\(\mathbb{Z}{}/k\mathbb{Z}\)-finiteness for certain \(S^ 1\)-spaces (Q1802991)

Wall's finiteness obstruction [\textit{C. T. C. Wall}, Ann. Math., II. Ser. 81, 56-69 (1965; Zbl 0152.219)] has been described in the equivariant setting to the effect that for a compact Lie group \(G\) and a finitely dominated \(G\)-CW complex \(X\), the obstruction \(w^ G(X)\) vanishes if and only if \(X\) is \(G\)-homotopy equivalent to a finite \(G\)-CW complex [see the book of \textit{W. Lueck}, Transformation groups and algebraic \(K\)- theory, Lecture Notes in Mathematics 1408 (1989; Zbl 0679.57022), as well as the survey article by \textit{P. Andrzejewski}, ``Algebraic topology'', Proc. Conf., Poznań/Pol. 1989, Lect. Notes Math. 1474, 20-37 (1991; Zbl 0741.57012) for information on different approaches to finiteness problems]. The paper under review provides two results on the finiteness problem for the circle group \(G=S^ 1\). The first result describes conditions under which for a connected \(G\)-CW complex \(X\) with finitely many orbit types, there exists a proper subgroup \(K\) of \(G\) and a \(K\)-CW complex \(Y\) such that the twisted product \(G\times_ K Y\) is \(G\)-homotopy equivalent to \(X\). The second result gives a condition under which for a connected \(G\)- space \(X\), the vanishing of \(w^ H(X)\) implies the vanishing of \(w^ H(X)\).