TY - JOUR
T1 - Generalized manifolds in products of curves
AU - Koyama, Akira
AU - Krasinkiewicz, Józef
AU - Spiez, Stanislaw
PY - 2011/3
Y1 - 2011/3
N2 - The intent of this article is to distinguish and study some ndimensional 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 rank H1(X) ≥ n. If rankH1(X) = n, then X is an n-torus. Moreover, if rank H1(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H1(X). If rank H1(X) < ∞, 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 H2(X) = 0.
AB - The intent of this article is to distinguish and study some ndimensional 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 rank H1(X) ≥ n. If rankH1(X) = n, then X is an n-torus. Moreover, if rank H1(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H1(X). If rank H1(X) < ∞, 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 H2(X) = 0.
KW - Embeddings
KW - Locally connected compacta
KW - Products of curves
KW - Weak manifolds
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U2 - 10.1090/S0002-9947-2010-05157-8
DO - 10.1090/S0002-9947-2010-05157-8
M3 - Article
AN - SCOPUS:79951842551
VL - 363
SP - 1509
EP - 1532
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 3
ER -