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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
##### Keywords:
collapsible polyhedron; $$PL$$ embedding; product of trees
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##### References:
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