Heegaard Splittings and a Theorem of Livesay

Abstract

Abstract. Let M be a nonorientable 3-manifold which is double covered by S2 X I. We give a short proof of the theorem of Livesay [1] that M is homeomorphic to P2 x / (where P2 denotes the projective plane).0. Let M be a compact connected nonorientable 3-manifold with dM consisting of two copies of P2 and 7A,(M) = Z2. Using the analysis of the Heegaard splittings of 53 in [4], we prove the following Theorem [1]. // the orientable two-fold cover of M is homeomorphic to S2 X I then there is an annulus A embedded in M with A n dM = dA consisting of two simple closed noncontractible curves, one in each component of dM.As in [1] the fact that M is homeomorphic to P2 X / follows immediately. We divide the proof of this theorem into a number of steps, working throughout in the PL category.1. Let r: M -» P be a retraction such that r restricted to each component of dM is a homeomorphism. r can be chosen transverse to a simple closed noncontractible curve a in P. Then a-"'(a) is a compact 2-manifold embedded in M. Exactly as in [1] it follows that r~\a) contains a component K which is orientable, one-sided and has 3A" equal to two noncontractible simple closed curves, one in each component of dM.Let p: S2 X I -» M be the double covering and let g: S2 X I -» 52 X / be the covering transformation. Let L denote p~liK). If W is the closure of a component of S2 X I -L, then W u gW = S2 X I and W n gW = L.Clearly we can assume without loss of generality that K is incompressible, i.e., there is no disk D embedded in M with D n K = dD and 3D a noncontractible curve in K. Also it will be supposed that genus K > 0, i.e., K is not an annulus.2. We show that K incompressible implies W is a handlebody. By [2] there are disjoint simple closed noncontractible curves C,, . . . , Cm in dW such that the normal closure of the elements of tr^dW) given by joining each C, to the base point along some path for 1 < i < aai is Ker d> (where d>; <nx(3 W)^> trx(W) is induced by the inclusion map).

Topological rigidity and H1–negative involutions on tori

“…But this involution on S n is standard. For, when n = 2 this was proved by Kérékjartò, Brouwer, and Eilenberg (see [23]); when n = 3 it was proved by Hirsch-Smale and Livesay (see [46]).…”

Smooth s-Cobordisms of Elliptic 3-Manifolds

“…e.g. [19,31,38]). For n = 4, work of Freedman [17] yielded a topological s-cobordism theorem for W with a relatively small fundamental group, e.g.…”