A contact-geometric variant of Smale's contraction

Abstract

Motivated by Smale's contraction of the compactly supported diffeomorphism group of the two-disc, we study contact forms on an open rotationally symmetric Darboux ball that agree with the standard contact form outside a compact set and whose Reeb flows have no trapped orbits. Such forms are called vertically convex. In dimensions three and five, we identify the quotient of their space by compactly supported diffeomorphisms with a contractible monodromy space and prove an equivariant product splitting. Consequently, the orbit of the standard contact form is a strong deformation retract. It follows that the space of vertically convex contact forms is contractible in dimension three and homotopy equivalent to the compactly supported diffeomorphism group, hence connected, in dimension five. The analogous subspace of forms defining the standard contact structure has the homotopy type of the corresponding compactly supported contactomorphism group in dimension five and is contractible in dimension three.