A generalization of Borsuk's pasting theorem and its application (Q1601652)

Generalizing the classical Borsuk pasting theorem for ANR's, the author proves that for any maps \(\alpha:A\to X\), \(f:A\to B\) between finite-dimensional A(N)R-compacta the pushout of the diagram \(X\overset{\alpha}{\leftarrow} A\overset{f}{\to} B\) is an A(N)R, provided \(\alpha\) is injective on the singularity set \(S_f=\{x\in A:f^{-1}f(x)\neq x\}\) of \(f\). The author applies this result to construct, given a prime \(p\), an AR-compactum \(M_p\) with \(\dim M_p=\dim_{\mathbb Z_{(p)}} M_p=\dim_{\mathbb Z_p} M_p=4\) and \(\dim_{\mathbb Q} M_p=\dim_{\mathbb Z_{p^\infty}} M_p=\dim_{\mathbb Z_q} M_p=3\) where \(q\neq p\) is prime. Next, he observes that for different primes \(p,q\) the product \(M_p\times M_q\) does not comply to the logarithmic law: \(7=\dim M_p\times M_q<\dim M_p+\dim M_q=8\). The paper ends with the question on the existence of 3-dimensional AR-compacta \(X,Y\) with \(\dim X\times Y\leq 5\).