Faculty Publications

Title

Pl involutions of some 3-manifolds

Document Type

Article

Keywords

Connected sum, Lens space, PL involution

Journal/Book/Conference Title

Proceedings of the American Mathematical Society

Volume

33

Issue

2

First Page

576

Last Page

581

Abstract

Let /it and /i2 be PL involutions of connected, oriented, closed, irreducible 3-manifolds Mt and A/2, respectively. Let a, i = l, 2, be a fixed point of/;, such that near a, the fixed point sets of A, are of the same dimension. Then we obtain a PL involution h-i # hâ on Mx # A/2 induced by hâ by taking the connected sum of Mi and M2 along neighborhoods of aâ. In this paper, we study the possibility for a PL involution /(on M, # A/2 having a 2-dimensional fixed point set F0 to be of the form /i, # h2, where A/, are lens spaces. It is shown that: (1) if F0 is orientable, then M1= M2 and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then.V/\ = A/2=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 Fis the disjoint union of a Klein bottle and two points. © 1972 American Mathematical Society.

Original Publication Date

1-1-1972

DOI of published version

10.1090/S0002-9939-1972-0295363-7

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