Constructing compacta of different extensional dimensions (Q2739253)

The fundamental idea of extension theory is indicated by the expression \(\dim X\leq K\) (the author uses \(\text{e-dim }X\leq K\)). This simply means that \(X\) is a topological space, \(K\) is a CW-complex, and every map of a closed subset of \(X\) to \(K\) extends to a map of \(X\) to \(K\). We often say that \(K\) is an absolute extensor for \(X\). For example, if \(X\) is a metrizable space, then the covering dimension of \(X\) is \(\leq n\) if and only if \(\dim X\leq S^n\). NEWLINENEWLINENEWLINEThe author generalizes a result due to A. Dranishnikov involving truncated cohomology \(T^*\). Let us quote the main result, Theorem 1.2, of this paper: NEWLINENEWLINENEWLINETheorem. Let \(K\) and \(P\) be countable simplicial complexes and let \(T^*\) be a truncated cohomology generated by \(L\). NEWLINENEWLINENEWLINE(a) If \(T^0\) is continuous, \(T^0(P)\neq 0\) and \(T^k(K)=0\) for all \(k<0\), then there exists a compactum \(X\) with \(\dim X\leq K\) such that \(X\) admits an essential map to \(P\). NEWLINENEWLINENEWLINE(b) If \(T^0\) is strongly continuous, \(T^0(P)\neq 0\) and \(T^k(K)=0\) for all \(k\leq 0\), then there exists a compactum \(X\) such that \(P<\dim X\leq K\). NEWLINENEWLINENEWLINEIn part (b), the notation, \(P<\dim X\), which seems not to be standard, should be interpreted as the statement that, \(\dim X\leq P\) is false. On p. 81, below Theorem 1.2, it is stated that the condition \(P<\dim X\) is stronger than the existence of an essential map to \(P\). We think this means that for some closed subspace \(A\) of \(X\), there is an essential map of \(A\) to \(P\). NEWLINENEWLINENEWLINEThe author shows that the statement (b) leads to a proof of the existence of infinite dimensional metrizable compacta whose cohomological dimension modulo \(\mathbb{Z}\) equals 2.