Date: 16 Sep 2006

Involutions and isotopies of lens spaces

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Abstract

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.