Involutions on and other -manifolds

Author:
Jeffrey L. Tollefson

Journal:
Trans. Amer. Math. Soc. **183** (1973), 139-152

MSC:
Primary 57A10; Secondary 55A10

MathSciNet review:
0326738

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Abstract: This paper exploits the following observation concerning involutions on nonreducible 3-manifolds: If the dimension of the fixed point set of a PL involution is less than or equal to one then there exists a pair of disjoint 2-spheres that do not bound 3-cells and whose union is invariant under the given involution. The classification of all PL involutions of is obtained. In particular, admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus with 1-dimensional fixed point set. Furthermore, there are just four fixed point free -actions and just one fixed point free -action on for each positive integer *k* (again, up to conjugation). The above observation is also used to obtain a general description of compact, irreducible 3-manifolds that admit two-sided embeddings of the projective plane.

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0326738-0

Keywords:
Three-manifolds,
involution,
cyclic group action

Article copyright:
© Copyright 1973
American Mathematical Society