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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Euclidean $ n$-space modulo an $ (n-1)$-cell

Author: J. L. Bryant
Journal: Trans. Amer. Math. Soc. 179 (1973), 181-192
MSC: Primary 57A15; Secondary 57A30
MathSciNet review: 0324703
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Abstract: This paper, together with another paper by the author titled similarly, provides a complete answer to a conjecture raised by Andrews and Curtis: if D is a k-cell topologically embedded in euclidean n-space $ {E^n}$, then $ {E^n}/D \times {E^1}$ is homeomorphic to $ {E^{n + 1}}$. Although there is at present only one case outstanding ( $ n \geqslant 4$ and $ k = n - 1$), the proof we give here works whenever $ n \geqslant 4$. We resolve this conjecture (for $ n \geqslant 4$) by proving a stronger result: if $ Y \times {E^1} \approx {E^{n + 1}}$ and if D is a k-cell in Y, then $ Y/D \times {E^1} \approx {E^{n + 1}}$. This theorem was proved by Glaser for $ k \leqslant n - 2$ and has as a corollary: if K is a collapsible polyhedron topologically embedded in $ {E^n}$, then $ {E^n}/K \times {E^1} \approx {E^{n + 1}}$. Our method of proof uses radial engulfing and a well-known procedure devised by Bing.

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Keywords: Euclidean space, usc decompositions of $ {E^n}$, cartesian factors of $ {E^n}$, collapsible polyhedron
Article copyright: © Copyright 1973 American Mathematical Society