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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.
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