Betti numbers of locally standard 2-torus manifolds
A locally standard \(2\)-torus manifold \(M\) is a smooth connected closed \(n\)-manifold with a locally standard \((\mathbb{Z}_{2})^{n}\)-action, i.e., locally looks like the standard \((\mathbb{Z}_{2})^{n}\)-action on \(\mathbb{R}^{n}\). It is well known that the orbit space \(Q\) of a locally standard \(2\)-torus manifold admits the structure of an \(n\)-dimensional manifold with corners. Due to the result by \textit{Z. Lü} and \textit{M. Masuda} [Colloq. Math. 115, No. 2, 171--188 (2009; Zbl 1165.57023)], all locally standard torus manifolds can be recovered by the following three data up to equivariant homeomorphism: (1) \(n\)-dimensional manifold with corner \(Q\); (2) a characteristic function \(\lambda:\mathcal{F}(Q)\to (\mathbb{Z}_{2})^{n}\), where \(\mathcal{F}(Q)\) is the set of facets of \(Q\): (3) a principal \((\mathbb{Z}_{2})^{n}\)-bundle \(\xi\) over \(Q\) (this may be regarded as an element in \(H^{1}(Q;(\mathbb{Z}_{2})^{n})\)). Denote the locally standard \(2\)-torus manifold which recovers from these data by the symbol \(M(Q;\lambda,\xi)\). For example, the small cover (defined in [\textit{M. W. Davis} and \textit{T. Januszkiewicz}, Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)]) is a locally standard \(2\)-torus manifold and can be denoted by \(M(P;\lambda)\) for a simple convex polytope \(P\) and a characteristic function \(\lambda\). Note that if \(P\) is a simple convex polytope then \(H^{1}(P;(\mathbb{Z}_{2})^{n})=\{0\}\); hence, the principal \((\mathbb{Z}_{2})^{n}\)-bundle over \(P\) is always trivial. In the paper under review, the authors compute the Betti numbers of a locally standard \(2\)-torus manifold \(M(Q;\lambda,\xi)\) such that the boundary of \(Q\) is isomorphic to the boundary of a simple convex polytope \(P\). As a consequence, they show that the Betti numbers of \(M(Q;\lambda,\xi)\) can be computed by the \(h\)-vectors of \(Q\) and the Betti numbers of \(E\), where \(E\) is the total space of the principal \((\mathbb{Z}_{2})^{n}\)-bundle \(\xi\). Namely, the Betti numbers of \(M(Q;\lambda,\xi)\) are determined by the number of faces on the boundary of \(Q\) and the Betti numbers of \(E\). Because such a manifold with corners \(Q\) can be constructed by the connected sum of a simple convex polytope \(P\) and a closed \(n\)-dimensional manifold, the authors remark that the locally standard \(2\)-torus manifold \(M(Q;\lambda,\xi)\) is obtained by the gluing along the free orbits of the small cover \(M(P,\lambda)\) and a closed manifold \(\widehat{E}\) with free \((\mathbb{Z}_{2})^{n}\)-action obtained by filling the boundaries of \(E\) by \(2^{n}\) copies of \(P\) for \(n>2\). The Betti numbers of \(M(P,\lambda)\) have already been computed by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)], i.e., \(h\)-vectors of \(P\), the Mayer-Vietoris exact sequence is one of the key tools to compute the Betti numbers of \(M(Q;\lambda,\xi)\).
scientific article; zbMATH DE number 6343671