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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Embeddings of open manifolds


Author: Nancy Cardim
Journal: Trans. Amer. Math. Soc. 351 (1999), 2353-2373
MSC (1991): Primary 57N37; Secondary 57N35, 57N45
Published electronically: January 27, 1999
MathSciNet review: 1641091
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Abstract: Let $TOP(M)$ be the simplicial group of homeomorphisms of $ M$. The following theorems are proved.

Theorem A. Let $M$ be a topological manifold of dim $\geq $ 5 with a finite number of tame ends $\varepsilon _{i}$, $1\leq i\leq k$. Let $TOP^{ep}(M)$ be the simplicial group of end preserving homeomorphisms of $M$. Let $W_{i}$ be a periodic neighborhood of each end in $M$, and let $p_{i}: W_{i} \to \mathbb{R}$ be manifold approximate fibrations. Then there exists a map $f: TOP^{ep}(M) \to \prod _{i}\ TOP^{ep}(W_{i})$ such that the homotopy fiber of $f$ is equivalent to $TOP_{cs}(M)$, the simplicial group of homeomorphisms of $M$ which have compact support.

Theorem B. Let $M$ be a compact topological manifold of dim $\geq $ 5, with connected boundary $\partial M$, and denote the interior of $M$ by $Int\ M$. Let $f: TOP(M)\to TOP(Int\ M)$ be the restriction map and let $\mathcal{G}$ be the homotopy fiber of $f$ over $id_{Int\ M}$. Then $\pi _{i}\ \mathcal{G}$ is isomorphic to $\pi _{i}\ \mathcal{C} (\partial M)$ for $i > 0$, where $ \mathcal{C} (\partial M)$ is the concordance space of $\partial M$.

Theorem C. Let $q_{0}: W \to \mathbb{R}$ be a manifold approximate fibration with dim $W \geq $ 5. Then there exist maps $\alpha : \pi _{i} \ TOP^{ep}(W) \to \pi _{i} \ TOP(\hat W)$ and $\beta : \pi _{i}\ TOP(\hat W) \to \pi _{i}\ TOP^{ep}(W)$ for $i >1$, such that $\beta \circ \alpha \simeq id$, where $\hat W$ is a compact and connected manifold and $W$ is the infinite cyclic cover of $\hat W$.


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Additional Information

Nancy Cardim
Affiliation: Univeridade Federal Fluminense - UFF, Instituto de Matemática, Departamento de Análise, Niterói, RJ, 24020-005 - Brazil
Email: ganancy@vm.uff.br

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02430-7
PII: S 0002-9947(99)02430-7
Keywords: Open manifolds, homeomorphisms of open manifolds, tame ends, manifold approximate fibrations, controlled homeomorphisms
Received by editor(s): November 20, 1996
Published electronically: January 27, 1999
Additional Notes: Partially suported by the CNPq of Brazil
Article copyright: © Copyright 1999 American Mathematical Society



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