Minimal atlases of closed contact manifolds

TL;DR

This paper investigates the minimal number of contact charts needed to cover closed contact manifolds, providing exact values for 3-manifolds and bounds for higher dimensions, revealing new insights into contact topology.

Contribution

It establishes an upper bound for the minimal contact chart number and computes exact values for all closed contact 3-manifolds, introducing new classifications.

Findings

C(M,ξ) ≤ dim M + 1 for all closed contact manifolds

C(M,ξ) = 2 for tight S^3

C(M,ξ) = 3 for overtwisted S^3 and connected sums of S^2×S^1

Abstract

We study the minimal number C(M,\xi) of contact charts that one needs to cover a closed connected contact manifold (M,\xi). Our basic result is C(M,\xi) \le \dim M + 1. We compute C(M,\xi) for all closed connected contact 3-manifolds: C (M,\xi) = 2 if M = S^3 and \xi is tight, 3 if M = S^3 and \xi is 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},\xi) \ge 3.