Contact structures on elliptic $3$-manifolds

Abstract

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on S 3 (after possibly conjugating by an isometry) preserves the standard contact structure. We also relate universally tight contact structures on 3-manifolds covered by S 3 to the isomorphism SO(4) = (SU(2) ◊ SU(2))/±1. The main tool used is equivariant framings of 3-manifolds. A contact structure on a 3-dimensional manifold M is a smooth, totally non- integrable tangent plane field, i.e., a tangent plane field locally of the form = ker( ) for a 1-form such that ^ d is everywhere non-degenerate. We shall assume that M is oriented. We say is positive if the orientation on M agrees with that induced by the volume form ^ d . Observe that the orientation of ^d does not depend on the sign of , and is thus determined by (even though = ker( ) only locally). A central role in understanding 3-dimensional manifolds has been played by co- dimension one structures - surfaces, foliations and laminations - in these manifolds. Without additional conditions such structures always exist, and are of not much consequence. However, the presence of essential co-dimension one structures - incompressible surfaces, taut foliations and essential laminations, leads to deep