Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 

 

Euclidean $ (q+r)$-space modulo an $ r$-plane of collapsible $ p$-complexes


Author: Leslie C. Glaser
Journal: Trans. Amer. Math. Soc. 157 (1971), 261-278
MSC: Primary 54.78
MathSciNet review: 0276943
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following general decomposition result is obtained: Suppose $ {K^p}(p \geqq 1)$ is a finite collapsible $ p$-complex topologically embedded as a subset of a separable metric space $ {X^q}$ where, for some $ r \geqq 1,{X^q} \times {E^r}$ is homeomorphic to Euclidean $ (q + r)$-space $ {E^{q + r}}$. Then the Cartesian product of the quotient space $ {X^q}/{K^p}$ with $ {E^r}$ is topologically $ {E^{q + r}}$ provided that $ q \geqq 3$ and, for each simplex $ {\Delta ^k} \in {K^p},({X^q} \times {E^r},{\Delta ^k} \times ({[0,1]^{r - 1}} \times 0))$ is homeomorphic, as pairs, to

$\displaystyle ({E^{q + r}},{[0,1]^{k + r - 1}} \times (0, \ldots ,0)).$

It is known that this condition is satisfied if $ q - p \geqq 2$ and $ q + r \geqq 5$. This result implies that if $ {K^k}$ is a finite collapsible $ k$-complex topologically embedded as a subset of Euclidean $ n$-space $ {E^n}$, then the Cartesian product of the quotient space $ {E^n}/{K^k}$ with $ {E^1}$ is topologically $ {E^{n + 1}}$ provided either (i) $ n \leqq 3$, (ii) $ n - k \geqq 2$, or (iii) each simplex of $ {K^k}$ is flat in $ {E^{n + 1}}$.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54.78

Retrieve articles in all journals with MSC: 54.78


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0276943-5
Keywords: Euclidean space, $ p$-cell, collapsible complex, separable metric space, quotient space, Cartesian product, a map bounded on the $ {E^r}$ factor, $ k$-flat, $ r$-fold suspension, pseudo-isotopy
Article copyright: © Copyright 1971 American Mathematical Society