Geodesic
Torus actions of complexity 1 and their local properties
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 23-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $G_{4,2}$, the complete flag manifold $F_3$, and quasitoric manifolds with an induced action of a subtorus of complexity $1$.
Mots-clés : torus action
Keywords: torus representation, Grassmann manifold, complete flag manifold, quasitoric manifold, bundle classification, Hopf bundle, sponge, space of periodic tridiagonal matrices. 
@article{TM_2018_302_a1,
     author = {Anton A. Ayzenberg},
     title = {Torus actions of complexity 1 and their local properties},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {23--40},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_302_a1/}
}
TY  - JOUR
AU  - Anton A. Ayzenberg
TI  - Torus actions of complexity 1 and their local properties
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2018
SP  - 23
EP  - 40
VL  - 302
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2018_302_a1/
LA  - ru
ID  - TM_2018_302_a1
ER  - 
%0 Journal Article
%A Anton A. Ayzenberg
%T Torus actions of complexity 1 and their local properties
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2018
%P 23-40
%V 302
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2018_302_a1/
%G ru
%F TM_2018_302_a1
Anton A. Ayzenberg. Torus actions of complexity 1 and their local properties. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 23-40. http://geodesic.mathdoc.fr/item/TM_2018_302_a1/

[1] Ayzenberg A., “Topological model for $h''$ vectors of simplicial manifolds”, Bol. Soc. Mat. Mex. Ser. 3, 23:1 (2017), 413–421, arXiv: 1502.05499 [math.AT] | DOI | MR

[2] Ayzenberg A., Space of isospectral periodic tridiagonal matrices, 2018, arXiv: 1803.11433 [math.AT]

[3] Bialynicki-Birula A., “Some theorems on actions of algebraic groups”, Ann. Math. Ser. 2, 98:3 (1973), 480–497 | DOI | MR

[4] Bredon G. E., Introduction to compact transformation groups, Pure Appl. Math., 46, Acad. Press, New York, 1972 | MR

[5] Buchstaber V. M., Panov T. E., Toric topology, Math. Surv. Monogr., 204, Am. Math. Soc., Providence, RI, 2015 | DOI | MR

[6] Buchstaber V. M., Ray N., “An invitation to toric topology: Vertex four of a remarkable tetrahedron”, Toric topology, Int. Conf. (Osaka, 2006), Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 1–27 | DOI | MR

[7] Buchstaber V. M., Terzić S., “$(2n,k)$-manifolds and applications”, Okounkov bodies and applications, Abstr. Workshop (May 25–31, 2014), Oberwolfach Rep., 11, no. 2, 2014, 1459–1513 | DOI | MR

[8] Buchstaber V. M., Terzić S., “Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $\mathbb CP^5$”, Moscow Math. J., 16:2 (2016), 237–273, arXiv: 1410.2482 [math.AT] | MR

[9] Buchstaber V. M., Terzić S., Toric topology of the complex Grassmann manifolds, 2018, arXiv: 1802.06449 [math.AT]

[10] Church Ph. T., Lamotke K., “Almost free actions on manifolds”, Bull. Aust. Math. Soc., 10 (1974), 177–196 | DOI | MR

[11] Davis M. W., Januszkiewicz T., “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR

[12] Fintushel R., “Classification of circle actions on 4-manifolds”, Trans. Amer. Math. Soc., 242 (1978), 377–390 | MR

[13] Goresky M., Kottwitz R., MacPherson R., “Equivariant cohomology, Koszul duality, and the localization theorem”, Invent. math., 131:1 (1998), 25–83 | DOI | MR

[14] Karshon Y., Tolman S., “Classification of Hamiltonian torus actions with two-dimensional quotients”, Geom. Topol., 18:2 (2014), 669–716, arXiv: 1109.6873 [math.SG] | DOI | MR

[15] Karshon Y., Tolman S., 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

[16] Masuda M., Panov T., “On the cohomology of torus manifolds”, Osaka J. Math., 43:3 (2006), 711–746, arXiv: math/0306100 [math.AT] | MR

[17] Orlik P., Raymond F., “Actions of the torus on 4-manifolds. I”, Trans. Amer. Math. Soc., 152 (1970), 531–559 | MR

[18] Orlik P., Raymond F., “Actions of the torus on 4-manifolds. II”, Topology, 13 (1974), 89–112 | DOI | MR

[19] D. A. Timashëv, “$G$-varieties of complexity 1”, Russ. Math. Surv., 51:3 (1996), 567–568 | DOI | DOI | MR

[20] D. A. Timashev, “Classification of $G$-varieties of complexity 1”, Izv. Math., 61:2 (1997), 363–397 | DOI | DOI | MR

[21] È. B. Vinberg, “Discrete linear groups generated by reflections”, Izv. Math., 5:5 (1971), 1083–1119 | DOI | MR

[22] Yoshida T., “Local torus actions modeled on the standard representation”, Adv. Math., 227:5 (2011), 1914–1955, arXiv: 0710.2166 [math.GT] | DOI | MR