Vol. 30, No. 1, 1969

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Covering manifolds with cells

Richard Paul Osborne and J. L. Stern

Vol. 30 (1969), No. 1, 201–207

In attempting to triangulate a topological manifold, one would like to be able to cover a manifold with closed cells whose intersections are nice. This paper is a study of minimal coverings of manifolds by open cells and a method of improving the interseclions as the connectivity allows. The principal theorem is the following.

Theorem 1: If Mn is a k-connected topological n-manifold (without boundary) and q is the minimum of k and n-3, then Mn can be covered by p open cells if p(q + 1) > n. Futhermore, these cells may be chosen so that the intersection of any collection of these cells is (q-1)-connected.

As a consequence of this theorem it is shown that a contractible open n-manifold (n 5) is the union of two open cells whose intersection is a contractible open manifold. One might note for instance that a 3-connected 10-manifold can be covered by 3 open cells whose intersections are 2-connected.

Mathematical Subject Classification
Primary: 57.01
Received: 13 May 1968
Published: 1 July 1969
Richard Paul Osborne
J. L. Stern