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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
Keywords:
fundamental group; torus manifolds
Full Text:
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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