Dimension of the product and classical formulae of dimension theory

Details Dimension of the product and classical formulae of dimension theory Dimension of the product and classical formulae of dimension theory

2011-12-08

arxiv.org/abs/1112.1974v1

Mathematics

Algebraic Topology

Scientific paper

Let $f : X \lo Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that $\dim X \leq \dim Y +\dim f$ where $\dim f =\sup \{\dim f^{-1}(y): y \in Y \}$. The first author conjectured that {\em $\dim Y + \dim f$ in Hurewicz's theorem can be replaced by $\sup \{\dim (Y \times f^{-1}(y)): y \in Y \}$}. We disprove this conjecture. As a by-product of the machinery presented in the paper we answer in negative the following problem posed by the first author: {\em Can for compact $X$ the Menger-Urysohn formula $\dim X \leq \dim A + \dim B +1$ be improved to $\dim X \leq \dim (A \times B) +1$ ?} On a positive side we show that both conjectures holds true for compacta $X$ satisfying the equality $dim(X\times X)=2\dim X$.

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