On 3-manifolds with locally standard \((\mathbb Z_2)^3\)-actions (Q1942057)

An effective \((\mathbb{Z}_{2})^{n}\)-action on a closed \(n\)-dimensional manifold \(M\) is said to be a locally standard if it locally looks like the standard representation of \((\mathbb{Z}_{2})^{n}\) on \(\mathbb{R}^{n}\). A typical example of manifolds with such group actions is a small cover (e.g. a real projective space) defined by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)] or more generally a \(2\)-torus manifold (e.g. a standard sphere) defined by \textit{Z. Lü} and \textit{M. Masuda} [Colloq. Math. 115, No. 2, 171--188 (2009; Zbl 1165.57023)]. In the paper under review, the authors study which homology \(3\)-sphere admits a locally standard \((\mathbb{Z}_{2})^{3}\)-action and prove the following two theorems: Theorem 1.1. A \(\mathbb{Z}_{2}\)-homology \(3\)-sphere \(M\) admits a locally standard \((\mathbb{Z}_{2})^{3}\)-action if and only if \(M\) is homeomorphic to a connected sum \(N\#\cdots \# N\) of \(8\) copies of a \(\mathbb{Z}_{2}\)-homology \(3\)-sphere \(N\). In particular, if \(M\) is irreducible, then \(M\) is homeomorphic to the standard sphere \(S^{3}\). Theorem 1.2. An orientable rational homology \(3\)-sphere \(M\) with \(H_{1}(M;\mathbb{Z}_{2})\simeq \mathbb{Z}_{2}\) admits a locally standard \((\mathbb{Z}_{2})^{3}\)-action if and only if \(M\) is homeomorphic to a connected sum \(\mathbb{R} P^{3}\# N\#\cdots \# N\) of a real projective space \(\mathbb{R} P^{3}\) and \(8\) copies of a \(\mathbb{Z}_{2}\)-homology \(3\)-sphere \(N\). The outline of the proof of the 1st result is as follows. To prove this result, the authors introduce a cut-and-paste operation on the orbit spaces. This operation may be regarded as a combinatorial interpretation of the equivariant gluing operation around invariant submanifolds. Using a cut-and-paste operation, they show that the orbit space must be the connected sum of a \(3\)-ball \(B\) and a \(\mathbb{Z}_{2}\)-homology \(3\)-sphere \(Q'\), i.e., a \(\mathbb{Z}_{2}\)-homology \(3\)-ball, say \(B\# Q'\). Moreover, they prove that the locally standard \((\mathbb{Z}_{2})^{3}\)-manifold over \(B\) is only the standard \(3\)-sphere \(S^{3}\) with the standard \((\mathbb{Z}_{2})^{3}\)-action and such a manifold over \(Q'\) is only \((\mathbb{Z}_{2})^{3}\times Q'\) with the free \((\mathbb{Z}_{2})^{3}\)-action on \((\mathbb{Z}_{2})^{3}\)-factor. Therefore, the manifold over \(B\# Q'\) is equivariantly homeomorphic to the equivariant connected sum of a free orbit (i.e., \(2^{3}(=8)\) points) of \(S^{3}\) and that of \((\mathbb{Z}_{2})^{3}\times Q'\). This manifold is nothing but a connected sum \(N\#\cdots \# N\) of \(8\) copies of a \(\mathbb{Z}_{2}\)-homology \(3\)-sphere \(N(\simeq Q')\). By using the similar method, the 2nd result is proved.