Involutions on 𝑆¹×𝑆² and other 3-manifolds

Tollefson, Jeffrey L.

doi:10.1090/S0002-9947-1973-0326738-0

Involutions on $S^{1}\times S^{2}$ and other $3$-manifolds
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Trans. Amer. Math. Soc. 183 (1973), 139-152 Request permission

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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