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Torus Actions of Complexity 1 and Their Local Properties

  • Anton A. Ayzenberg
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Abstract

We consider an effective action of a compact (n − 1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n − 1 has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold G4,2, the complete flag manifold F3, and quasitoric manifolds with an induced action of a subtorus of complexity 1.

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References

  1. 1.
    A. Ayzenberg, “Topological model for h″ vectors of simplicial manifolds,” Bol. Soc. Mat. Mex., Ser. 3, 23 (1), 413–421 (2017); arXiv: 1502.05499 [math.AT].MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Ayzenberg, “Space of isospectral periodic tridiagonal matrices,” arXiv: 1803.11433 [math.AT].Google Scholar
  3. 3.
    A. Bialynicki-Birula, “Some theorems on actions of algebraic groups,” Ann. Math., Ser. 2, 98 (3), 480–497 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. E. Bredon, Introduction to Compact Transformation Groups (Academic, New York, 1972), Pure Appl. Math. 46.zbMATHGoogle Scholar
  5. 5.
    V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.CrossRefzbMATHGoogle Scholar
  6. 6.
    V. M. Buchstaber and N. Ray, “An invitation to toric topology: Vertex four of a remarkable tetrahedron,” in Toric Topology: Int. Conf., Osaka, 2006 (Am. Math. Soc., Providence, RI, 2008), Contemp. Math. 460, pp. 1–27.Google Scholar
  7. 7.
    V. M. Buchstaber “(jointly with S. Terzić), “(2n, k)-manifolds and applications,” in “Okounkov bodies and applications: Abstr. Workshop, May 25–31, 2014,” Oberwolfach Rep. 11 (2), 1459–1513 (2014), pp. 1469–1472.Google Scholar
  8. 8.
    V. M. Buchstaber and S. Terzić, “Topology and geometry of the canonical action of T 4 on the complex Grassmannian G4,2 and the complex projective space CP5,” Moscow Math. J. 16 (2), 237–273 (2016); arXiv: 1410.2482 [math.AT].MathSciNetzbMATHGoogle Scholar
  9. 9.
    V. M. Buchstaber and S. Terzić, “Toric topology of the complex Grassmann manifolds,” arXiv: 1802.06449 [math.AT].Google Scholar
  10. 10.
    Ph. T. Church and K. Lamotke, “Almost free actions on manifolds,” Bull. Aust. Math. Soc. 10, 177–196 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. W. Davis and T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions,” Duke Math. J. 62 (2), 417–451 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. Fintushel, “Classification of circle actions on 4-manifolds,” Trans. Am. Math. Soc. 242, 377–390 (1978).MathSciNetzbMATHGoogle Scholar
  13. 13.
    M. Goresky, R. Kottwitz, and R. MacPherson, “Equivariant cohomology, Koszul duality, and the localization theorem,” Invent. Math. 131 (1), 25–83 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Y. Karshon and S. Tolman, “Classification of Hamiltonian torus actions with two-dimensional quotients,” Geom. Topol. 18 (2), 669–716 (2014); arXiv: 1109.6873 [math.SG].MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Y. Karshon (joint with S. Tolman), “Topology of complexity one quotients,” Talk at the Int. Conf. “Algebraic Topology, Combinatorics, and Mathematical Physics” on occasion of V. Buchstaber’s 75th birthday (Moscow, May 2018), http://www.mathnet.ru/present20512.Google Scholar
  16. 16.
    M. Masuda and T. Panov, “On the cohomology of torus manifolds,” Osaka J. Math. 43 (3), 711–746 (2006); arXiv: math/0306100 [math.AT].MathSciNetzbMATHGoogle Scholar
  17. 17.
    P. Orlik and F. Raymond, “Actions of the torus on 4-manifolds. I,” Trans. Am. Math. Soc. 152, 531–559 (1970).MathSciNetzbMATHGoogle Scholar
  18. 18.
    P. Orlik and F. Raymond, “Actions of the torus on 4-manifolds. II,” Topology 13, 89–112 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    D. A. Timashëv, “G-varieties of complexity 1,” Russ. Math. Surv. 51 (3), 567–568 (1996) [transl. from Usp. Mat. Nauk 51 (3), 213–214 (1996)].CrossRefzbMATHGoogle Scholar
  20. 20.
    D. A. Timashev, “Classification of G-varieties of complexity 1,” Izv. Math. 61 (2), 363–397 (1997) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 61 (2), 127–162 (1997)].Google Scholar
  21. 21.
    E. B. Vinberg, “Discrete linear groups generated by reflections,” Izv. Math. 5 (5), 1083–1119 (1971) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 35 (5), 1072–1112 (1971)].CrossRefzbMATHGoogle Scholar
  22. 22.
    T. Yoshida, “Local torus actions modeled on the standard representation,” Adv. Math. 227 (5), 1914–1955 (2011); arXiv: 0710.2166 [math.GT].MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Faculty of Computer ScienceNational Research University “Higher School of Economics,”MoscowRussia

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