POLAR ACTIONS ON NONNEGATIVELY CURVED MANIFOLDS

FUQUAN FANG

In Riemannian geometry, a classical and main theme is to understand the geometry and topology

of manifolds with definite curvature sign. In this talk, I will in brief introduce a special but very

rich action on a riemannian manifold, polar action. Rigidity theorems will be introduced, based

on a joint work with K. Grove also partially with G. Thorbergsson.

Capital Normal University, BeijingE-mail address: fuquan fang@yahoo.com

ALGEBRAIC MODELS FOR CONFIGURATION SPACES

DONALD STANLEY

The rational homotopy type of a topological space is contained in its model which is a commuta-

tive differential graded algebra. We are interested in models of F (M,k) the ordered configuration

space of k points in a manifold M. For certain classes of manifolds models for F (M,k) have been

discovered. Examples include complex projective algebraic manifolds and configurations of two

points in two connected closed manifolds. We give a model for the ordered configuration space of

three points in any closed manifold.

Regina University, CanadaE-mail address: Donald.Stanley@uregina.ca

ON THE COHOMOLOGY OF TORIC ORIGAMI MANIFOLDS

HAOZHI ZENG

Toric origami manifolds, introduced by A. Canas da Silva, V. Guillemin and A. R. Pires,

are generalization of toric symplectic manifolds. In this talk we will discuss the cohomology

of orientable toric origami manifolds with acyclic proper faces. This talk is based on the joint

work with Anton Ayzenberg, Mikiya Masuda and Seonjeong Park.

Osaka City University, JapanE-mail address: zenghaozhi@icloud.com

SMALL COVERS OF ASSOCIAHEDRA AND COMBINATORIALREALIZATION OF CYCLES

ALEXANDER GAIFULLIN

A classical question by Steenrod (late 1940s) was whether it is possible to realize an integral

homology class of a topological space by a continuous image of the fundamental class of an oriented

smooth closed manifold. (Homology classes satisfying this condition are called realizable.) This

question was answered by Thom (1954) who showed that there exist non-realizable homology

classes but a certain multiple of any homology class is realizable.

In 2007 the speaker found an explicit combinatorial procedure that, for a given singular cycle in

a topological space, constructs a manifold realizing a multiple of the homology class representing

by this cycle. Moreover, this construction allowed us to prove that, for every n, there exists an

oriented smooth closed manifold Mn that satisfy the following Universal Realization of Cycles

property (or the URC-property): A multiple of any n-dimensional integral homology class of any

topological space can be realized by an image of the fundamental class of a non-ramified finite-

sheeted covering over Mn. Several series of examples of URC-manifolds (i.e. manifolds satisfying

the URC-property) were found by the speaker in 2013. The simplest of them was a small cover of

the permutohedron. In the talk we shall present a modification of the explicit procedure for the

realization of cycles that will allow us to find URC-manifolds that are even simpler than a small

cover of the permutohedron. In particular, we shall show that a small cover of the associahedron

is also a URC-manifold.

Steklov Mathematical Institute, Russian Academy of SciencesE-mail address: agaif@mi.ras.ru

ON THE COHOMOLOGY RING OF A REAL MOMENT-ANGLE MANIFOLD

LI CAI

In this talk, I will describe a necessary and sufficient condition that when a real moment-angle

complex (RMAC) is a topological manifold, which is based on M. Davis’s work. From this we

get a necessary and sufficient condition of a moment-angle complex (MAC) being a topological

manifold. Then I will give a description of the cohomology ring of a RMAC, from which we deduce

the cohomology rings of a class of polyhedral products, including MACs.

School of Mathematics and Systems Science, Chinese Academy of SciencesE-mail address: l-cai@amss.ac.cn

COMBINATORICS OF TRIANGULATED SPHERES AND TOPOLOGY OFMOMENT-ANGLE COMPLEXES

IVAN LIMONCHENKO

In 2003 I.Baskakov found an example of a 2-dimensional simplicial complex K s.t. its moment-

angle complex ZK is not formal, having a nontrivial triple Massey product. We introduce a wide

class of simple polytopes P of dimensions greater than 2, whose nerve complexesK = ∂P ∗ have this

property, based on the Baskakov construction and the result of V.Buchstaber and V.Volodin which

presents any flag nestohedron as a 2-truncated cube, that is a consecutive cut of codimension 2 faces

in a cube of the appropriate dimension. Nestohedra is a rich classical family of simple polytopes

well known in discrete and combinatorial geometry and in representation theory. For instance, it

contains the so called graph-associahedra (e.g. permutohedra, stellahedra, associahedra or Stasheff

polytopes, cyclohedra or Bott-Taubes polytopes and etc.); for such polytopes P we present some

results on additive structure in the cohomology ring H∗(ZP ), namely bigraded Betti numbers of

the type β−i,2(i+1)(P ) and torsion computations. These Betti numbers give us an estimate for the

minimal number of multiplicative generators in the Pontryagin algebra H∗(ΩZP ).

Next, we compare this case to the one when the moment-angle complex ZK is formal and has

cohomology length 1 or 2. For these cases we give algebraic characterizations on the face ring

Q[K] over rationals and present a number of examples of ZK , for which the homotopy types and

bigraded Betti numbers can be calculated explicitly. In the case of cohomology length 2 when K

is not a triangulated sphere we give examples of moment-angle complexes ZK with no torsion in

integral cohomology that are not homotopy equivalent to connected sums of sphere products. In

the class of polytopal spheres and formal moment-angle manifolds no such example is known yet.

The work was supported by the Russian Foundation for Basic Research grant no. 14-01-00537a

and the grant MK-600.2014.1 from the President of Russia.

Department of Geometry and Topology, Faculty of Mechanics and Mathematics, Moscow S-tate University, Leninskiye Gory, Moscow 119992, Russia

E-mail address: ilimonchenko@gmail.com

BIGRADED BETTI NUMBERS OF FULLERENES

NIKOLAY EROKHOVETS

The talk is based on the joint work with Victor Buchstaber.

Toric topology associates to each simple n-polytope P with facets F1, . . . , Fm an (m + n)-

dimensional moment-angle manifold ZP with canonical action of the torus Tm = (S1)m.

This gives a tool to study the combinatorics of P in terms of the algebraic topology of ZP and

visa versa. Define the multi-graded ring with mdeg(x) = (−i, 2ω), i ∈ Z>0, ω ⊂ [m]:

R∗(P ) = Λ[u1, . . . , um]⊗ Z[v1, . . . , vm]/(uivi, v2j , vi1 . . . vik : Fi1 ∩ · · · ∩ Fik = ∅),

mdegui = (−1, 2i), mdegvi = (0, 2i), dui = vi, dvi = 0.

Theorem. [Bu-Pa] There is a ring isomorphism H[R∗(P )] ' H∗(ZP ,Z).

Let Pω =⋃

i∈ω Fi for a subset ω ⊂ [m]. It is known that β−i,2ω = rk H |ω|−i−1(Pω,Z). Set

β−i,2j =∑|ω|=j β

−i,2ω. A multigraded Poincare duality implies β−i,2ω = β−(m−n−i),2([m]\ω).

For a 3-polytope P 6= ∆3 nonzero Betti numbers are β0,2∅ = β−(m−3),2[m] = 1,

β−i,2ω = rk H0(Pω,Z) = β−(m−3−i),2([m]\ω) = rk H1(P[m]\ω,Z), |ω| = i + 1 = 2, . . . ,m − 3. For

|ω| = i + 1 the number β−i,2ω + 1 is equal to the number of connected components of the set

Pω ⊂ P . Set h = m− 3. Then (1− t2)h(1 + ht2 + ht4 + t6) = 1− β−1,4t4 +h∑

j=3

(−1)j−1(β−(j−1),2j −

β−(j−2),2j)t2j + (−1)h−1β−(h−1),2(h+1)t2(h+1) + (−1)ht2(h+3).

Let P be a simple convex 3-polytope. A k-belt is a cyclic sequence (F1, . . . , Fk) of facets, such

that Fi1 ∩ · · · ∩ Fir 6= ∅ if and only if i1, . . . , ir ∈ 1, 2, . . . , k − 1, k, k, 1.A fullerene (see [DeSS13]) is a simple convex 3-polytope that has facets only pentagons and

hexagons. Any fullerene P has p5 = 12 pentagons and p6 6= 1 hexagons.

Theorem 1. Any fullerene P has no 3-belts [Bu-Er].

Corollary 1. We have β−1,4 = (9+p6)(8+p6)2

, β−1,6(P ) = 0, and β−2,6 = (6+p6)(8+p6)(10+p6)3

.

Theorem 2. Any fullerene P has no 4-belts.

Corollary 2. We have β−2,8(P ) = 0, β−3,8 = (4+p6)(7+p6)(9+p6)(10+p6)8

, and the product map

H3(ZP )⊗H3(ZP )→ H6(ZP ) is trivial .

Theorem 3. Any fullerene P has 12 + k five-belts, where 12 belts surround pentagons and k

belts consist of hexagons with any hexagon intersecting neighbours by opposite edges. Moreover,

if k > 0 then P consists of two dodecahedral caps and k hexagonal 5-belts between them.

Corollary 3. For a fullerene P we have β−3,10 = 12 + k, k > 0.

The work is supported by the Russian President grant MK-600.2014.1 and the RFBR grant

14-01-31398-a.

2 NIKOLAY EROKHOVETS

References

[Bu-Er] V.M.Buchstaber, N.Erokhovets, Graph-truncations of simple polytopes, Proc. of Steklov Math Inst, MAIK,Moscow, V. 289, 2015.

[Bu-Pa] V.M.Buchstaber, T.E.Panov, Toric Topology, AMS Math. Surv. and Mon. V. 204, 2015. 518 pp.[DeSS13] M.Deza, M.Dutour Sikiric, M.I.Shtogrin, Fullerenes and disk-fullerenes, Russian Math. Surveys,

68:4(2013), 665-720.

Lomonosov Moscow State UniversityE-mail address: erochovetsn@hotmail.com

THE BETTI NUMBERS OF REAL TORIC VARIETIES ASSOCIATED TOWEYL GROUPS OF REGULAR TYPES

SUYONG CHOI

We compute the (rational) Betti number of real toric varieties associated to Weyl groups of

regular types. The formula for the rational Betti numbers of toric varieties associated to Weyl

groups of type A is established by Henderson in 2010. In this talk, we give the complete formula

for the rational Betti numbers of toric varieties associated to Weyl groups of type B and D. This

formula also holds for the Betti numbers with coefficient G, where 2 is invertible in G.

Ajou University, KoreaE-mail address: schoi@ajou.ac.kr

ON FANO THREEFOLDS WITH SEMI-FREE C∗-ACTIONS, I

QILIN YANG AND DAN ZAFFRAN

Let X be a Fano threefold and C∗ ×X → X an algebraic action. Then X has a S1-invariant

Kahler structure and the corresponding S1-action admits an equivariant moment map which is

at the same time a perfect Bott-Morse function. We will initiate a program to classify the Fano

threefolds with semifree C∗-actions using the Morse theory and holomorphic Lefschetz fixed point

formula as the main tools. In this paper we give a complete list of all possible Fano threefolds

without the interior isolated fixed points for any semifree C∗-actions. Moreover when the actions

whose fixed point sets have only two connected components and a few of the rest cases, we give

the realizations of the semifree C∗-actions.

Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, CHINAE-mail address: yqil@mail.sysu.edu.cn

Department of Mathematical Science, Dan Dahle Building, 101, Florida Institute of Technol-ogy 150 W. University Blvd Melbourne, Florida 32901, U. S. A.

E-mail address: dzaffran@fit.edu

2010 Mathematics Subject Classification. 14J45, 32M05, 53C55, 53D20, 57R20.Key words and phrases. Fano threefold, algebraic action, Hamiltonian action, moment map, Morse theory,

holomorphic Lefschetz formula, equivariant localization.

THE FOUNDATION OF (2n, k) - MANIFOLDS

SVJETLANA TERZIC

We introduce the class of toric (2n, k)-manifolds, which are special class of closed, smooth

manifolds M2n equipped with a smooth effective action of the compact torus T k, 1 ≤ k ≤ n

and an open T k-equivariant map µ : M2n → Rk whose image is a convex polytope, where Rk is

considered with the trivial T k-action. These manifolds we axiomatize by requiring the additional

relations between the smooth structure of a manifold, the given torus action and an almost moment

map µ.

The class of (2n, k) manifolds contains many interesting and important manifolds such as qu-

asitoric manifolds, complex Grassmann manifolds Gk+1,q and full flag manifolds Fk+1 equipped

with the canonical action of the torus T k+1, and the complex projective spaces CP n, n =(k+1q

)−1

equipped with T k+1-action obtained as the composition of the representation T k+1 → T n+1 given

by the q-th symmetric power and the standard action of T n+1 on CP n. The description of the

toric structure of a quasitoric manifold M2n essentially uses the fact that the orbit space M2n/T n

is homeomorphic to P n, where P n is a simple polytope. On the other hand for the complex Grass-

mann manifolds Gk+1,q or the complex flag manifolds Fk+1 there is an almost moment map µ

whose image is not a simple polytope and the orbit spaces Gk+1,q/Tk and Fk+1/T

k are essentially

different from the corresponding polytopes. The reason is that it may happen that some convex

polytopes over a vertices of P k which are not the faces of P k are of the structural importance for

the description of the orbit space as well.

In that context our axiomatization of (2n, k) manifolds enables us to divide M2n into T k-

invariant subspaces such that for each of them the almost moment map induces on its orbit

space locally trivial fibre bundle whose base is a polytope over some vertices of P k. These spaces

we call an admissible spaces, the corresponding polytopes admissible polytopes, while the fibers

of the corresponding bundles we call the spaces of parameters. The admissible polytopes form

an abstract complex C(M2n, P k) whose topology is defined to be the quotient topology of the

map f : M2n → C(M2n, P k) such that π f = µ, where π : C(M2n, P k) → P k is the canonical

projection. All of this leads to the space obtained by trivial gluing to each polytope of C(M2n, P k)

its corresponding space of parameters. This space is naturally endowed with T k-action and we

prove that its orbit space is homeomorphic to M2n/T k. Under some additional assumption on

the spaces of parameters we show that the orbit space M2n/T k is homeomorphic to Sk−1 ∗ F , for

some closed subspace F ⊂M2n.

The talk is based on the results obtained jointly with Victor M. Buchstaber.

Faculty of Mathematics and Natural Sciences, University of MontenegroE-mail address: sterzic@ac.me

EQUIVARIANT COMPLEX BORDISM OF GKM-MANIFOLDS

ALASTAIR DARBY

We consider GKM-manifolds admitting an equivariant stably complex structure and the well-

known labelled graphs that arise from them that encode the fixed point data. By defining the

GKM-graphs axiomatically we show, using the universal toric genus, that any abstract GKM-

graph is the fixed point data for some stably complex GKM-graph up to equivariant complex

bordism. We then give complete bordism invariants for the abstract graphs and classify a large

number of them.

Fudan UniversityE-mail address: adarby@fudan.edu.cn

CIRCLE ACTIONS ON SYMPLECTIC MANIFOLDS WITH SOMEMINIMALITY CONDITIONS

HUI LI

We consider a compact symplectic manifold with a Hamiltonian circle action. Assume that

the even Betti numbers of the manifold are “minimal”, or the connected components of the fixed

point set of the action satisfies a “minimal condition”, we determine some global invariants of the

manifold, and for some cases, we determine the diffeomorphism or symplectomorphism type of

the manifold.

Soochow UniversityE-mail address: hui.li@suda.edu.cn

PONTRYAGIN ALGEBRAS OF SOME MOMENT-ANGLE COMPLEXES

YAKOV VEREVKIN

We consider the problem of describing the Pontryagin algebra (loop homology) of moment-angle

complexes and manifolds. The moment-angle complex ZK is a cell complex built of products of

polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural

torus action and plays an important role in toric topology. In the case when K is a triangulation

of a sphere, ZK is a topological manifold, which has interesting geometric structures.

Generators of the Pontryagin algebra H∗(ΩZK) when K is a flag complex have been described

in the work of Grbic, Panov, Theriault and Wu. Describing relations is often a difficult problem,

even when K has a few vertices. Here we describe these relations in the case when K is the

boundary of a pentagon or a hexagon. In this case, it is known that ZK is a connected sum of

products of spheres with two spheres in each product. Therefore H∗(ΩZK) is a one-relator algebra

and we describe this one relation explicitly, therefore giving a new homotopy-theoretical proof of

McGavran’s result. An interesting feature of our relation is that it includes iterated Whitehead

products which vanish under the Hurewicz homomorphism. Therefore, the form of this relation

cannot be deduced solely from the result of McGavran.

Department of Geometry and Topology, Faculty of Mechanics and Mathematics, Moscow S-tate University

E-mail address: verevkin j.a@mail.ru

ORBIFOLD TOWERS AND THEIR INTEGRAL COHOMOLOGY RING

JONGBAEK SONG

Toric manifolds over product of simplices∏k

i=1 ∆ni were studied by Y. Civan, N. Ray, and many

subsequent authors. In particular, S. Choi, M. Masuda and D. Y. Suh described a certain free

torus action on the moment angle complex Z =∏k

i=1 S2ni+1, and derive a tower structure on the

resulting toric manifold, called a generalized Bott tower.

We construct an orbifold analogue of generalized Bott towers by replacing free torus action into

locally free torus action. In general, the resulting space is an orbibundle, but additional restrictions

on the torus action gives us a genuine fibration whose fiber is a weighted projective space. Finally,

we compute their integral cohomology ring and see how they differ from the cohomology ring of

generalized Bott towers.

This is a joint work with Anthony Bahri and Soumen Sarkar

KAIST, Daejeon, KoreaE-mail address: jongbaek.song@gmail.com

INVARIANT COMPLEX STRUCTURES ON NILMANIFOLDS

DMITRY MILLIONSCHCHIKOV

A left-invariant complex structure on nilmanifold G/Γ that corresponds to some real simply

connected Lie group G can be defined as an almost-complex structure J on the tangent Lie

algebra g of G (J2 = −1) satisfying the well-known integrability condition:

[JX, JY ] = [X, Y ] + J [JX, Y ] + J [X, JY ], ∀X, Y ∈ g.

We study the properties of nilpotent Lie algebra g, which arise because of the existence of a left

invariant complex structure J on G. We present a family D(n) of naturally graded Lie algebras

with the nil-index s(D(n)):

s(D(n)) =

[2

3dimD(n)

].

Theorem. Let g be a nilpotent Lie algebra with an integrable complex structure and dim g ≥ 8.

gk = [g, gk−1] stands for the k-th ideal of the descending central sequence of the Lie algebra g.

Then we have the following estimates:

dim g− dim g4 ≥ 5, dim g− dim g6 ≥ 8.

.

References

[1] D.V. Millionshchikov, “Complex structures on nilpotent Lie algebras and descending central series”,http://arxiv.org/abs/1412.0361.

Lomonosov Moscow State University, Department of Mechanics and Mathematics, Leninskiegory, 1, Moscow, 119992, Russia

E-mail address: million@mech.math.msu.su

Supported by the grant of Russian Scientific Foundation N 14-11-00414.

ON THE COHOMOLOGY OF PARTIAL QUOTIENTS OF MOMENT-ANGLEMANIFOLDS

TARAS PANOV

We describe the cohomology of the quotient ZK/H of a moment-angle complex ZK by a freely

acting subtorus H in Tm by establishing a ring isomorphism of H ∗(ZK/H,R) with an appropriate

Tor-algebra of the face ring R[K], with coefficients in an arbitrary commutative ring R with unit.

The quotients ZK/H include moment-angle manifolds themselves, projective toric manifolds (the

result was known for both these cases), and also ‘projective’ moment-angle manifolds. The latter

admit non-Kaehler complex-analytic structures as LVM-manifolds. We prove the collapse of the

corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with

respect to ‘strongly homotopy multiplicative’ maps in the category DASH, following Gugenheim-

May and Munkholm.

Moscow State UniversityE-mail address: taras@in.ru

CODIMENSION-1 PL EMBEDDINGS OF A FAMILY OF MANIFOLDS INTOSPHERES

BO CHEN

In toric topology, small covers and quasi-toric manifolds are manifolds with torus actions which

could be reconstructed from their orbit-spaces(simple polytopes). In this talk, we focus on the

embedding problem of such manifolds with special orbit-spacesspecial simple polytopes whose 2-

faces are all of even number of vertices. Name such polytope by even polytope. We constructed

codimension-1 embeddings of such manifolds into spheres. The construction is mainly depend on

embeddings of even polytopes in to balls in “nice” positions.

Huazhong University of Science and TechnologyE-mail address: bobchen@hust.edu.cn

SECONDARY POLYTOPES AND CHOW STABILITY OF TORIC VARIETIES

NAOTO YOTSUTANI

Chow stability is one notion of Mumford’s Geometric Invariant Theory for studying the moduli

space of polarized varieties. Gelfand, Kapranov and Zelevinsky detected that Chow stability of

polarized toric varieties is completely determined by its inherent ‘secondary polytope’, which is a

polytope whose vertices correspond to regular triangulations of the associated (Delzant) polytope.

In this talk, we would like to discuss combinatorial framework for the Chow form of a (not-

necessaliry-smooth) projective toric variety and its applications.

Fudan UniversityE-mail address: naoto-yotsutani@fudan.edu.cn

COHOMOLOGICAL RIGIDITY OF MOMENT-ANGLE MANIFOLDS

FEIFEI FAN

In this talk, we give a survey of the works that study the two rigidity problems in toric topology:

(1) Suppose ZK1 and ZK2 are two moment-angle manifolds such that H∗(ZK1)∼= H∗(ZK2). Are

ZK1 and ZK1 homeomorphic?

(2) Let K1 and K2 be simplicial spheres, and let ZK1 and ZK2 be their respective moment-angle

manifolds. When a graded ring isomorphism H∗(ZK1)∼= H∗(ZK2) implies a combinatorial

equivalence K1 ≈ K2?

School of Mathematics and Systems Science, Chinese Academy of SciencesE-mail address: fanfeifei@mail.nankai.edu.cn

COHOMOLOGY OF TORUS MANIFOLDS AND SOCLES OF FACE RINGS

ANTON AYZENBERG

Let X be a closed 2n-manifolds with an effective action of compact n-torus T . Let S be a

simplicial poset associated with this action and λ be a charateristic function. When X is a

smooth toric variety or a quasitoric manifold, it is well known that H∗(X) ∼= Z[S]/Θ, where Z[S]

is the face ring, and Θ is the ideal generated by certain linear forms (which are in turn determined

by λ). In general there exists a map ρ : Z[S]/Θ → H∗(X) which may be neither injective nor

surjective. In my talk I will discuss the properties of this map.

When all proper faces of the orbit space are acyclic, the dual poset is a homology manifold. In

this case both kernel and cokernel of ρ can be described explicitly. In particular, I proved that the

kernel lies in the socle of Z[S]/Θ. Such socles, for S being a homology manifold, are well-studied

in commutative algebra and maybe adapted for the needs of toric topology.

If the assumption of proper face acyclicity is dropped, in the joint work with Mikiya Masuda

we proved that the kernel of ρ lies in the iterated socles of Z[S]/Θ. This notion will be defined in

the talk.

National Research University Higher School of Economics, MoscowE-mail address: ayzenberga@gmail.com