Generalized manifolds in products of curves

Abstract

Abstract. 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 H 1 (X) ≥ n. If rank H 1 (X) = n, then X is an n-torus. Moreover, if 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 ≥ 2n − rank H 1 (X). If rank H 1 (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 H 2 (X) = 0.

Surfaces in products of two curves

“…We refer the reader to [7] for the proofs of similar properties established there for ramified n-manifolds.…”

“…It suffices to consider the case where pr 1 (M ) is a one-point union of circles S 1 , S 2 , · · · , S k , k ≥ 1. Since pr 1 (M ) = |pr 1 (L)|, by Theorem 2.9 of [7], there is a 1-cell σ ∈ pr 1 (L) contained in S 1 . Then, for every 1-cell τ ∈ L/σ, we have |L/τ | = S 1 because σ ⊂ |L/τ | and the latter set is a union of disjoint circles.…”

“…From the proof of Theorem 2.9 in [7], we have that M = |L|, where L is an irreducible pseudo-2-manifold subcomplex of K 1 K 2 . In particular, M is the union of 2-cells of L. The projections pr i : P 1 × P 2 → P i induce natural functions K 1 K 2 → K i .…”

“…Each cell borders upon the next one along in turn: 7 There is an action of Z 2n ∼ = Z 2 Z n on M , with (o, o) being a unique fixed point, such that the sequence represents a cyclic orbit of the 2-cell ov 0 × ov 1 , and each cell is followed by its image under a generating homeomorphism. (The generating homeomorphism is the restriction to M of the homeomorphism h : P n × P n → P n × P n defined by h(x, y) = (ρ(y), ρ(x)), where ρ : P n → P n is the rotation around o through the angle 2π n .…”

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

“…11-01-00822, Russian Academy of Sciences program "Mathematical methods of construction and analysis of models of complex systems", Russian Government project 11.G34.31.0053 and Federal Program "Scientific and scientificpedagogical staff of innovative Russia 2009-2013". Proposition 1.3 ([17], [24]). The cone over every compact n-polyhedron embeds in a product of n + 1 trees.…”