TY - JOUR

T1 - On dimensionally exotic maps

AU - Dranishnikov, Alexander

AU - Levin, Michael

N1 - Funding Information:
The first author was supported by NSF grant DMS-0904278. The second author was supported by ISF grant 836/08.
Publisher Copyright:
© 2014, Hebrew University Magnes Press.

PY - 2014/10/2

Y1 - 2014/10/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84908503576&partnerID=8YFLogxK

U2 - 10.1007/s11856-014-1056-5

DO - 10.1007/s11856-014-1056-5

M3 - Article

AN - SCOPUS:84908503576

VL - 201

SP - 967

EP - 987

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -