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Differentiable engulfing and coverings of manifolds MacLean, Douglas W.

Abstract

There are now engulfing theorems for topological, piecewise linear, and differentiable manifolds. Differentiable engulfing so far was reduced to piecewise linear engulfing using the J. H. C. Whitehead triangulation of a differentiable manifold and J. R. Munkres' theory of obstructions to the smoothing of piecewise-differentiable homeomorphisms. In the first part of the thesis we observe that the method of proof of M. H. A. Newman's topological engulfing theorem applies, up to a local lemma, simultaneously to all three categories of manifolds. We prove this local lemma in the differentiable case and thus obtain a differentiable engulfing theorem which has a direct proof. Then we solve the problem of the existence of a stretching diffeomorphism between complementary subcomplexes of a simplicial complex in Euclidean space which is crucial for all applications of engulfing. Next we prove a theorem concerning the uniqueness of open differentiable cylinders which is the differentiable analogue of the uniqueness theorem for open cones. A consequence of this theorem is that if M₁ and M₂ are compact differentiable manifolds with diffeomorphic interiors then M₁x R and M₂xR are diffeomorphic, where (R denotes the real line. Another consequence is that if a differentiable manifold is the monotone union of open differentiable cells it is diffeomorphic to Euclidean space. We present several applications of differentiable engulfing which actually hold in all three categories of manifolds. Our methods are such that they apply also to noncompact manifolds. Theorem: [formulae omitted] This theorem has several corollaries. For instance, if M is a k-connected differentiable manifold of dimension n without boundary, k ≤ n - 3 if k>0, and if [formula omitted] then M may be covered by m open differentiable n-cells. Using this result, we give a new and direct proof of the uniqueness of the differentiable structure of Euclidean n-space for n ≥ 5. Finally, we prove a general h-cobordism theorem. Theorem: Let M be a connected differentiable manifold of dimension n, n ≥ 5, with two connected boundary components N₁, and N₂ such that the inclusion of N₁ into M is a homotopy equivalence, i = 1,2. Then there is a diffeomorphism of N₁x(0,oo) onto M - N₂.

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