‘}V4f31_] RETURNING MATERIALS: P1ace in book drop to LJBRARJES remove this checkout from ‘J-llflfll-IL. your record. fifl§§_w111 be charged if book is returned after the date stamped beWOw. _ \ fem _.__,_.—-'—A 2 2p — ACTIONS ON THE 2-DIMENSIONAL AND THE SOLID KLEIN BOTTLES BY Fawaz Mohammad Abudiak A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1984 ABSTRACT Z 2p - ACTIONS ON THE 2-DIMENSIONAL AND THE SOLID KLEIN BOTTLES BY Fawaz Mohammad Abudiak In this thesis PL homeomorphisms of periods p and 2p are classified on both the 2—dimensional Klein bottle K2 and the solid Klein bottle K, where p is an odd prime number. It is shown that up to weak equivalence there is only one class of homeomorphisms of period p on K2 and only three equivalence classes of homeomorphisms of period 2p on K2, distinguished by the fixed point . th sets of their p powers. Also, free cyclic actions of odd period are class- ified on K as well as cyclic actions of period 2p. In the first case it is shown that, up to weak equival- ence, only one such action exists, while in the second case there are three such homeomorphisms, distinguished by the fixed point sets of their pth power. Finally, semi-free action on K are classified for any finite period n. It is shown that these exist only for n equals two and for all odd values of n such that F(hn) = a. TO MY PARENTS ii ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor Kyung Whan Kwun, my thesis advisor, for suggesting the problem and for his continuous support, patience, and encouragement during the completion of the problem. iii TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . CHAPTER 0. PRELIMENARIES AND DEFINITIONS. . . . . . CHAPTER 1. 252p - ACTIONS ON THE Z-DIM KLEIN BOTTLE SECTION 1.1. Zip - ACTIONS ON THE 2-DIM KLEIN BOTTLE. . . . . . . . . . . . . . SECTION 1.2. 222p - ACTIONS ON THE 2-DIM KLEIN BOTTLE. . . . . . . . . . . . . . CHAPTER 2. Zfizp - ACTIONS ON THE SOLID KLEIN BOTTLE K . . . . . . . . . . . . . . FR . . SECTION 2 1 EE 212k+1 ACTIONS ON K SECTION 2.2. fl: - ACTIONS ON K . . . . . . ZP SECTION 2.3. SEMI-FREE ACTIONS ON K . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . iv Page INTRODUCTION Let X and X' be topological spaces. Two homeomorphisms h and g on X and X' respectively are said to be weakly equivalent (written h mm 9) if there exists a homeomorphism t :X + X' such that t-lgt = hl for some positive integer i # 1. All maps and spaces considered in this thesis are in the piecewise linear category. In this thesis we classify piecewise linear homeo- morphisms of periods p and 2p, p an odd prime, on both the 2-dim and the solid Klein bottles. Chapter 1 deals with the homeomorphisms of periods p and 2p on the 2—dim Klein bottle K2, p odd prime. Proposition 1 of section 1.1 asserts that up to weak equivalence there is a: unique homeomorphism of period p on K2. Theorem 1 of section 1.2 gives all the homeomorphisms of period 2p on K2 up to weak equival- ence, p odd prime. In fact there are three such homeomorphisms hi distinguished by the fixed point sets of hip, i = 1, 2, 3. Chapter 2 is divided into three sections. Section 2.1 deals with the free homeomorphisms of odd period on the solid Klein bottle K. Proposition 3 of this section states that there is only one such homeomorphism up to weak equivalence. Section 2.2 provides a complete classification of homeomorphisms of period 2p on K, p odd prime. The main theorem of this section is the classification Theorem 1 which asserts that up to weak equivalence there are only three such homeomorphisms hi, distinguished by F(hip), i = 1, 2, 3. Finally section 2.3 deals with the semi-free periodic actions on K. These actions are given in Theorem 1 of that section. CHAPTER 0 PRELIMINARIES AND DEFINITIONS Throughout this thesis, we work in the PL (piece- wise linear) category and all spaces and maps will be piecewise linear. . . . . . n A homeomorphism h : X + X is periodic if h = identity for n >1 in Z5. If n = 2, h is said to be an involution. Let h be a periodic map of a space X. The cyclic group generated by h will be denoted by <11>. If h is periodic on X, then the orbit space of h is the quotient space obtained by identifying x with hl(x) for all i and all x in X. The orbit space of h will be denoted by X/<11>. The identification map ph : X + X/‘