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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized manifolds in products of curves


Authors: Akira Koyama, Józef Krasinkiewicz and Stanisław Spież
Journal: Trans. Amer. Math. Soc. 363 (2011), 1509-1532
MSC (2010): Primary 54E45, 55N05, 57N35; Secondary 55M10, 57Q05
Published electronically: October 8, 2010
MathSciNet review: 2737275
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Abstract: The intent of this article is to distinguish and study some $ n$-dimensional compacta (such as weak $ n$-manifolds) with respect to embeddability into products of $ n$ curves. We show that if $ X$ is a locally connected weak $ n$-manifold lying in a product of $ n$ curves, then $ \operatorname{rank} H^{1}(X)\ge n$. If $ \operatorname{rank} H^{1}(X)=n$, then $ X$ is an $ n$-torus. Moreover, if $ \operatorname{rank} H^{1}(X)<2n$, then $ X$ can be presented as a product of an $ m$-torus and a weak $ (n-m)$-manifold, where $ m\ge 2n-\operatorname{rank} H^{1}(X)$. If $ \operatorname{rank} H^{1}(X)<\infty $, then $ X$ is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak $ 2$-manifold $ X$ lying in a product of two graphs such that $ H^{2}(X)=0$.


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Additional Information

Akira Koyama
Affiliation: Department of Mathematics, Faculty of Science, Shizuoka University, Suruga, Shizuoka, 422-8529, Japan
Email: sakoyam@ipc.shizuoka.ac.jp

Józef Krasinkiewicz
Affiliation: The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland
Email: jokra@impan.pl

Stanisław Spież
Affiliation: The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland
Email: spiez@impan.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05157-8
PII: S 0002-9947(2010)05157-8
Keywords: Embeddings, locally connected compacta, weak manifolds, products of curves
Received by editor(s): April 3, 2008
Received by editor(s) in revised form: June 4, 2009, June 18, 2009, and July 2, 2009
Published electronically: October 8, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.