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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Sergey A. Melikhov and Justyna Zajac
Journal: Proc. Amer. Math. Soc. 141 (2013), 2519-2535
MSC (2010): Primary 54C25, 57Q35; Secondary 55P57, 06A07, 57M20, 55M15
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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Additional Information

Sergey A. Melikhov
Affiliation: Steklov Mathematical Institute of the Russian Academy of Sciences, ul. Gubkina 8, Moscow 119991, Russia
Email: melikhov@mi.ras.ru

Justyna Zajac
Affiliation: Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Email: jzajac@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11524-8
PII: S 0002-9939(2013)11524-8
Received by editor(s): May 17, 2011
Received by editor(s) in revised form: September 29, 2011, October 7, 2011, and October 8, 2011
Published electronically: February 21, 2013
Additional Notes: The first author is supported by Russian Foundation for Basic Research Grant No. 11-01-00822, Russian Government project 11.G34.31.0053 and Federal Program “Scientific and Scientific-Pedagogical Staff of Innovative Russia” 2009–2013
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.