The fundamental group of locally standard \(T\)-manifolds.

*(English)*Zbl 1401.57046A \(2n\)-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 A. Hattori and 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 [M. Wiemeler, Math. Z. 273, No. 3–4, 1063–1084 (2013; Zbl 1269.57014) and 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.

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.

Reviewer: Shintaro Kuroki (Okayama)

##### MSC:

57S25 | Groups acting on specific manifolds |

57R19 | Algebraic topology on manifolds and differential topology |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

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\textit{H. Zeng}, Algebr. Geom. Topol. 18, No. 5, 3031--3035 (2018; Zbl 1401.57046)

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##### References:

[1] | 10.1090/surv/204 |

[2] | 10.1090/gsm/124 |

[3] | 10.1215/S0012-7094-91-06217-4 · Zbl 0733.52006 |

[4] | 10.1515/9781400882526 · Zbl 0813.14039 |

[5] | ; Hattori, Osaka J. Math., 40, 1, (2003) |

[6] | 10.1007/BF01704912 · Zbl 0624.93016 |

[7] | 10.4310/MRL.2013.v20.n5.a6 · Zbl 1295.53088 |

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[9] | 10.1007/s11401-017-1034-4 · Zbl 1381.55003 |

[10] | 10.1007/s00209-012-1044-6 · Zbl 1269.57014 |

[11] | 10.1016/j.aim.2011.04.007 · Zbl 1226.57037 |

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