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Involutions on $ S\sp{1}\times S\sp{2}$ and other $ 3$-manifolds


Author: Jeffrey L. Tollefson
Journal: Trans. Amer. Math. Soc. 183 (1973), 139-152
MSC: Primary 57A10; Secondary 55A10
DOI: https://doi.org/10.1090/S0002-9947-1973-0326738-0
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 $ {S^1} \times {S^2}$ is obtained. In particular, $ {S^1} \times {S^2}$ admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus $ {S^1} \times {D^2}$ with 1-dimensional fixed point set. Furthermore, there are just four fixed point free $ {Z_{2k}}$-actions and just one fixed point free $ {Z_{2k + 1}}$-action on $ {S^1} \times {S^2}$ 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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Additional Information

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

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