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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The_Equivariant_Surgery_on_Manifolds_and_Its_Application_to_Classify_PL_Involutions_of_the_Klein_Bottle_K.pdf | 99.35 KB |