FREE ODD PERIODIC ACTIONS ON THE SOLID KLEIN BOTTLE

Transcription

1 An-Najah J. Res. Vol. 1 ( 1989 ) Number 6 Fawas M. Abudiak FREE ODD PERIODIC ACTIONS ON THE SOLID LEIN BOTTLE ey words : Free acion, Periodic acion Solid lein Bole. Fawas M. Abudiak * V.' ZZ..).a11,L.A.;15TY1 Luiz I 11 4.;1 c.:j141 r:; ez...a.11 Z.1.1.2%; _sji /-4 J.' Absrac The cyclic acions of odd period and empy fixed poin se are sudied on he solid lein Bole. I is shown ha up o weak equivalence here is only one such acion. 1 Noaion and Preliminaries D2, R,S 1 denoe he uni disk [ xer2 : x 1 ], he field of real numbers and he uni circle. A 3 manifold M is irreducible if every 2 sphere in M bounds a 3 cell in M. The cyclic group generaed by he periodic map h is denoed by < h >. If h is periodic on a space X, hen he orbi space of h is he quoien space obained by idenifying each x wih h' ( x ) for all i. The orbi space of h will be denoed by X/h. The idenificaion map p h : X > X/h is called he orbi Assisan Prof., Deparmen of Mahemaics, Faculy of Science. An-Najah Naional Universiy 17

2 Free Odd Periodic Pf nh X if X/ map. Two acions of <h> and <f> on X are said o be weakly equivalen if here is a homeomorphism of X such ha <h -1 > = < f > and ha h 1 =f for some i. Equivalenly, h and f are weakly equivalen if here are homeomorphisms and T ha make he diagram commuaive, i. e. p h = pf. The se [ xex : h( x) = x of fixed poins of h will be denoed by F ( h). The solid lein Bole is he space obained form D2xR by idenifying ( z, ) wih (, + 1). An elemen of his space wih represanaive ( z, ) will be denoed by [ z, ]. 2 Resuls. The follwing is he main resul : Theorem A. Up o weak equivalence here is exacly one free Z2r+1 acion on he solid lein Bole. Firs we prove he following wo Lemmas. Lemma 1. If h : is a PL homeomorphism of period 2r + 1 on, hen F ( h ) is eiher empy or a simple closed curve ( homeomorphic o S 1 ). Proof. Le n = 2r + 1. Then n can be wrien as n pl l p22 Pff, m where p,..., pm are disinc odd primes and i, 2, m are posiive inegers. If m = 1, hen h is of period p i ' on k which is a homology 1 sphere. Hence, we find ha, by Ill,F(h) is eiher 0 or a homology 1 sphere. 18

3 An-Najah J. Res. Vol. I ( 1989 ) Number 6 Fawas M. Abudiak Since F ( h ) can no be wo dimensional (for = 2 ) i is eiher 0 or ( home omorphic o ) S 1. If m = 2, hen n = pil p22 and hp1 1 is of period p212 As above F ( hpl i ) is eiher 0 or S 1. If F ( hpl i ) = 0, hen F ( h ) =. Because F( hpll ) is invarian under h, hen if F ( hp1 1 ) S 1 and F ( h) = 0, hen h is of period pi ' on F( hp1 1 ) S 1. So, by [ 1, F ( h ) S 1. Now suppose ha he resul is rue for m = i. Le C = p i l pi, Then he period of h is cpi Then h is of period c on k, hence, by he inducion hypohesis, F ( hpi ) is eigher 0 or S 1. If F ( hpi ) = 0, hen F ( h ) =0. If F ( li ± 1 ) S 1, hen by [ 1 ], F ( h ) Si or 0. Remark. The proof above shows ha if F (h) S 1, hen F(h i )=F(h) S 1 for all 1 < i < 2r + 1. Lemma 2. Le h : be a homeomorphism of period 2r + 1. If h acs freely on, hen / h Proof : Le B = / h and le p : B be he orbi map. Since h acs freely on, hen is a regular 2r + 1 covering of B by [ 4 ] Hence p ( n-i() ) ( which is infinie cyclic ) is a normal subgroub of index 2r + 1 in 7 1( B ) - So we have a shor exac sequence 0 --> Z > 1 ( B ) Z 2r 1 -->0 Since B is covered by a conracible space and no nonrivial finie group can ac freely on a finie dimensional conracible space [ 3 ], ( B ) has no orsion subgroup. Le a be he image of a generaor of Z under f and le b be such ha g(b) is a generaor of Z2r1.1 Since p ( 7 1() ) is normal in 7 1 (B), bab-1 e < a >. So bab-1 = a or a-1. If bab-1 = a-1, hen 7 (B) / [ 7 1 (B), 7 1 (B) I which is isomorphic o H 1 (B) is finie ( for he cose b b+ [ 7r i (B), 7ri (B) ] is of order 19

4 Free Odd Periodic 2r + 1 ). Here. [ 7r / (B), (B) I is he commuaor subgroup. Hence he Euler characerisic, 9(,( B) =2_ 0 ( - 1 )'r 1 = 1 + r2 because r l = o = r3 and r3 = o because B is nonorienable Bu his implies ha 9(( B ) 1, conradicing he fac ha X( B) = o. Hence bab-1 a 1 and we mus have bab-1 = a and so YrB ) is abelian. From he fundamenal heorem of abelian groups we have 73-1 (B) = Z+Tor(n- i (B) = Z. Noe ha B is compac, nonorienable, irreducible wih a wo dimensional lein Bole as is boundary. Moreover, B conains no 2 sided projecive plane p 2, because if B conains such p 2, hen p -1 1:12 will be a 2 sphere S, and since is irreducible, S bounds a 3 cell C. Then p ( C ) is a 3 manifold bounded by p2. Now, by [ 2 ] heorem 11 7, B is homeomorphic o. Wih hese wo lemmas a hand we now urn o prove our heorem. Proof of heorem A. Le h 1 > be defined by ' h i ([z,]) =- [z,+ 2r 2r+ I The map h 1 is a homeomorphism of period 2r + 1 and F ( h i ) = 0 for all 1 i 2r. Hence, by lemma 2, /h 1. Now Le h : --> be any homeomorphism of period 2r + 1 such ha F ( h i ) = 0 for 1 i -S 2r. Lemma 2 implies ha /h. Le p 1 : > IC/11 1 and p : ---> 1C/h be he orbi maps.. p 1 and p are ( 2r + 1) covering projecions of Jh i and / h, respecively. Le :/h 1 /h be a homeomorphism Since ic', and p are ( 2r + 1) covering projecions of and since 77 1 ( / h ) ( is infinie cyclic ) has a unique normal subgropup of index 2r + 1, here is a homeomorphism > making he diagram 20

5 An-Najah J. Res. Vol. 1 ( 1989 ) Number 6 Fawas M. Abudiak i ba /h Commuaive. Now by he commuaiviy of he diagram. h ) pa. P 1 > < We obain p1 = p1 I hi. Tha is I hi is a nonrivial covering ransformaion on wih respec o p1. Hence,--1 for some 1 i 2r,which means ha h is weakly equivalen o h 1. This complees he proof. References 1 E. E. Floyed, On periodic maps and Euler characerisic of associaed spaces, Trans. Amer. Mah. 72 ( 1952), J. Hemple, 3 manifolds, Ann. of Mah. sudies 86, princeon Universiy press, princeon, New Jersey, S. T. Hu, Homoopy heory, Academic press, New York W. S. Massey, Algebraic Topology : An Inroducion, Springer Verlage, GTM # 56,