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Contractible polyhedra in products of trees and absolute retracts in products of dendrites. (English) Zbl 1288.54015
The authors study the problem of embeddability of \(n\)-dimensional compacta into products of \(n\) dendrites or trees. The main result is Theorem 1.8 saying that each collapsible \(n\)-dimensional polyhedron \(PL\) embeds into the product of \(n\) trees and this product collapses on the image of the embedding. This theorem implies Corollary 1.9 saying that for a compact \(n\)-dimensional polyhedron \(X\) the following conditions are equivalent: (1) \(X\) \(PL\) embeds into the product of \(n\) trees, (2) \(X\) \(PL\) embeds into the product of an \((n-1)\)-dimensional polyhedron and a tree, (3) \(X\) collapses onto an \((n-1)\)-polyhedron, (4) \(X\) \(PL\) embeds into a collapsible \(n\)-dimensional polyhedron. Answering a problem of Koyama, Krasinkiewicz and Spie\(\dot{\mathrm z}\), the authors prove that Sklyarenko’s compact space \(X\) (i.e., the one-point compactification of the mapping telescope of the direct sequence consisting of the square maps of the unit circle) is a 2-dimensional AR space such that for every \(k\geq 0\) the product \(X\times [0,1]^k\) quasi-embeds into a product of \(2+k\) dendrites but does not embed into the product of any \(2+k\) curves.
MSC:
54C25 Embedding
57Q35 Embeddings and immersions in PL-topology
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