Contractible polyhedra in products of trees and absolute retracts in products of dendrites

Melikhov, Sergey; Zaja̧c, Justyna

doi:10.1090/S0002-9939-2013-11524-8

Contractible polyhedra in products of trees and absolute retracts in products of dendrites
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by Sergey A. Melikhov and Justyna Zaja̧c

Proc. Amer. Math. Soc. 141 (2013), 2519-2535

DOI: https://doi.org/10.1090/S0002-9939-2013-11524-8

Published electronically: February 21, 2013

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Abstract:

We show that a compact $n$-polyhedron PL embeds in a product of $n$ trees if and only if it collapses onto an $(n-1)$-polyhedron. If the $n$-polyhedron is contractible and $n\ne 3$ (or $n=3$ and the Andrews–Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding.

In contrast, there exists a $2$-dimensional compact absolute retract $X$ such that $X\times I^k$ does not embed in any product of $2+k$ dendrites for each $k$.

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