On the topology of bi-cyclopermutohedra (Q6099748)

\textit{G. Yu. Panina} [Proc. Steklov Inst. Math. 288, No. 1, 132--144 (2015; Zbl 1322.51013)] has introduced an \((n-2)\)-dimensional regular CW complex whose \(k\)-cells are labeled by cyclically ordered partitions of \(\{1, 2, \ldots, n + 1\}]\) into \((n + 1-k)\) non-empty parts, where \((n + 1-k) > 2\) -- the boundary relations in the complex corresponding to the orientation preserving refinement of partitions -- called a cyclopermutohedron and denoted by \(CP_{n+1}\). Using discrete Morse theory, \textit{I. Nekrasov} et al. [Eur. J. Math. 2 , no. 3, 835--852 (2016; Zbl 1361.51015)] showed that the homology groups \(H_i(CP_{n+1})\) are torsion free for all \(i\geq 0\) and computed their Betti numbers. \(CP_{n+1}\) admits a free \({\mathbb Z}_2\) action; the quotient space \(CP_{n+1}/{\mathbb Z}_2\) is called a bi-cyclopermutohedron and denoted by \(QP_{n+1}\). The aim of this paper is to compute the \({\mathbb Z}_2\)- and the \({\mathbb Z}\)-homology groups of \(QP_{n+1}\).