Abstract
We call a value y = f(x) of a map f: X → Y dimensionally regular if dimX ≤ dim(Y × f −1(y)). It was shown in [6] that if a map f: X → Y between compact metric spaces does not have dimensionally regular values, then X is a Boltyanskii compactum, i.e., a compactum satisfying the equality dim(X × X) = 2dim X − 1. In this paper we prove that every Boltyanskii compactum X of dimension dim X ≥ 6 admits a map f: X → Y without dimensionally regular values. We show that the converse does not hold by constructing a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.
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The first author was supported by NSF grant DMS-0904278.
The second author was supported by ISF grant 836/08.
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Dranishnikov, A., Levin, M. On dimensionally exotic maps. Isr. J. Math. 201, 967–987 (2014). https://doi.org/10.1007/s11856-014-1056-5
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DOI: https://doi.org/10.1007/s11856-014-1056-5