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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ {\rm PL}$ involutions of some $ 3$-manifolds


Author: Myung Mi Myung
Journal: Proc. Amer. Math. Soc. 33 (1972), 576-581
MSC: Primary 57C99; Secondary 55C35, 57E30
MathSciNet review: 0295363
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Abstract: Let $ {h_1}$ and $ {h_2}$ be PL involutions of connected, oriented, closed, irreducible 3-manifolds $ {M_1}$ and $ {M_2}$, respectively. Let $ {a_i},i = 1,2$, be a fixed point of $ {h_i}$ such that near $ {a_i}$ the fixed point sets of $ {h_i}$ are of the same dimension. Then we obtain a PL involution $ {h_1}\char93 {h_2}$ on $ {M_1}\char93 {M_2}$ induced by $ {h_i}$ by taking the connected sum of $ {M_1}$ and $ {M_2}$ along neighborhoods of $ {a_i}$. In this paper, we study the possibility for a PL involution h on $ {M_1}\char93 {M_2}$ having a 2-dimensional fixed point set $ {F_0}$ to be of the form $ {h_1}\char93 {h_2}$, where $ {M_i}$ are lens spaces. It is shown that: (1) if $ {F_0}$ is orientable, then $ {M_1} = - {M_2}$ and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then $ {M_1} = {M_2} = {\text{a}}$ projective 3-space, and in this case, F is the disjoint union of two projective planes and h is unique up to PL equivalences, (3) if F contains a Klein bottle K, then F is the disjoint union of a Klein bottle and two points.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0295363-7
PII: S 0002-9939(1972)0295363-7
Keywords: PL involution, lens space, connected sum
Article copyright: © Copyright 1972 American Mathematical Society