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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  MacLean, Douglas W. 
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.

Subject  Differential topology 
Genre  Thesis/Dissertation 
Type  Text 
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  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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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.