Bubbly continua with the shape of spheres (Q6545502)

This paper concerns the question posed by \textit{W. Kuperberg} [Fundam. Math. 83, 7--23 (1973; Zbl 0269.54020)]: if a \(q\)-dimensional compactum \(X\) is without small \(q\)-dimensional cycles (WSC\({}_q\)), then is it true that any family of mutually disjoint \(q\)-bubbles contained in \(X\) is finite?\N\NA compactum \(X\) is \(q\)-bubble if \(H^q(X) \neq 0\) and \(H^q(Y)=0\) for every proper closed subset \(Y\) of \(X\), and \(X\) is WSC\({}_q\) if \(H^q(X)\) is finitely generated. Here \(H^\ast(X)\) denotes the Alexander-Spanier cohomology with integer coefficients.\N\NThe main result of the paper states: for each \(q\geq 2\) and for every sequence of pairwise relatively prime integers greater than \(1\), there is a \(q\)-dimensional compactum that has the shape of the \(q\)-sphere \(S^q\) and contains infinitely many mutually disjoint \(q\)-bubbles. Note here that for each \(q\), there are as many compacta as there are real numbers.