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The fundamental group of locally standard \(T\)-manifolds. (English) Zbl 1401.57046
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 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.
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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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
[8] 10.2140/agt.2015.15.2393 · Zbl 1325.53108
[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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