Minimal atlases on 3-manifolds

Abstract

AbstractIn this paper we investigate the minimal number of charts (spaces homeomorphic to open subsets of ℝ3) needed to cover a closed 3-manifold M3. We prove that this number is two if the Bockstein βω1(M3)∈H2(M3;ℤ) of the first Stiefel-Whitney class of M3 is zero and three if it is not. We state several properties equivalent to βω1(M3) = 0, for example the condition of being covered by two orientable subspaces, and the condition that all torsion elements of H1(M3;ℤ) be represented by orientation-preserving loops. We also show that the non-orientable 3-manifolds which can be covered with two charts are precisely those manifolds which can be described as sewn-up r-link exteriors. These are manifolds obtained by removing from S3 the interiors of two disjoint handlebodies of genus r and then identifying the two boundary components by a homeomorphism.