Last update: November 21, 2012

Toric Topology 2012 in Osaka

Dates : November 16 (Fri) -- November 19 (Mon), 2012

Venue : Lecture Room 301 in €Κ€, Osaka City University

Our building (€Κ€) is No. 29 on the campus map and next to the Guesthouse (No. 25 on the campus map) ,

This meeting is an activity of the bilateral program between Japan and Russia: gTopology and geometry of torus actions and combinatorics of orbit quotientsh

Previous meeting: Toric Topology 2011 in Osaka

After our meeting the 39th conference on Transformation Groups will be held in Tokyo from Nov. 23rd to 25th.

Related future conference (June 17-22, 2013) : Algebraic Topology and Abelian Functions Conference in honour of Victor Buchstaber on the occasion of his 70th birthday (Moscow)

Visitors and participants ( shows speakers)

@@ ABE Hiraku (Tokyo Metropolitan Univ., Japan) 11/11-23

@AYZENBERG Anton (Moscow State Univ. Russia) 11/17-29 (11/23-27 Tokyo)

@@ CAI Li (Kyushu Univ., Japan) 11/15-30

@CHOI Suyoung (Ajou Univ., Korea) 11/16-20

@FUKUKAWA Yukiko (Osaka City Univ. Japan)

@@ HARA Yasuhiro (Osaka Univ. Japan)

@@ HASUI Sho (Kyoto Univ. Japan)

@HATANAKA Miho (Osaka City Univ. Japan)

@HIGASHITANI Akihiro (Osaka Univ. Japan)

@@ HORIGUCHI Tatsuya (Osaka City Univ. Japan)

@@ IRIYE Kouyemon (Osaka Pref. Univ. Japan)

@ISHIDA Hiroaki (OCAMI, Japan)

@KAJI Shizuo (Yamaguchi, Univ. Japan) 11/15-20

@@ KAWACHI Yasuhiro (Japan)

@@ KIMURA Yoshiyuki (Osaka City Univ. Japan)

@KISHIMOTO Daisuke (Kyoto Univ. Japan) 11/17-19

@KAMISHIMA Yoshinobu (Tokyo Metropolitan Univ. Japan) 11/17-19

@@ KOMORI Yohei (Waseda Univ. Japan)

@@ KUWATA Hideya (Osaka City Univ. Japan)

@@ LEE Eunjeong (KAIST, Korea) 11/15-21

@LIMONCHENKO Ivan (Moscow State Univ., Russia) 11/15-29 (11/23-27 Tokyo)

@LU Zhi (Fudan Univ., China) 11/15-19

@@ MASUDA Mikiya (Osaka City Univ. Japan)

@MATSUMURA Tomoo (KAIST, Korea) 11/11-20

@MURAI Satoshi (Yamaguchi Univ. Japan) 11/16-18

@@ NAGASE Teruko (Doshisha Univ. Japan)

@@ NUMATA Hasuhide (Shinshu Univ. Japan)

@PANOV Taras (Moscow State Univ., Russia) 10/22-12/2 (11/23-27 Tokyo)

@PARK Hanchul (Ajou Univ. Korea) 11/16-20

@@ PARK Kyoung Sook (Ajou Univ. Korea) 11/16-19

@PARK Seonjeong (KAIST, Korea) 11/15-21

@SARKAR Soumen (KAIST, Korea) 11/11-26 (11/22-26 Tokyo)

@@ SATO Hiroshi (Gifu Shotoku Gakuen Univ.)

@SATO Takashi (Kyoto Univ. Japan)

@@ SONG Jongbaek (KAIST, Korea) 11/15-21

@@ SUH Dong Youp (KAIST, Korea) 11/15-19

@SUN Yi (Fudan Univ. China) 11/15-19

@@ SUYAMA Yusuke (Osaka City Univ. Japan)

@@ TAMBOUR Jerome (KAIST, Korea) 11/14-22 (Cancelled)

@UMEMOTO Yuriko (Osaka City Univ. Japan)

@USTINOVSKY Yury (Steklov Institute., Russia) 11/8-26 (11/23-25 Tokyo)

@@ YAMAGUCHI Khohei (The University of Electro-Communications, Japan)

@@ YOSHIDA Takahiko (Meiji Univ. Japan) 11/16-17

@YU Li (Nanjing Univ., China) 11/17-22

@@ ZENG Haozhi (Osaka City Univ.)

Titles and abstracts

AYZENBERG Anton

Title: Polyhedral joins and products

Abstract: For each simplicial complex K on m vertices and simplicial complexes L1,L2,...,Lm we associate new simplicial complex K(L1,L2,...,Lm) defined similar to the polyhedral product functor. This construction gives the structure of an operad on the set of all simplicial complexes. We will describe the homotopy type of K(L1,L2,...,Lm) and show for which choice of K, L1,...,Lm the complex K(L1,L2,...,Lm) is a sphere.

CHOI Suyoung

Title: On the toric rigidity of simple polytopes

A simple polytope (resp, fan) is said to be (toric) cohomologically rigid if its combinatorial type can be decided by the cohomology rings of its supporting quasitoric manifold (resp, fan). The notion of cohomological rigidity for fans was originally introduced by Masuda and Suh, and, later, it was re-defined for simple polytope by C-Panov-Suh. Since the cohomology ring of quasitoric manifold determines the Tor-algebra of its orbit polytope (due to C-Panov-Suh), Buchstaber suggested to define the notion of algebraic version of rigidity: a simple polytope is said to be (toric) algebraically rigid if its combinatorial type can be decided by its Tor-algebra. With slightly weaker condition, C and Kim has been introduced the notion of combinatorial rigidity: a simple polytope is said to be (toric) combinatorially rigid if its combinatorial type can be decided by its bigraded Betti numbers. Let $P$ be a simple polytope (supportable quaistoric manifolds). Statement $A$ : $P$ is combinatorially rigid. Statement $B$ : $P$ is algebraically rigid. STatement $C$ : $P$ is cohomologically rigid. Due to the work C-Panov-Suh, it is obvious that $A \Rightarrow B \Rightarrow C$. However, the converse directions has been unknown. In this talk, I would like to present the examples which proves the converse does not hold: there is a polytope which is cohomologically rigid but not algebraically, and there is a polytope which is algebraically rigid but not cohomologically. I expect that these examples can be solutions to the question by Panov at the toric conference held in Osaka in November 2011.

FUKUKAWA Yukiko

Title: The ring structure of the cohomology ring of the Peterson variety

The Peterson variety Y is the subvariety of the flag variety defined as a special class of the Hessenberg variety, and some S^1 subtorus in T^n acts on Y. The S^1-fixed point set of Y is isolated and is the subset of the T^n-fixed point set of the flag variety. In [1] I, M. Masuda and H. Ishida determine the ring structure of the T^n-equivariant cohomology ring of the flag variety by using GKM graph. Since the action of S^1 on Y dose not satisfy the GKM conditions, we can not use GKM theory directly when we calculate the ring structure of the S^1-equivariant cohomology ring of Y. But by treating the S^1-equivariant cohomology ring of Y as the subring of the T^n-equivariant cohomology ring of the flag variety, we can determine the ring structure of the (S^1-equivariant) cohomology ring of Y by using the combinatorial date. My talk will be based on the joint work with M. Harada and M. Masuda.

[1] Y. Fukukawa, M. Masuda and H. Ishida, The cohomology ring of the GKM graph of the flag manifold of classical type. arXiv:1104.1832

HATANAKA Miho

Title: The uniqueness of decompositions of a (quasi)toric manifold into products of real 2 or 4 dimensional (quasi)toric manifolds

Abstract: The uniqueness of a direct decomposition of a closed smooth manifold does not hold in general up to diffeomorphism. However, it holds for decompositions of (quasi)toric manifolds into products of real 2 or 4 dimensional (quasi)toric manifolds up to diffeomorphism, which I will talk about.

HIGASHITANI Akihiro

Title: Edge contractions and minimal flag simplicial spheres

Abstract: Abstract: In this talk, we will introduce edge contractions for simplicial complexes and discuss the flag simplicial spheres which do not preserve the flagness by edge contraction for every edge.

ISHIDA Hiroaki

Title: Complex manifolds with maximal torus actions

Abstract: Let $M$ be a compact connected complex manifold of complex dimension $n$ with an effective $T^m$-action preserving the complex structure. We call the $T^m$-action is maximal if there exists an orbit of real dimension $2m-2n$. In this talk, we give a one-to-one correspondence between the family of compact connected complex manifolds with maximal torus actions and the family of pairs $(\Delta, \mathfrak{h})$ of nonsingular fan in $\mathbb{R}^m$ and subspace of $\mathbb{C}^m$ satisfying certain conditions. As an application, we show that the moment-angle complex $Z_K$ for a simplicial poset $K$ (may have a ghost vertex) admits a complex structure invariant under the natural torus action if and only if $K$ is a star-shaped sphere and $\dim Z_K$ is even. Moreover, we show that such a complex structure on $Z_K$ coincides with one of complex structures constructed by Panov and Ustinovsky.

KAJI Shizuo

Title: Cohomology of a GKM-graph with symmetry

Abstract: By GKM theory, for a manifold with a good torus action one can associate a graph which encodes topological information such as the equivariant cohomology. If the torus action extends to a Lie group action, the graph is equipped with the Weyl group symmetry. I will consider an abstraction of this setting; starting with a GKM graph with symmetry, I will deduce some properties for the cohomology of the graph.

KAMISHIMA Yoshinobu

Title: On the holomorphic torus-Bott tower of aspherical manifolds

Abstract: The purpose of this note is to explain the recent results concerning holomorphic torus-nilBott tower (the fiber is a complex torus). We have introduced this notion two years ago when the structure was not clear at that time. The first result is that the Kaehler Bott manifold will be biholomorphic to a complex euclidean space form. Secondly, we discuss examples of torus-nilBott manifolds which admit a non-Kaehler geometric structure, that is a LcK structure, a complex contact structure. This is partially a joint work with M. Nakayama.

KISHIMOTO Daisuke

Title: The conjecture of Bahri, Bendersky, Cohen and Gitler on polyhedral products

Abstract : I will explain an affirmative resolution to the conjecture of Bahri, Bendersky, Cohen and Gitler on wedge decompositions of polyhedral products of shifted complexes. This is a joint work with K. Iriye.

LIMONCHENKO Ivan

Title: Nontrivial torsion in cohomology rings of some moment-angle manifolds

Abstract: Our aim is to find a combinatorial characterisation of a simplicial complex $K$, such that there is no torsion in the cohomology ring of the corresponding moment-angle complex~$\mathcal Z_K$. In this talk we discuss a few examples of such complexes, well-known in combinatorial geometry. Then some recent results, concerning the nontrivial torsion case will be introduced.

LU Zhi

Title: On the lifting problem from small covers to quasitoric manifolds

Abstract: We shall consider the lifting problem from small covers to quasitoric manifolds from the view-point of cobordism, and obtain that the lifting from small covers to quasitoric manifolds is realizable in the sense of cobordism.

MATSUMURA Tomoo

Title: Schubert Calculus of Weighted Grassmannians (joint work with H. Abe)

Abstract: After a brief introduction to Schubert calculus, we will explain our results on the computation of Schubert structure constants for weighted Grassmannians. Namely, the equivariant cohomology of weighted Grassmannians has a natural Schubert basis (weighted Schubert classes) and the structure constants of the ring are computed in terms of the Knutson-Tao jigsaw puzzles. We also managed to prove that the structure constants are positive in a sense of Graham, i.e. they are polynomials in certain parameters with non-negative coefficients. If time allows, we will also explain the twisting of Schur functions that allows us to present the cohomology rings by a certain variant of symmetric polynomials.

MURAI Satoshi

Title: Stacked triangulations and face numbers

Abstract: A simplicial d-polytope is said to be r-stacked if it can be@tria ngulated without introducing faces of dimension < d-r. For example, the tria ngular bipyramid is 1-stacked but regular octahedron is not 1-stacked. In this talk, I will discuss the r-stackedness of simplicial polytopes when r is small. In particular, I will show that the r-stackedness of a simplicial polytope only depends on its face numbers when r is small relative to its dimension.

PARK Hanchul

Title: A new graph invariant arises in toric topology

Abstract: In this talk, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number and Euler characteristic of the real toric variety associated to a graph associahedron $P_{B(G)}$. They can be calculated by a purely combinatorial method (in terms of graphs) and are named $a_i(G)$ and $b(G)$, respectively. To our surprise, for specific families of the graph G, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers. This is a joint work with Suyoung Choi of Ajou University.

PARK Seonjeong

Title: Strong cohomological rigidity problems for projective bundles over a quasitoric manifold

Abstract: A quasitoric manifold M is said to be strongly cohomological rigid if any graded ring isomorphism on the cohomology ring of M is induced by a self-diffeomorphism on M. If E is the Whitney sum of complex line bundles over a quasitoric manifold, then the projectivization P(E) is also quasitoric manifold. In this talk, I will introduce some affirmative partial answers to the strong cohomological rigidity problems for such a quasitoric manifold P(E). This talk is based on the joint work with Suyoung Choi.

PANOV Taras

Title: Moment-angle complexes corresponding to chordal graphs

Abstract: Let K be a flag simplicial complex, Z_K the corresponding moment-angle complex, and k a field. We prove that the following conditions are equivalent: - the face ring k[K] is a Golod; - the multiplication in H^*(Z_K) is trivial; - the 1-skeleton of K is a chordal graph; - \zk is Z_K is homotopy equivalent to a wedge of spheres. This gives a complete characterisation of flag complexes K whose corresponding Z_K has homotopy type of a wedge of spheres. Based on a joint work with Jelena Grbic, Stephen Theriaul and Jie Wu.

SARKAR Soumen

Title: On equivariant cobordism of lens spaces (Joint work with Prof. Dong Youp Suh)

Abstract: We give a new definition of lens spaces. We discuss the equivariant cobordism of lens spaces with respect to the natural compact torus actions. The main tool is the quasitoric theory.

SATO Takashi

Title: The cohomology ring of the GKM graph of the flag manifold of type F_4

Abstract: In this talk, we determine the equivariant cohomology ring of F_4/T combinatorially by the GKM theory.

SUN Yi

Title: "Buchstaber invariant of universal complexes

Abstract: A universal complex is introduced by Davis and Januszkiewicz to give an equivalent definition of colorings of simpicial complexes. It also can derive an equivalent definiton of Buchstaber invariant and a condition of judgment for the lifting problem of colorings. It has been proved that real universal complexes cannot be nondegenerately mapped into complex universal complexes when dimension is greater than 3 by Anton Ayzenberg. And I've proved that there is a nondegenerate map from 4-dimensional real universal complex to 5-dimensional complex universal complex, and want to generalize the similar property to higher dimension.

TAMBOUR Jerome (Cancelled)

Title: A toric-like foliation on complex manifolds and Dehn-Sommerville equations

Abstract: Following the works of Lopez de Medrano, Verjovsky and Meersseman, Bosio described in 2001 a construction of non kahler compact complex manifolds. Those manifolds, known as LVMB manifolds, are parametrized by simplicial complete fans and, when the fan is rational, one can also define a foliation on a LVMB manifold N whose leaves are closed and diffeomorphic to compact tori. Moreover, the leaves space can naturally be identified with a compact toric variety. The Betti numbers of the cohomology of this torus variety are closely related to the combinatorics of the fan describing N. In the non rational case, one can still define a foliation on a LVMB manifold N but the leaves are not closed anymore. The leaves space being non Hausdorff, it is more convenient to study instead the basic cohomology of the foliation. In the case where the fan is shellable, computations have been made by Battaglia and Zaffran. In this talk, using Ishidafs complexes (developed to compute the cohomology of toric varieties), we will present the computation in the general case. As a consequence of this, we obtain the Dehn-Sommerville equations for starshaped spheres (i.e. spheres which are the underlying complex of a complete simplicial fan.

UMEMOTO Yuriko

Title: On the growth functions of hyperbolic Coxeter groups

Abstract: After an introduction to growth of hyperbolic Coxeter groups of finite volume, some results about the growth rate and its arithmetic nature are presented which deal with the cocompact and cofinite cases in three dimensions.

USTINOVSKY Yury

Title: Complex geometry of moment-angle-manifolds

Abstract: In this talk we discuss complex structures on moment-angle-manifolds, a well-known in toric topology family of topological spaces. We study its complex geometry, namely describe field of meromorphic functions, construct special holomorphic foliations, and transversally kahlerian metrics, describe analytic subsets and Dolbeault cohomology.

YU Li

Title: Free Z_p-torus actions in dimension 2 and 3

Abstract: For any finite index normal subgroup N of a finitely presented group G, we obtain some lower bounds of the rank of the first homology of N (with mod p coefficients) in terms of some invariants of G and the quotient group G/N. Using this, we confirm the Halperin-Carlsson Conjecture for any free Zp-torus actions (p is any prime) on 2-dimensional finite CW-complexes and any free Z2-torus actions on compact 3-manifolds.

Accommodations Osaka City University Guest House, Kansai Kenshu Center

Useful link The Manchester Toric Topology Page

Please contact Mikiya Masuda (e-mail: masuda[at]sci.osaka-cu.ac.jp) if you have any question.