The Equivariant Surgery on Manifolds and Its Application to Classify PL Involutions of the Klein Bottle K

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Publication Year: 
1979
Publisher: 
A Theseis presented to the Faculty of Science, Department of Mathematics University of jordan
Description: 

This thesis is aimed to expose a general technique in classifying piecewise linear involutions on 3-dimensinal manifolds, the P-equiveriant surgery developed by Tollefson [10] and Tollefson and Kim [11], see also Livesay [3] and [4] and Tao [9]. We will also give some examples of this general technique and apply a simple version of this idea to classify the PL involutions on K. The idea is, if h is an involution on a 3-manifold M, we look for an appropriate surface S properly  embedded in M for which h(S) = S or h(S) ∩ S  ≠ Ø , and then cut M along SUh(S) to get a manifold M' and an induced involution h’:M’->M’, where h’ is easier to classify than h.
Lamma1: Let h: k->k be a PL involution. Then we can always find a 2-sided separating simple closed curve J such that h(J)=J.
Theorem 2: Up to PL equivalence there are five PL involutions on K with fixed point sets homeomorphic to (i) S1US1, (i i) S1US0, (iii) S1, (iv) S 0  or (v)Ø.

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