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Branched Covers and Contact Structures

1987; American Mathematical Society; Volume: 101; Issue: 2 Linguagem: Inglês

10.2307/2046007

1088-6826

Jesús Gonzalo,

Mathematical Dynamics and Fractals

The object of this paper is to prove the following Theorem.Every closed, orientable three-manifold has a parallelization by three contact forms.1. Introduction.Throughout this paper, M stands for a closed, oriented threemanifold.Definition.A 1-form to on an odd-dimensional manifold is called a contact form if w A dto A • • • A du is a volume form.Remark.For any function / and any 1-form to, we have the identity fee A (d(fto))" = f" + xto A (dec)".So if / has no zeros then /to is a contact form if and only if to is a contact form.This means that the property of being a contact form is determined by the hyperplane distribution of the kernels of to at each point.Definition.A contact structure is a hyperplane distribution which is locally the kernel of contact forms, i.e. every point has a neighborhood and a contact form on that neighborhood which annihilates the hyperplanes of the distribution.Definition.Let Nx, N2 be manifolds of the same odd dimension, with contact forms to,, to2 respectively.Then tox is equivalent to to2 if there is a diffeomorphism 4>: Nx -> N2 and there is a nowhere zero function / on Nx such that the equality O*to2 = ftox is satisfied.Equivalently, the diffeomorphism takes the hyperplane distribution Ker to x into Kerto2.In R3, with coordinates x, y, z, the standard contact form is xdy + dz.Let S3 *■+ R4 be the usual inclusion, with component functions xx, x2, x3, xA; then the standard contact form on the sphere S3 is xxdx2 -x2dxx + x3dx4 -x4dx3.Definition.Let /V be a manifold of dimension 2« + 1.An almost contact structure on N is a pair (to, Q), where to is a 1-form on N and ñ is a 2-form on TV, such that to A ñ" is a volume form.The space of almost contact structures on N is homotopy equivalent to the space of reductions of the structure group of TN to U(n), acting on R2" + 1 in the obvious way determined by the factorization R2"4"1 = R © C.In 1966 Chern [3] posed the question of the existence of a contact form on Sl X S2 # RP3.