Celebratio Mathematica

C. Hodgson and J. H. Rubinstein: “Involutions and isotopies of lens spaces,” pp. 60–96 in Knot theory and manifolds (Vancouver, BC, 2–4 June 1983). Edited by D. Rolfsen. Lecture Notes in Mathematics 1144. Springer (Berlin), 1985. MR 823282 Zbl 0605.57022 incollection

In this paper we study the topology of the three-dimensional lens spaces by regarding them as two-fold branched coverings. The main result obtained is a classification of the smooth involutions on lens spaces having one-dimensional fixed point sets. We show that each such involution is conjugate, by a diffeomorphism isotopic to the identity, to an isometry of the lens space (given the standard spherical metric).

Using this classification of involutions, we deduce that genus one Heegaard splittings of lens spaces are unique up to isotopy. We apply this result to give a new proof of the classification of lens spaces up to diffeomorphism. We also compute the group of isotopy classes of diffeomorphisms of each lens space.

Dale P. O. Rolfsen	Related
Craig David Hodgson	Related

C. D. Hodgson, J. H. Rubinstein, H. Segerman, and S. Tillmann: “Veering triangulations admit strict angle structures,” Geom. Topol. 15 : 4 (2011), pp. 2073–2089. MR 2860987 Zbl 1246.57034 article

Henry Segerman	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

C. D. Hodgson, J. H. Rubinstein, and H. Segerman: “Triangulations of hyperbolic 3-manifolds admitting strict angle structures,” J. Topol. 5 : 4 (2012), pp. 887–908. Zbl 06121970 ArXiv 1111.3168 article

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a ‘geometric’ triangulation of the manifold). Under a mild homology assumption on the manifold, we construct topological ideal triangulations that admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis, and an example illustrating that the triangulations produced using our methods are not generally geometric.

Henry Segerman	Related
Craig David Hodgson	Related

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J. Hyam Rubinstein

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