\(p\)-local stable splitting of quasitoric manifolds (Q315864)

A quasitoric manifold \(M\) is an even dimensional manifold equipped with a half dimensional torus action whose orbit space has the structure of a simple convex polytope \(P\). In the paper under review, the authors study the \(p\)-localized suspension \(\Sigma M_{(p)}\) of a quasitoric manifold \(M\). The main result can be stated as follows: \(\Sigma M_{(p)}\) is homotopy equivalent to \(X_{1}\vee \cdots \vee X_{p-1}\) such that for each \(i\), \(\widetilde{H}_{*}(X_{i};\mathbb{Z})=0\) unless \(\ast\equiv 2i+1\mod 2(p-1)\). This result may be regarded as a generalization of Mimura-Nishida-Toda's result for \(M=\mathbb{C}P^n\) (\(\mathbb{C}P^n\) is the easiest example of a quasitoric manifold). To show the main theorem, the authors construct a self map \(\underline{u}:M\to M\), by using the moment-angle complex \(\mathcal{Z}_{K(P)}\), which satisfies the condition of \textit{M. Mimura} et al. [J. Math. Soc. Japan 23, 593--624 (1971; Zbl 0217.48801)], and use a general result about the homotopy decomposition of the \(p\)-localized suspension by Mimura-Nishida-Toda. As a corollary of this result, they also obtain that two quasitoric manifolds \(M\), \(N\) with the same orbit space \(P\) satisfy that there is a homotopy equivalence \(\Sigma M_{(p)}\simeq \Sigma N_{(p)}\) for \(p>n\). Note that two quasitoric manifolds \(M\), \(N\) have the same orbit space \(P\) if and only if their equivariant cohomology rings are isomorphic as a graded rings. They also show that for the projection \(\pi:\mathcal{Z}_{K(P)}\to M\) and \(p>n\) the map \(\Sigma \pi_{(p)}:\Sigma (\mathcal{Z}_{K(P)})_{(p)}\to \Sigma M_{(p)}\) is null homotopic, and study the non-triviality of \(\pi_{(p)}\) itself and \(\Sigma^{\infty}\pi\) (in particular) for quasitoric manifolds over products of simplicies.