A short proof of nonhomogeneity of the pseudo-circle

A SHOR T PR OOF OF NONHOMO GENEITY OF THE PSEUDO-CIR CLE KR YSTYNA KUPERBERG AND KEVIN GAMMON De dic ate d to James T. R o gers, Jr. on the o c c asion of his 65th birthday Abstra ct. The p seudo-circle is known to be nonhomogeneous. The original proofs of th is fact w ere disco vered ind ep en dently b y L. F earnley [6] and J.T. Rogers, J r. [17]. The purp ose of this paper is to provide an alternative, very short proof based on a result of D. Bellamy and W. Lewis [4]. 1. Introduction A pseudo-ar c is a hereditarily indecomp osable, c hainable con tinuum. In 194 8, E.E. Moise [16] constructed a ps eu do-arc as an indecomp osable con tin uum homeomorphic to eac h of its sub- con tinua. Moise correctly conjectured that th e hereditarily indecomp osable con tinuum giv en b y B. Knaster [11] in 1922 is a pseudo-arc. Also in 1948, R.H. Bing [1] prov ed th at Moise’s exam- ple is homogeneous. In 1951, Bing [2] p ro v ed that ev ery hereditarily indecomp osable chainable con tinuum is a pseudo-arc and that all pseudo-arcs are homeomorphic. In 1959, Bing [3] ga ve another charact erization of the p seudo-arc: a homogeneous c hainable contin uum . The history of many other asp ects of the pseudo-arc can b e found in su rv ey p ap ers b y W. Lewis [14] and [15 ]. In 195 1, Bing [2] describ ed a p seudo-cir cle , a planar hereditarily indecomp osable circularly c h ainable con tinuum whic h separates the plane and wh ose ev ery prop er su b cont inuum is a pseud o- arc. It has b een sho wn b y L. F earnley in [6] and J. T. Rogers, Jr. in [17] that the pseudo-circle is not h omogeneous. F earnley also pr o ved that the pseudo-circle is uniqu e [5 ] and [7]. T he fact that the pseud o-circle is not homogeneous also follo ws from more general theorems prov ed in [8], [10], [13], and [18]. This p ap er offers y et another, v er y sh ort p ro of, a consequence of a r esult of D. Bellam y and W. Lewis [4]. Similarly as in [18], an infinite co ve ring sp ace of a plane s eparating cont inuum is used. 2. Preliminaries Throughout the pap er, a c ontinuum will r efer to a nondegenerate compact and connected m et- ric space. A con tinuum is i nde c omp osable if it is n ot th e union of tw o prop er su b con tinua. A con tinuum is her e ditarily inde c omp osable if ev ery sub con tin uum is also in d ecomp osable. F or a p oint a in X , the c omp osant K ( a ) of a in X is the union of all prop er sub cont inua of X cont ain- ing a . An indecomp osable contin uum con tains uncounta bly many pairwise disjoint comp osan ts, see [12] Theorem 7, page 21 2. 2000 Mathematics Subje ct Classific ation. 54F15; 54F50. Key wor ds and phr ases. p seudo-circle, pseudo-arc, homogeneous, comp osan t, indecomp osable contin uum. 1 2 KR YSTYNA KU PERBERG AND K EVIN GAMMON A top ologica l space X is homo gene ous if for any tw o points in X there is a homeomorphism of X on to itself mapp ing one p oin t on to the other. Let C denote the pseudo-circle. W e may assume that C is co nta ined in a planar annulus A in suc h a wa y that the winding n umber of eac h circular chain in the sequ en ce of cro oke d circular c h ains defining C is one. Any h omeomorphism h : C → C extends to a con tin uous map f : A → A of d egree ± 1. (First extend h to a map U → A for some closed annular neighborh o o d U of C , then comp ose a retraction of A on to U w ith this extension.) Let e A b e the un iv ersal co vering sp ace of A with pr o jection p . F or an y e x ∈ e A and e y ∈ p − 1 ( f ( p ( e x ))) there is a map e f suc h that the diagram e f e A − → e A p ↓ ↓ p A − → A f comm u tes and e f ( e x ) = e y ; see for example [9], Theorem 16.3. Let b A b e the disc that is a tw o-p oint compactificatio n of e A . Denote the t w o added p oin ts of the compactificatio n by a and b . The map e f extends un iquely to a m ap F : b A → b A . Let e C = p − 1 ( C ), and let P = e C ∪ { a, b } , a t wo -p oint compactificat ion of p − 1 ( C ). D. Bellam y and W. Lewis considered this set in [4] and prov ed that P is a pseudo-arc. Denote by H th e restriction of F to P and n ote that (1) either H ( a ) = a and H ( b ) = b , or H ( a ) = b and H ( b ) = a , (2) e f ( e C ) = e C and hence H ( P ) = P , (3) e f | e C is one-to-one. Th us Lemma. H is a home om orphism fr om P to P . 3. Proof of no nhomogeneity of the pseudo-circle Theorem 1. The pseudo-cir cle is not homo gene ous. Pr o of. Let K ( a ) and K ( b ) b e the comp osan ts of a and b , resp ectiv ely , in the pseudo-arc P . Let e x and e y b e tw o p oints in P su c h that e x ∈ ( K ( a ) ∪ K ( b )) − { a, b } and e y ∈ P − ( K ( a ) ∪ K ( b )). If C w ere homogeneous, then there would b e a homeomorph ism h : C → C taking x = p ( e x ) on to y = p ( e y ). Th en there w ould b e a homeomorphism H : P → P as d escrib ed in section 2 taking e x on to e y . This is not p ossible since under eve ry such h omeomorphism, the s et K ( a ) ∪ K ( b ) is in v arian t; the image of a comp osan t is a comp osant.  Remark. It is not imp ortant for th is pro of that K ( a ) and K ( b ) are n ot the same set, but th e authors are grateful to D. Bellam y and W. Lewis for sho wing that K ( a ) and K ( b ) were indeed differen t co mp osan ts. Theorem 2. If for some x , the c omp osant K ( a ) interse cts the fib er p − 1 ( x ) , then it c ontains p − 1 ( x ) . A S HOR T PROOF OF NONHOMOGENEITY OF THE PSEUDO-CIRCLE 3 Pr o of. If y ∈ p − 1 ( x ) ∩ K ( a ), then by the definition of a comp osan t, there is a prop er sub con tin uum W of P that con tains b oth a and y . Let g : e C → e C b e a deck transf ormation suc h that p − 1 ( x ) = { g n ( y ) } n ∈ Z , Z b eing the set of intege rs. Denote b y G the exte nsion of g to P . The set W n = G n ( W ) is a con tinuum conta ining a and g n ( y ). Thus p − 1 ( x ) ⊂ K ( a ).  Note that T h eorem 2 and Remark ab o v e imply that p ( K ( a ) − { a } ) ∩ p ( K ( b ) − { b } ) = ∅ . Question. Ca n the sets p ( K ( a ) − { a } ) and p ( K ( b ) − { b } ) b e use d to classify the c omp osants of the pseudo-cir cle C ? The authors w ould lik e to thank Jim Rogers, Da vid Bellam y , and W a yn e Lewis for th eir com- men ts. Referen ces [1] R.H. Bing, A homo gene ous inde c omp osable plane c ontinuum , Duke Math. J. 15 (1948), 729-742. [2] R.H. Bing, Conc erning her e di tarily inde c omp osable c ontinua , P acific J. Math. 1 (1951), 43-51. [3] R.H. Bing, Each homo gene ous nonde gener ate chainable c ontinuum i s a pseudo-ar c , Proc. Amer. Math. S oc. 10 (1959), 345-346 [4] D.P . Bellam y and W. Lewis, An orientation r eversing home omorphism of the plane wi th i nvariant pseudo-ar c , Proc. A mer. Math. So c. 114 (1992), 1145-1149. [5] L. F earnley , The pseudo-cir cle i s unique , Bull. A mer. Math. So c. 75 (1969), 398-401. [6] L. F earnley , The pseudo-cir cle i s not homo gene ous , Bull. Amer. Math. So c. 75 (1969), 554-558. [7] L. F earnley , The pseudo-cir cle i s unique , T rans. Amer. Math. Soc. 149 (1970), 45-64. [8] C.L. Hagopian, The fixe d-p oint pr op erty for almost chainable homo gene ous c ontinua , Illinois J. Math. 20 (1976), 650-652. [9] S.T. Hu, H omotop y Theory , Elsevier Science and T ec hnology Bo oks, 1959. [10] J. K ennedy , and J.T. Rogers, Jr., Orbits of the pseudo cir cle , T rans. A mer. Math. So c. 296 (1986), 327-340. [11] B. K naster, Un c ontinu dont tout so´ us-c ontinu est ind´ ec omp osable , F und. Math. 3 (1922), 247-286. [12] K. Kuratowski, T opology , V ol. I I, Academic Press, 1968. [13] W. Lewis, Almost chainable homo gene ous c ontinua ar e chainable , Houston J. Math. 7 ( 1981), 373-377. [14] W. Lewis, T he pseudo-ar c , Con temp. Math. 117 (1991), 103-123. [15] W. Lewis, T he pseudo-ar c , Bol. So c. Mat. Mexicana 5 (1999), 25-77. [16] E.E. Moise, An i nde c omp osable plane c ontinuum which is home omorphic to e ach of its nonde gener ate sub c on- tinua , T rans. Amer. Math. S oc. 63 (1948), 581-594. [17] J.T. Rogers, Jr., The pseudo-cir cle is not homo gene ous , T rans. Amer. Math. Soc. 148 (1970), 417-42 8. [18] J.T. Rogers, Jr., Homo gene ous, sep ar ating pl ane c ontinua ar e de c omp osable , Michigan Math. J. 28 (1981), 317-322. Dep ar tmen t of Ma thema tics and St a ti stics, Auburn University, Auburn, AL 36849, USA E-mail addr ess : kuperkm@auburn .edu Dep ar tmen t of Ma thema tics and St a ti stics, Auburn University, Auburn, AL 36849, USA E-mail addr ess : gammokb@auburn .edu