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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.
57S25 Groups acting on specific manifolds
57R19 Algebraic topology on manifolds and differential topology
14F35 Homotopy theory and fundamental groups in algebraic geometry
Full Text: DOI
[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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