Minimal atlases of closed contact manifolds

Abstract

We study the minimal number C(M,ξ) of contact charts that one needs to cover a closed connected contact manifold (M,ξ). Our basic result is C(M,ξ) \le \dim M + 1. We compute C(M,ξ) for all closed connected contact 3-manifolds: C (M,ξ) = 2 if M = S^3 and ξis tight, 3 if M = S^3 and ξis overtwisted or if M = #_k (S^2 \times S^1), 4 otherwise. We also show that on every sphere S^{2n+1} there exists a contact structure with C(S^{2n+1},ξ) \ge 3.

40 pages, 18 figures