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
HTML articles powered by AMS MathViewer

by Jeffrey L. Tollefson

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

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

PDF | 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.

References

D. B. A. Epstein, Projective planes in $3$-manifolds, Proc. London Math. Soc. (3) 11 (1961), 469–484. MR 152997, DOI 10.1112/plms/s3-11.1.469

Finite cyclic group actions on

John Hempel and William Jaco, Fundamental groups of $3$-manifolds which are extensions, Ann. of Math. (2) 95 (1972), 86–98. MR 287550, DOI 10.2307/1970856

The structure of

manifold groups

Kyung Whan Kwun, Piecewise linear involutions of $S^{1}\times S^{2}$, Michigan Math. J. 16 (1969), 93–96. MR 242161
G. R. Livesay, Involutions with two fixed points on the three-sphere, Ann. of Math. (2) 78 (1963), 582–593. MR 155323, DOI 10.2307/1970543
G. R. Livesay, Fixed point free involutions on the $3$-sphere, Ann. of Math. (2) 72 (1960), 603–611. MR 116343, DOI 10.2307/1970232

Homology

28

W. H. Row Jr., Irreducible $3$-manifolds whose orientable covers are not prime, Proc. Amer. Math. Soc. 34 (1972), 541–545. MR 296947, DOI 10.1090/S0002-9939-1972-0296947-2
P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690–711. MR 177, DOI 10.2307/1968950
Yoko Tao, On fixed point free involutions of $S^{1}\times S^{2}$, Osaka Math. J. 14 (1962), 145–152. MR 140092
Jeffrey L. Tollefson, Free involutions on non-prime $3$-manifolds, Osaka Math. J. 7 (1970), 161–164. MR 266184
Friedhelm Waldhausen, Über Involutionen der $3$-Sphäre, Topology 8 (1969), 81–91 (German). MR 236916, DOI 10.1016/0040-9383(69)90033-0
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
John Stallings, On fibering certain $3$-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375