. Burgess’s Theorem for Surfaces. No w w e mo v e to the sp ecific set- ting of a cont inuum with infi nitely m an y complement ary domains. Theorem 3.7 (Burgess’s Theorem for S urfaces) . L et X b e a sub c ontinuum of a close d su rfac e S , and let ( U n ) ∞ n =1 b e a se quenc e of distinct c omplemen- tary domains of X . If X = lim n →∞ ∂ U n , then X is either inde c omp osable or 2-inde c omp osable. Pr o of. By [2 , Th eorem 1], we must only show th at X is not the essen tial union of three pr op er sub cont inua. By wa y of con tradiction, su pp ose that X = X 1 ∪ X 2 ∪ X 3 , where eac h X i is a prop er sub con tin uum of X not con tained in S j 6 = i X j . W e will fir st fin d app ropriate closed disks D 1 , D 2 , and D 3 . Designated p oint s in D 1 and D 3 will fun ction as the end p oints of a cat’s cradle throu gh D 2 . By the definitions of X 1 , X 2 , and X 3 , there exist closed disk s D 1 , D 2 , D 3 ⊂ S suc h that (1) the inte rior of D i in tersects X i , and (2) D i is d isjoin t from D j ∪ X j when i 6 = j . Because of the increasing densit y of ( U n ) ∞ n =1 , U n in tersects eac h D i when n is large enough. By passing to a sub sequence, assume that U n ∩ D i 6 = ∅ for eac h n ∈ N and i ∈ { 1 , 2 , 3 } . Then for eac h n ∈ N let A n ⊂ U n b e an arc suc h th at, for eac h i ∈ { 1 , 2 , 3 } , (1) A n ∩ D i 6 = ∅ , (2) n o prop er sub-arc of A n in tersects eac h D i , and (3) A n ∩ ∂ D i is finite. Note that the last condition can b e achiev ed since eac h U n is op en. It is evident that Int( A n ) in tersects only one D i for eac h n . By p assin g to a subsequen ce and relab eling the d isks, w e can assume without loss of generalit y th at Int( A n ) ∩ D 2 6 = ∅ . Extend eac h A n to an arc ˜ A n with arcs in INDECOMPOSABLE CON TINUA IN SURF A CES 7 D 1 and D 3 to obtain a cat’s cradle from d 1 to d 3 through D 2 , where d i is a designated p oin t of D i . Notice that S ∞ n =1 ˜ A n is s till disjoint from X 2 , sin ce ˜ A n \ ( D 1 ∪ D 3 ) ⊂ U n is disj oin t from X and ˜ A n ∩ ( D 1 ∪ D 3 ) ⊂ D 1 ∪ D 3 is disjoin t fr om X 2 . Let ( A ′ n i ) ∞ i =1 b e the linearly ordered sequ ence of compact arcs in  S ∞ n =1 ˜ A n  ∩ D 2 guaran teed by Lemma 3.6. First, notice that X 2 separates A n i ∩ D 2 from A n j ∩ D 2 in D 2 when i 6 = j . T o see this, observe that A n i ∩ D 2 ⊂ U n i , and analogously A n j ⊂ U n j . These are d istinct complemen tary domains of X , so X separates A n i from A n j in S and thus in D 2 . Since X ∩ D 2 ⊂ X 2 , w e see that X 2 separates A n i from A n j in D 2 . Accordingly , c ho ose p oin ts x 1 , . . . , x 4 g +3 ∈ X 2 so that (1) eac h x i lies b et w een A ′ n i and A ′ n i +1 in D 2 , and (2) the set S j 6 = i A n j do es n ot separate x i from A n i in D 2 . Since X 2 is connected and X 2 ∩ S 4 g +4 j =1 A n j = ∅ , there exists an arc J ⊂ S \ S 4 g +4 j =1 A n j con taining { x i } 4 g +3 i =1 . By c hoice of x i , there are disjoint arcs ( A ′′ n i ) 4 g +3 i =1 ⊂ D 2 whic h join J to A n i without in tersecting an y other A ′ n j . By collapsing the arc J to a p oin t, the graph K 3 , 4 g +3 is obtained in G = J ∪ 4 g +3 [ i =1 ( ˜ A n i ∪ A ′′ n i ) , i.e., K 3 , 4 g +3 is a minor of G . Lemma 3.2 concludes that the gen us of S is at least g + 1, con tradicting our assump tion that the gen us is g .  3.3. A P artial Con v erse. As can b e exp ected, Theorem 3.7 has a partial con v erse: If X is indecomp osable, then there exists a sequence ( U n ) ∞ n =1 of complemen tary domains of X , not necessarily distinct, such th at lim n →∞ ∂ U n = X . T h is fact will b e used in the next section, so we p ro v e it here. The pro of of this pr op erty follo ws closely the outline of [3 , Theorem 2.10], mo dified sligh tly to allo w for a finite degree of multic oher e nc e . Definition 3.8 (m ulticoheren t) . A connected top ological sp ace X is multi- c oher ent of de gr e e k if, f or an y p air of closed, connected su bsets A and B such that A ∪ B = X , the inte rsection A ∩ B consists of at most k comp onen ts. Closed su rfaces and pu nctured closed sur faces are examples of finitely m ulticoheren t spaces. Let S b e a fin itely multico herent s pace, and sup p ose that A ⊂ S is a closed set whic h separates p from q f or p oin ts p, q ∈ S . Then, by [12, Th eorem 1], th ere is a closed subset B ⊂ A w hic h has at most k comp onents whic h also separates p from q . W e use this pr op erty in the follo wing pro of. Here, if A and B are compact non-empty subsets of a metric sp ace ( X, d ), H d ( A, B ) r epresen ts the Hausdorff distance b etw een A and B in the hypersp ace of non-empt y compact sub sets of X . See [10] for details. 8 C. P . CURR Y Theorem 3.9. L et X b e an inde c omp osable c ontinuum in the close d sur- fac e S . Then ther e exists a se quenc e ( U n ) ∞ n =1 of (not ne c essarily distinct) c omplementary dom ains of X such tha t lim n →∞ ∂ U n = X . Pr o of. This is clear if X is a p oint , s o assume X is a non-degenerate in d e- comp osable con tin uum. F or the purp oses of this pr o of, sup p ose that S is equipp ed with a metric d in which th e set B ǫ ( p ) = { x ∈ S | d ( p, x ) < ǫ } is s imply connected wh en ǫ ∈ (0 , 1). Let p , q , and r lie in d ifferen t com- p osant s of X . F or eac h n ∈ N , define Q n = the comp onent of X \ B 1 /n ( p ) conta ining q , and R n = the comp onent of X \ B 1 /n ( p ) conta ining r. Notice th at lim n →∞ Q n = lim n →∞ R n = X , by density of comp osan ts. Since Q n and R n are differen t comp onen ts of X \ B 1 /n ( p ), they are separated in S \ B 1 /n ( p ) b y S \ ( B 1 /n ( p ) ∪ X ). Thus, Q n and R n are closed and separated in the normal space S \ B 1 /n ( p ), so there is a sub set K n , closed in S \ B 1 /n ( p ), of S \ ( B 1 /n ( p ) ∪ X ) wh ic h separates Q n and R n . Sin ce S \ B 1 /n ( p ) is finitely m ulticoheren t, sa y of degree k , then a su bset L 1 n ∪ . . . ∪ L k n is a closed separator (in S \ B 1 /n ( p )) of Q n and R n , wh er e the elemen ts of the union are disjoint, closed, and connected (though p erhaps some are empt y). W e may assume that they are ordered so that H d ( L 1 n , X ) ≤ . . . ≤ H d ( L k n , X ) , where H d ( ∅ , X ) can b e regarded as ∞ . Moreo v er, since eac h L i n is disj oin t from X and connected, there exists a complement ary domain U i n suc h th at L i n ⊂ U i n . (The set U i n can b e an y complemen tary domain of X if L i n is empty .) The sequence ( U 1 n ) ∞ n =1 formed in this w a y is the requ ir ed sequ en ce of complemen tary domains. W e w ill sh o w that any conv er gent su b sequence of ( ∂ U 1 n ) ∞ n =1 con v erges to X , implying th at the sequence itself conv erges to X . First w e demonstrate that X ⊂ lim inf n →∞ S k i =1 ∂ U i n . Cho ose x ∈ X \ { p } and 0 < ǫ < d ( x, p ). Let N ∈ N such th at, for all n ≥ N (1) Q n ∩ B ǫ ( x ) 6 = ∅ , (2) R n ∩ B ǫ ( x ) 6 = ∅ , and (3) B 1 /n ( p ) ∩ B ǫ ( x ) = ∅ . F or n ≥ N , c ho ose q n ∈ Q n ∩ B ǫ ( x ) and r n ∈ R n ∩ B ǫ ( x ). Let A n ⊂ B ǫ ( x ) ⊂ S \ B 1 /n ( p ) b e an arc joining q n to r n . Th en A n ∩ k [ i =1 L i n 6 = ∅ , INDECOMPOSABLE CON TINUA IN SURF A CES 9 since S k i =1 L i n separates q n from r n in S \ B 1 /n ( p ). Since L i n ⊂ U i n for eac h i and q n , r n are n ot in any U i n (they lie in X ), A n ∩ k [ i =1 ∂ U i n 6 = ∅ , implying that k [ i =1 ∂ U i n ∩ B ǫ ( x ) 6 = ∅ . This is true for all n ≥ N , so x ∈ lim inf n →∞ S k i =1 ∂ U i n , and w e ha v e that X = lim n →∞ S k i =1 ∂ U i n . No w, let u s consid er the individual limits lim n →∞ ∂ U i n for some fixed i ≤ k . By passin g to a su bsequence, w e ma y assu me that the limit X i = lim n →∞ ∂ U i n exists f or eac h i ≤ k . Th en, since lim n →∞ S k i =1 ∂ U i n = X , we see th at S k i =1 X i = X . Ho w ev er, this is a finite union of cont inua, and X is in decomp osable, s o at least one X i is n ot a prop er su b contin u um of X . By con tin uit y , w e s ee th at H d ( X 1 , X ) ≤ H d ( X i , X ) for all i ≤ k , so X 1 = lim n →∞ ∂ U 1 n = X .  4. Characteriza tion of Shield ed Inde composa ble Continua A sub con tin uum X of a closed su rface S is called shielde d if, for ev ery complemen tary domain U of X , ∂ U 6 = X . W e extend Theorem 2.6, wh ic h is a c haracterizatio n of planar indecomp osable conti nua, to shielded s ub con- tin ua of closed su rfaces. The assumption that a conti nuum is s h ielded already imparts some com- plexit y . F or instance, an y sequence of complemen tary domains whose b ound - aries conv erge to the con tin uum must consist of infi nitely many distinct el- emen ts. W e use this p rop erty to bridge the gap left b y the r elativ ely we ak separation pr op erties of compact su rfaces. The pro ofs in this section will d ep end up on the existence of particularly w ell-b eha v ed homeomorphism s of simply connected domains in a surface to the u n it disk D in th e plane. Sp ecifically , we w ish for a n ull sequ en ce of crosscuts in a domain to corr esp ond to a null sequence of crosscuts in D . W e will prov e that conformal isomorp hisms hav e this pr op erty , so for the remainder w e will assume that the closed surface S is endo w ed with a conformal stru cture and that d is the corresp ondin g m etric. Lemma 4.1. If X is a non-de gene r ate c ontinuum in S , then al l of its simply c onne cte d c omplementary domains ar e c onformal ly isomorphic to the unit disk. R emark. Note that, in con trast to the planar case, some complemen tary domains may not b e simply connected. 10 C. P . CURR Y Pr o of. Let P b e the un iv ersal co v ering space of S , with corresp onding confor- mal co v ering map π : P → S . Recall that, by the Un iformization Theorem [9, Theorem 1.1], P is conformally isomorph ic to a simply connected subset of the Riemann sphere. If ˆ U is a comp onen t of π − 1 ( U ), then ˆ U is simply connected and π | ˆ U is a co v ering map. In fact, π | ˆ U is a conformal isomor- phism since its trivial fun dament al group is isomorph ic to the group of d ec k transformations for π | ˆ U . Ho w ev er, ˆ U misses π − 1 ( X ), so ˆ U is conformally isomorphic to the u nit d isk by th e classical Riemann mapping theorem.  The p ro of of the follo win g lemma is iden tical to the pro of of [3, L emma 3.4], and is includ ed here for completeness. Lemma 4.2. L e t U b e a simply c onne cte d op en subset of S with non- de gener ate b oundary. L et φ : U → D b e a c onform al isomorphism. Then the image of a nul l se quenc e ( K n ) ∞ n =1 of cr osscuts of U is a nul l se q uenc e of cr osscuts in D . Pr o of. Let ( K n ) ∞ n =1 b e a null sequence of crosscuts of U w ith image sequence ( A n ) ∞ n =1 = ( φ ( K n )) ∞ n =1 . Without loss of generalit y , assume that ( K n ) ∞ n =1 con v erges to a p oint x ∈ ∂ U . By passing to a sub sequence, we m a y assu m e that the image sequence con v erges to a con tin uum L ⊂ D . Since ( K n ) ∞ n =1 do es n ot accum ulate on a subset of U , w e s ee that L ⊂ ∂ D . Let t ∈ L . There exists a c hain of crosscuts ( A ′ n ) ∞ n =1 of D con v erging to t whic h maps to a n ull sequence ( K ′ n ) ∞ n =1 of crosscuts of U by φ − 1 . (See [9, Lemma 17.9]; the pro of do es n ot rely on the planarit y of U .) W e ma y assume that ( K ′ n ) ∞ n =1 con v erges to a p oin t of ∂ U by passing to a subsequence. F or eac h m ∈ N , that ( A n ) ∞ n =1 accum ulates on t implies that all bu t fi nitely man y A n in tersect the crosscut n eighb orh o o d of A ′ m corresp ondin g to t . Also, sin ce ( K n ) ∞ n =1 forms a n ull sequence in U and conv er ges to x , w e see that all b u t finitely man y K n (th us A n ) lie en tirely within the crosscut n eigh b orho o d of K ′ m (th us A ′ m ) corresp onding to t . Ho w ev er, the crosscut neigh b orh o o ds of A ′ m form a null sequen ce as m → ∞ , so ( A n ) ∞ n =1 form a null sequen ce.  Lemma 4.3. Supp ose U ⊂ S is op e n, c onne cte d, and simply c onne c te d with non-de gener ate b oundary, and let φ : U → D b e a c onformal isomorphism. L et B 1 and B 2 b e disjoint close d disks me eting ∂ U , and let E i ⊂ ∂ D denote the set of endp oints of the cr osscuts of D which form φ (( ∂ B i ) ∩ U ) . If, for differ ent i and j , E i lies in a c omp onent of ∂ D \ E j , then ther e is a gener alize d cr osscut K of U which sep ar ates B 1 ∩ U f r om B 2 ∩ U in U . M or e over, if ∂ U is lo c al ly c onne cte d, then K is a cr osscut of U . Pr o of. Ident ical to pro of of Lemma 3.5 in [3].  Theorem 4.4. A shielde d sub c ontinuum X of a close d surfac e S is inde- c omp osable if and only if it has a se quenc e of c omplementary domains which satisfies the double-p ass c ondition. Pr o of. Supp ose fi rst that X is indecomp osable. By Theorem 3.9, there is a sequence ( U n ) ∞ n =1 of complementa ry d omains of X such th at lim n →∞ ∂ U n = INDECOMPOSABLE CON TINUA IN SURF A CES 11 X . F or eac h n ∈ N , let K n b e a generalized crosscut of U n . Let A n and B n b e shadows of K n so that A n ∪ B n = ∂ U n , with H d ( A n , X ) ≤ H d ( B n , X ). Then A n and B n are su b cont inua of X . Th ere is a su bsequence ( U n i ) ∞ i =1 so that ( A n i ) ∞ i =1 and ( B n i ) ∞ i =1 con v erge to sub cont inua A and B of X . W e see that A ∪ B = X , since A n i ∪ B n i = ∂ U n i . S ince X is ind ecomp osable, either A or B is not a prop er sub cont inuum of X . W e hav e H d ( X, A n i ) ≤ H d ( X, B n i ), so A = X . Th is is tru e for all c hoices of su bsequences ( U n i ) ∞ i =1 where ( A n i ) ∞ i =1 and ( B n i ) ∞ i =1 b oth con v erge, so ( U n ) ∞ n =1 satisfies the doub le-pass condition. No w, assume that a conti nuum X ⊂ S has a sequence ( U n ) ∞ n =1 of comple- men tary domains satisfying th e double-pass condition. By wa y of con tra- diction, sup p ose that ( X 1 , X 2 ) is a decomp osition of X . T here are disjoint closed disks D 1 , D 2 ⊂ S so th at, for i ∈ { 1 , 2 } , (1) the inte rior of D i in tersects X i , and (2) if i 6 = j , D i is d isjoin t from X j ∪ D j . Since X satisfies the double-pass condition, th er e exists N ≥ 1 such that, for all n ≥ N , no generalized crosscut of U n separates D 1 ∩ U fr om D 2 ∩ U in U n . Without loss of generalit y , N = 1. No w we make us e of th e assu mption that X is shielded. S in ce ∂ U n 6 = X for eac h n and lim n →∞ ∂ U n = X , no elemen t of ( U n ) ∞ n =1 app ears infi nitely often in th e sequence. Therefore, we can assume by passing to a s u bsequence that ( U n ) ∞ n =1 consists of differen t simply connected complementary domains of X . Let φ n : U n → D b e a conformal isomorph ism. According to L emma 4.3, the sets E n, 1 and E n, 2 , comprised of the endp oints of the crosscuts consti- tuting φ n (( ∂ D 1 ) ∩ U n ) and φ n (( ∂ D 2 ) ∩ U n ) separate eac h other in ∂ D . It is eviden t that D 1 ∩ U n and D 2 ∩ U n ma y or ma y n ot s eparate th e other in U n . By passing to a s ubsequence, we can assume that, for ev ery n ∈ N , either (1) D 1 ∩ U n separates D 2 ∩ U n in U n , or (2) n either separates the other in U n . W e will find a crosscut F n ⊂ D \ φ ( D 2 ) of D wh ich joins p oin ts of E n, 1 whic h separate E n, 2 in ∂ D . In the case that D 1 ∩ U n separates D 2 ∩ U n , a comp onent of ( ∂ D 1 ) ∩ U n also do es, so we can define F n as a comp onen t of φ (( ∂ D 1 ) ∩ U n ) w hic h separates φ n ( D 2 ∩ U n ) in D . In the second case, there are comp onents K 1 and K 2 of φ n (( ∂ D 1 ) ∩ U n ) whose endp oint s separate E 2 ,n in ∂ D . Let F n b e the u nion of an arc joining k 1 ∈ Int( K 1 ) to k 2 ∈ Int( K 2 ) with one comp onent eac h of K 1 \ { k 1 } and K 2 \ { k 2 } . Notice that φ − 1 n ( F n ) is in fact a crosscut of U n in either case, sin ce p oints of F n close to ∂ U n are in ∂ D 1 . Let A n b e an arc in U n joining comp onent s of D 2 ∩ U n whic h are sep arated b y φ − 1 n ( F n ), in tersecting φ − 1 n ( F n ) tr ansv ersely exactly once. Let C n ⊂ D 2 ∪ A n b e a simple closed curve con taining A n , formed b y joining th e endp oin ts of A n to a designated p oint a ∈ D 2 with line s egmen ts otherwise disjoint from C 1 ∪ . . . C n − 1 . 12 C. P . CURR Y Let Y = C 1 ∪ C 2 ∪ . . . ∪ C 2 g +1 , wh ere g is the genus of S . Th is is the one-p oin t union of 2 g + 1 simp le closed curv es, so S \ Y is n ot connected (see for in s tance [1, 5.14]). Since the arcs F 1 , . . . , F 2 g +1 in tersect Y exactly once tr ansv ersely and hav e their endp oint s in X 2 , w e conclude that eac h comp onent of S \ Y con tains p oints of X 1 . Ho we ve r, since Y is the union of arcs in S \ X and B 2 , w e see that Y is disj oin t from X 1 . Th is contradicts the assump tion that X 1 is connected.  R emark. Su pp ose that th e con tin uum in q u estion h as lo cally conn ected b ound ary comp onen ts. Then Lemma 4.3 provides a crosscut (rather than a generalized crosscut) to sho w th at X fails the double-pass condition. Hence, w e can observ e that a con tin uum w ith lo cally connected b oun dary comp o- nen ts is ind ecomp osable if and only if it satisfies the double-pass condition with crosscuts. 5. Concl usion Though the results p resen ted here are adv ances, the corresp onding the- orems for p lanar con tin ua are muc h stronger. Here we d iscuss stronger generalizat ions whic h ma y hold. 5.1. Burgess’s The orem. T h eorem 3.7 is a direct tr anslation of the corol- lary to [2, Th eorem 10]. In turn, that theorem f ollo w s fr om a fund amen tal theorem of C. E. Burgess, repr o duced b elo w. Theorem. [2, T heorem 9] L et H ⊂ S 2 b e close d, M ⊂ S 2 a c ontinuum, and M 1 , M 2 , and M 3 sub c ontinua of M . Supp ose that K 1 , K 2 , and K 3 ar e close d disks, disjoint fr om e ach other and H , and K i interse cts M j if and only if i = j . Then ther e do not exist thr e e c omplementar y domains of M ∪ H al l of which interse ct K 1 , K 2 , and K 3 . Question 5.1 . Is there an extension of this theorem to closed surfaces? Using Burgess’s theorem, one can pro v e that a planar contin u um which is the common b oundary of three of its complementa ry domains [7] or is the impression of one of its prime ends [11] is either indecomp osable or 2- indecomp osable. Also, a planar con tin uum w hic h is the limit of a disjoin t sequence of sh ado ws is indecomp osable [3]. Theorem 3.7 was prov en here without proving the analog of Burgess’s theorem, but interesti ng theorems w ould su rely follo w if Q uestion 5.1 h ad a p ositiv e answ er. 5.2. Kuratowski’s Theorem. In particular, the requirement of Theorem 3.7 that X ha v e infi nitely man y d istinct domains w hose b oun daries limit to the con tin uum is p r obably str on ger th an necessary for th e conclusion. This motiv ates the follo wing question. Question 5.2 . Is there a finite v ersion of T heorem 3.7 lik e K urato wski’s theorem? S p ecifically , if a con tin uum X in a closed su rface of gen us g is the common b ound ary of thr ee complemen tary domains, wh at is the maximal n ( g ) so that X is the union of n ( g ) indecomp osable con tin ua? INDECOMPOSABLE CON TINUA IN SURF A CES 13 A p artial answ er is illustrated b y an example. Let C 1 , C 2 , C 3 ⊂ C ∞ \ {∞} b e homeomorphic copies of the p seudo circle su c h that, for distinct i, j, k , (1) C i is con tained in the closure of the comp on ent of S 2 \ C j con taining ∞ , (2) C i ∩ C j is a single p oint not in C k , and (3) C i ∩ C k is in a differen t comp osan t of C k than C j ∩ C k . This is a con tin uum with 5 complementa ry domains. Tw o complemen tary domains, U and V , meet eac h element of { C 1 , C 2 , C 3 } . Th ere are thr ee complemen tary domains, W 1 , W 2 , and W 3 , wh er e W i is the b ounded com- plemen tary domain of C i . No p rop er su b conti nuum of C 1 ∪ C 2 ∪ C 3 can separate U f r om V b ecause of condition 3 , so ∂ U = ∂ V = C 1 ∪ C 2 ∪ C 3 . Remov e d isks D 1 , D ′ 1 ⊂ W 1 , D 2 ⊂ W 2 , and D 3 ⊂ W 3 , and paste cylinders joining ∂ D 1 to ∂ D 2 and ∂ D ′ 1 to ∂ D 3 in a w a y resulting in an orien table surf ace S of gen us 2. Th en C 1 ∪ C 2 ∪ C 3 is the common b ound ary of three domains, th ough it is not the union of j ust tw o indecomp osable con tin ua. This indicates that an extension of Ku rato wski’s theorem along these lines m ust tak e the gen us into account . 5.3. Characterization Theorem. Theorem 4.4 is significantly w eak er than Theorem 2.6 , since Theorem 2.6 holds for con tin ua wh ic h are b oundaries of connected op en sets in S 2 . The au th or susp ects the statemen t of T heo- rem 2.6 holds tru e in sur faces, w ith some qualification. F or instance, an essen tially em b edd ed s imple closed curv e in a torus satisfies the doub le-pass condition as it is stated h ere, but artificially so. One can define a gener alize d cr osscut with curves of a domain U as the disjoin t u nion of a generalized crosscut and a finite num b er of simple closed curv es. By definin g shado ws in analogy to shado ws of generalized crosscuts, one obtains an equiv alen t notion in simp ly connected domains – ad d ing disjoin t simple closed curv es to a generalize d crosscut do es not c hange the shado ws. Ho we ve r, generalized crosscuts with curv es allo ws stronger sepa- ration in domains wh ic h are not simply connected. Question 5.3 . Let X b e a con tin uum in a closed sur face S . I f for eve ry sequence ( C i ) ∞ i =1 of generalized crosscuts with cu r v es there exists a c hoice of shado ws ( S i ) ∞ i =1 , wh ere S i a shadow of C i , wh ic h con v erge to X , is X indecomp osable? 5.4. Higher Dimensions. On e ma y ask ho w useful recognition from the complemen t might b e in m ore general s paces. Sp ecifically , in R 3 , th is ap- proac h d o es not lo ok useful. F ormulati ons of prime end theory in R 3 ha v e had limited applicabilit y . F urther, M. Luba ´ nski constructed in [8] a family of abs olute n eigh b orho o d retracts w hic h can b e the common b ound ary of an y fin ite n umber of domains in R 3 . Ku rato wski [7, p . 560] n oted that this can b e extended to an in finite num b er of complementary domains. 14 C. P . CURR Y Referen ces [1] P . S. Aleksandrov. Combinatorial top olo gy. Vol. 1 . Graylock Press, Ro chester, N. Y., 1956. [2] C. E. Burgess. Con tinua and t h eir complemen tary domains in th e plane. Duke Math. J. , 18:901–917, 1951. [3] Clinton P . Curry , John C. Ma yer, and E. D. Tymc hatyn. Characterizing indecom- p osable p lane continua from t heir complements. Pro c. A mer. Math. So c., 2008, to app ear. [4] F rank Harary . Gr aph the ory . Addison-W esley Pub lishing Co., Reading, Mass.-Menlo P ark, Calif. -London, 1969. [5] J. Ho cking and G. Y oung. T op olo gy . A ddison-W esley , R eading, MS, 1961. [6] J. K rasinkiewicz. On internal comp osants of indecomp osable plane contin ua. F und. Math. , 84:255– 263, 1974. [7] K. Kuratow ski. T op olo gy , volume I I. A cademic Press, New Y ork, 1968. [8] M. Lu ba ´ nski. An example of an absolute n eighbou rh oo d retract, which is the common b oundary of th ree regions in the 3-d imensional Euclidean space. F und. Math. , 40:29– 38, 1953. [9] J. Milnor. Dynamics in One Complex V ariable . The Annals of Mathematics Stu dies. Princeton Universit y Press, Princeton, NJ, 3rd edition, 2006. [10] S. B. Nadler, Jr. Continuum the ory , vol ume 158 of Mono gr aphs and T extb o oks in Pur e and Applie d Mathematics . Marcel Dekker I n c., New Y ork, 1992. An in tro duction. [11] N. E. Rutt. Prime ends and ind ecomp osability . Bul l . A mer. Math. So c. , 41:265–273, 1935. [12] A. H . St on e. Incidence relations in multico herent spaces. I. T r ans. Amer. Math. So c. , 66:389– 406, 1949. (Clin ton P . Curry) Dep ar tment of Ma th ema ti cs, Universi ty of Alabama a t Birmingh am, Bi rmingham, AL 35294-1170 E-mail addr ess , Clinton P . Curry: clintonc@ua b.edu