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

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.