The fundamental group of locally standard \(T\)-manifolds
A \(2n\)-dimensional manifold with a half dimensional torus \(T^{n}\)-action is called a locally standard \(T\)-manifold if every point in \(M\) has a \(T\)-invariant open neighborhood which is equivariantly diffeomorphic to a \(T\)-invariant open set of a faithful representation space of \(T\). If a compact, oriented, locally standard \(T\)-manifold has a fixed point, then this manifold is called a torus manifold as introduced by \textit{A. Hattori} and \textit{M. Masuda} [Osaka J. Math. 40, No. 1, 1--68 (2003; Zbl 1034.57031)]. It is known that the fundamental group of a torus manifold is isomorphic to that of its orbit space \(M/T\) (by [\textit{M. Wiemeler}, Math. Z. 273, No. 3--4, 1063--1084 (2013; Zbl 1269.57014) and \textit{T. Yoshida}, Adv. Math. 227, No. 5, 1914--1955 (2011; Zbl 1226.57037)]). However, if there is no fixed point in a locally \(T\)-manifold, this result is not true. In the paper under review, the author computes the fundamental group of locally standard \(T\)-manifolds whose set of principal \(T\)-orbits has the trivial bundle structure, i.e., the author shows the following formula: \[ \pi_{1}(M)\simeq \pi_{1}(M/T)\times N/\widehat{N}, \] where \(N:=\mathrm{Hom}(S^{1},T)\) and \(\widehat{N}\) is the sublattice in \(N\) generated by the isotropy weights of the characteristic submanifolds, i.e., codimension-two invariant \(T\)-manifolds, in \(M\). Note that if there is a fixed point, then \(N=\widehat{N}\); therefore, this result contains the result of the fundamental group of torus manifolds.
scientific article; zbMATH DE number 6935828
scientific article