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 Differentiable engulfing and coverings of manifolds
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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 piecewisedifferentiable 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 kconnected 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 ncells. Using this result, we give a new and direct proof of the uniqueness of the differentiable structure of Euclidean nspace for n ≥ 5. Finally, we prove a general hcobordism 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.
Item Metadata
Title 
Differentiable engulfing and coverings of manifolds

Creator  
Publisher 
University of British Columbia

Date Issued 
1969

Description 
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 piecewisedifferentiable 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 kconnected 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 ncells. Using this result, we give a new and direct proof of the uniqueness of the differentiable structure of Euclidean nspace for n ≥ 5. Finally, we prove a general hcobordism 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.

Genre  
Type  
Language 
eng

Date Available 
20110621

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080489

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.