Combinatorial Constructions in Topology

University of Regina, August 17-21, 2015.

Titles and Abstracts:

Karim Adiprasito: Hodge Theory for combinatorial geometries Cancelled

A conjecture of Read predicts that the coefficients of the chromatic polynomial of any graph form a log-concave sequence. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space.

All known proofs use Hodge theory for projective varieties, and the more general conjecture of Rota for possibly "nonrealizable'' configurations is still open. We will argue here that the three pillars of Hodge theory (the Hard Lefschetz theorem, the Hodge-Riemann relations and the Hodge Index Theorem) continue to hold in a realm that goes far beyond that of projective algebraic geometry, and even Kähler geometry.

This combinatorial Hodge theory gives strong restrictions on combinatorial and numerical invariants of matroids, including Whitney numbers of matroids. The theorems on positivity, on the other hand, extend naturally to general finitary combinatorial geometries, and provide the first time a working intersection theory in this generality.

Alejandro Adem: Homotopy group actions and group cohomology

Let $$G$$ be a finite group, in this talk we will discuss the notion of a homotopy action of $$G$$ on a finite complex $$X$$. We will describe some natural cohomological invariants associated to this and how interesting geometric actions can arise. Examples and related work involving actions on spheres and their products will be described.

Anton Ayzenberg: $$h'$$- and $$h''$$-numbers in combinatorics and cohomology of torus manifolds

Let $$X$$ be a closed $$2n$$-manifold with a locally standard action of a compact $$n$$-torus $$T$$. The orbit space $$X/T$$ is a manifold with corners; let $$S$$ be a simplicial poset dual to $$X/T$$. The action determines a characteristic function, which assigns an integral vector to every facet of $$X/T$$. A natural question is: can we describe $$H^*(X)$$ in terms of $$X/T$$, characteristic function (and maybe something else)?

The answer for toric and quasitoric manifolds is well known. In these cases $$S$$ is a simplicial sphere, and we have $$H^*(X)\cong \mathbb{Z}[S]/\Theta$$, where $$\mathbb{Z}[S]$$ is the Stanley--Reisner ring of $$S$$; and $$\Theta$$ is an ideal generated by linear forms ($$\Theta$$ is determined by the characteristic function). Betti numbers of $$X$$ are the $$h$$-numbers of the simplicial sphere $$S$$.

Now let $$X$$ be a torus manifold such that all proper faces of its orbit space are acyclic and the free part of action is trivial. Then the poset dual to $$X/T$$ is a simplicial manifold. $$h'$$- and $$h''$$-numbers of simplicial manifolds were introduced in combinatorial commutative algebra as the generalizations of $$h$$-numbers of spheres. In my talk I want to explain the relation between $$H^*(X)$$ and this combinatorial theory.

Tony Bahri: On the integral cohomology rings of certain toric orbifolds

A criterion is described which ensures that a toric orbifold, determined by a simple polytope and a characteristic map, has torsion free cohomology concentrated in even degree. The description is shown to transform well under the simplicial wedge construction. A report of joint work with Nigel Ray, Soumen Sarkar and Jongbaek Song.

Marzieh Bayeh: Orbit diagrams and equivariant LS-category

Consider a topological group $$G$$ acting continuously on a Hausdorff topological space $$X$$. The notion of orbit type has been studied for decades and it is used in many different areas in geometry and topology. Using this concept, Hurder and Töben give a lower bound for equivariant LS-category. In this talk we introduce a new way to look at the orbits, which gives a better tool to find lower bounds for equivariant LS-category. Also we construct a counter examples to previous results.

Christin Bibby: Rational homotopy theory of chordal arrangements

To any graph, one may associate three flavors of hyperplane arrangements: linear (subspaces of a complex vector space), toric (subtori of a complex torus), or abelian (abelian subvarieties of a complex abelian variety). In the linear case, there is considerable literature on the rational homotopy theory of the complement, and the toric case is similar in flavor. The abelian case is more complicated due to lack of formality of the space. When the graph is chordal, we have a Koszul model and use quadratic-linear duality to compute the minimal model and show that the space is rationally $$K(\pi,1)$$. This is joint work with Justin Hilburn.

Pavle Blagojević: Cutting, embedding, bouncing characteristic classes (slides)

The properties of the regular representation bundles over the configuration space of $$k$$ distinct points in the Euclidean space has classically been studied extensively by F. Cohen, R. Cohen, Chisholm, Handel, Kuhn, Neisendorfer, V. Vassiliev, and many others.

Motivated by geometric problems we present new computations of twisted Euler classes, Stiefel-Whitney classes and their monomials as well as corresponding Chern classes of these bundles.

Thus, we not only extend and complete previous work, supplying for example a proof for a conjecture by Vassiliev, but also make progress in solving and extending variety of problems from Discrete Geometry, among them

• (1) the conjecture by Nandakumar and Ramana Rao that every convex polygon can be partitioned into $$k$$ convex parts of equal area and perimeter;
• (2) Borsuk's problem on the existence of "$$k$$-regular maps" between Euclidean spaces, which are required to map any $$k$$ distinct points to $$k$$ linearly independent vectors;
• (3) Ghomi and Tabachnikov problem about the existence of "$$\ell$$-skew smooth embeddings" from a smooth manifold $$M$$ to a Euclidean space $$E$$, which are required to map tangent spaces at $$\ell$$ distinct points of $$M$$ into $$\ell$$ skew subspaces of $$E$$.
(This lecture is based on the joint work with Bárány, Frederick Cohen, Wolfgang Lueck, Roman Karasev, Szüch and Gunter M. Ziegler)

Suyoung Choi: A new family of projective toric manifolds

Consider a polytope $$P$$ obtainable from consecutive wedge operations from an $$m$$-gon. In this talk, we completely classify toric manifolds over $$P$$ and prove that all of them are projective. As a consequence, we provide an infinite family of projective toric manifolds. This talk is based on the work jointly with Hanchul Park (KIAS).

Matthias Franz: Big polygon spaces

Polygon spaces are configuration spaces of polygons with prescribed edge lengths. I will present a related family of manifolds, called big polygon spaces. They come with a canonical torus action whose fixed point set is the corresponding polygon space. Big polygon spaces are interesting because for suitably chosen length vectors they provide the only known examples of torus actions on compact orientable manifolds to which the "GKM method" applies although their equivariant cohomology is not free. Time permitting, I will also present a related construction involving division algebras.

Alex Gonzalez: Partial groups and extensions

Partial groups are a generalization of groups recently introduced to solve some deep problems about classifying spaces of (finite) groups. Although partial groups are by definition algebraic objects, they carry a natural simplicial structure that I will exploit in order to present an extension theory of partial groups.

Tara Holm: Hamiltonian circle actions on symplectic four-manifolds

I will report on recent work with Liat Kessler on Hamiltonian circle actions on symplectic four-manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of two-sphere-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this joint work with Kessler, we have characterized Hamiltonian circle actions on them, up to (possibly non-equivariant) symplectomorphism. As a by-product, we provide an algorithm that determines the reduced form of a blowup form and which also provides a method for computing the Gromov width. We can also compute the circle equivariant cohomology of these manifolds, in terms of the fixed point set, which can include isolated fixed points and fixed surfaces.

Shintaro Kuroki: On a maximal dimension of the torus which acts on a GKM-manifold effectively

GKM manifold is an almost complex manifold with torus action whose fixed points and one-dimensional orbits have the structure of a graph. After the work of Goresky-Kottwitz-MacPherson (GKM) about the equivariant cohomology of GKM manifolds, Guillemin and Zara introduced a combinatorial counterpart of a GKM manifold, called a GKM graph. An (m,n)-type GKM graph $$(\Gamma,\mathcal{A},\nabla)$$ is defined as an m -valent graph $$\Gamma$$ with a label on edges, called an axial function $$\mathcal {A}:E(\Gamma)\to H^2(BT^n)$$, where $$n\le m$$. In this talk, I will define a finitely generated free abelian group $$\ mathcal{O}(\Gamma,\mathcal{A},\nabla)$$ for an $$(m,n)$$-type GKM graph $$(\Gamma,\mathcal{A},\nabla)$$ and show that its rank gives a necessarily and sufficient condition for the extension of this $$(m,n)$$-type GKM graph to an $$(m,l)$$-type GKM graph, where $$n\le l$$. If time permits, I will give a classification of all axial functions on the complete graph with 4-vertices and compute their $$\mathcal{O}(\Gamma, \mathcal{A},\nabla)$$. As a consequence of this, for example, we see that there is no effective $$T^3$$-action which is an extension of the $$T^2$$-action on the complex quadric $$Q_3(\simeq SO(5)/SO(3)\times SO(2))$$.

Pascal Lambrechts: On the rational homotopy of configuration spaces

Given a  manifold $$M$$, the space of (ordered) configuration of $$k$$ points in $$M$$ is defined as the space $\operatorname{Conf}(k,M) := \{ (x_1,\dots,x_k)\in M^k:x_i\not=x_j\textrm{ for }i\not= j \}.$ In this talk I will prsent various results about what can be said about the rational homotopy type of that space. In particular we show that the rational homotopy type of $$\operatorname{Conf}(3,M)$$ depends only on the rational homotopy type of $$M$$ when $$M$$ is compact without boundary and $$4$$-connected. We also exhibit nice explicit models for the stable rational homotopy type of $$\operatorname{Conf}(k,M)$$ when $$M$$ is a compact manifold with boundary. (joint work with Hector Cordova-Bulens and Don Stanley)

Zhi Lü: Elementary symmetric polynomials in Stanley-Reisner face ring

Stanley-Reisner face ring has played an import role on the algebraic combinatorics, algebraic geometry, toric geometry and toric topology. The Stanley-Reisner face ring of a simple $$n$$-polytope $$P$$ with $$m$$ facets $$F_1, ..., F_m$$ is defined as the quotient of the polynomial algebra $${\Bbb Z}[x_1, ..., x_m]$$ ${\Bbb Z}(P)={\Bbb Z}[x_1, ..., x_m]/\mathcal{I}_P$ where $$\mathcal{I}_P$$ is the ideal formed by the square-free monomials $$x_{i_1}\cdots x_{i_r}$$ with $$F_{i_1}\cap\cdots \cap F_{i_r}=\varnothing$$. In a similar way, we define the quotient $$\wedge[x_1, ..., x_m]/\mathcal{I}_P$$ of the exterior algebra $$\wedge[x_1, ..., x_m]$$, denoted by $$\wedge(P)$$, which is called the Stanley--Reisner exterior face ring of $$P$$. We know from the seminal work of M. Davis and T. Januszkiewicz in [Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 61 (1991), 417-451] that the elementary symmetric polynomial $$\sigma_i(x_1, ..., x_m)$$ in $${\Bbb Z}(P)$$ are exactly equivariant Chern classes of the moment-angle manifold $$\mathcal{Z}_P$$.

In this talk I will talk about how the divisibility and decomposability of elementary symmetric polynomial $$\sigma_n(x_1, ..., x_m)$$ in $${\Bbb Z}(P)$$ and $$\wedge(P)$$ influence on the combinatorics of $$P$$ and the topology of toric space over $$P$$. This can also be carried out in the cases of mod 2 face rings $${\Bbb Z}_2(P)$$ and $$\wedge(P)\otimes{\Bbb Z}_2$$. This is a joint work with Jun Ma and Yi Sun.

Mikiya Masuda: Cohomology of regular Hessenberg varieties and representations of symmetric groups

Mainak Poddar: Toric principal bundles

Gubeladze proved that algebraic vector bundles over an affine toric variety are trivial. This is a generalization of the Quillen-Suslin theorem (Serre conjecture) to a singular situation. Existence of trivialization is not known for algebraic principal bundles over an affine toric variety. We deal with this problem in the case of torus equivariant principal bundles. We will first explain the Tannakian approach to principal bundles. Then we will present a classification of torus equivariant principal $$G$$-bundles over a toric variety when $$G$$ is a reductive group. Some open problems will be mentioned. This is a joint work with Indranil Biswas and Arijit Dey.

Mentor Stafa: Polyhedral products and monodromy (slides)

Polyhedral products are a combinatorial construction involving topological spaces that enjoy many applications. In this talk we will study the monodromy action for a particular fibration. We will use the topology of polyhedral products to obtain the monodromy action as a faithful representation of a free product of finite groups to $$SL_n(Z)$$.

Hugh Thomas: A topological approach to the amplituhedron

Amplituhedra were recently introduced by Arkani-Hamed et al. in the context of $$N=4$$ supersymetric Yang-Mills theory. They include as a special case even-dimensional cyclic polytopes, but in general they are not polytopal. Rather, they are defined as the image of a map from one positive Grassmannian to another (including entire postive Grassmannians as further special cases). I will discuss a new candidate definition for the amplituhedron, developed with Nima Arkani-Hamed and Jaroslav Trnka, which gives a direct characterization of the points of the amplituhedron, without reference to a map from another positive Grassmannian. This characterizatiom includes some topological data (of a fairly elementary sort).

Ismar Volić: Some combinatorial problems arising from manifold calculus of functors

Manifold calculus of functors has in recent years been applied with great success to spaces of embeddings, most notably knot and link spaces. During this time, some interesting combinatorial problems arose as a consequence. After describing the general setup of manifold caclulus of functors and how it specializes to knots and links, I will describe some of these problems relating to certain spaces of diagrams and trees, homology of link spaces, and subspace arrangements. Time permitting, I will make some general comments about how functor calculus might provide a different point of view on some recent results surrounding the Tverberg Conjecture.

Jihyeon Jessie Yang: Newton-Okounkov bodies of Bott-Samelson varieties and toric degenerations to Bott towers

In the late 1950’s, R. Bott and H. Samelson introduced certain manifolds which they used to study the cohomology of the quotient $$K/T$$, the flag variety of a compact Lie group $$K$$, and the cohomology of the loop space of $$K$$. In the 1970’s, M. Demazure showed that these manifolds are algebraic varieties referred to as Bott-Samelson varieties (schemes in more general settings), and they are very closely related to Schubert varieties and representation theory of reductive groups. Particularly, Bott-Samelson varieties provide the resolutions of singularities of Schubert varieties and they became an important tool in geometric representation theory. In 1989, Bott noticed that each of Bott-Samelson varieties supports an action of a real torus of half the dimension of the space, which is a special case of the phenomenon of complete integrability. However, it turned out that this torus action is not holomorphic. In 1991, M. Grossberg introduced Bott towers, which are diffeomorphic to Bott-Samelson varieties, and each of them supports a holomorphic action of a real torus of half the dimension of the space. In 1994, applying results on Bott-towers to Bott-Samelson varieties, M. Grossberg and Y. Karshon obtained a Demazure-type character formula as a sum over lattice points inside certain polyhedral region (not necessarily polytope), called twisted cube, counted with signs. In 2010, B. Pasquier interpreted the relation between Bott-Samelson varieties and Bott towers in algebro-geometric view point. Namely, there is a one-parameter flat degeneration from a Bott-Samelson variety to a Bott tower. Using an explicit description of the fan which defines the Bott tower as a smooth toric variety, he proved the vanishing of some cohomology of line bundles on Bott-Samelson varieties. Recently, with M. Harada, the author found a rather elementary and combinatorial property with which the twisted cubes are genuine polytopes. Also we proved that with this property, the twisted cubes can be realizable as Newton-Okounkov bodies of Bott-Samelson varieties. In this talk, using the theory of Newton-Okounkov bodies, we will have a unified view point to the previous results described above and discuss about further applications.

Dong Youp Suh: Bott towers, CP-towers, and flagnized Bott towers

A Bott tower is a sequence of $$\mathbb{CP}^1$$-fiber bundles such that each stage of which is a toric manfold. One of the interesting open problem is whether the diffeomorphism class of Bott towers is determined by their cohomology rings, and this is called the cohomologial rigidity problem for Bott towers. The notion of Bott tower can be extended to generalized Bott tower which is similarly defined except that the fibers at $$i$$-th stage is $$\mathbb{CP}^{n_i}$$, and each stage of it is a toric manifold. The cohomological rigidity problem for generalized Bott towers is open, but there are some affirmative partial answers.

A CP-tower is similar to a generalized Bott tower, but we do not assume that each stage is a toric manifolds. The cohomological rigidity problem has the positive answer for the class of CP-towers of dimension less than or equal to 6, but it has a negative answer for dimension 8, which we will review briefly.

On the other hand certain Bott towers can be viewed as a degeneration of Bott-Samelson varieties which are not toric but diffeomorphic to Bott towers. Bott-Samelson varieties are CP-towers. In fact, Grossberg and Karshon showed that by modifying the complex structure of a Bott-Samelson variety there exists a one parameter family of Bott-Samelson varieties whose limit is a Bott tower. Even though the notion of Bott tower is extended to generalized Bott towers and CP-towers, there is no analogue of Bott-Samelson variety for such extended notions. Instead, in this talk, we define a flagnized Bott-Samelson variety as a generalized notion of Bott-Samelson variety, and show that there is a one parameter family of flagnized Bott-Samelson variety whose limit is a flagnized Bott tower, which is a sequence of flag manifold fiber bundles. A flagnized Bott tower is not a toric manifold, but a GKM manifold.

This talk is based on a joint work with Shintaro Kuroki, and an on-going project with Eunjeong Lee and Shintaro Kuroki.

He Wang: Resonance varieties and Chen ranks of braid-like groups (slides)

The resonance varieties of a finitely generated group $$G$$ are closed subvarieties of $$H^1(G,\mathbb{C})$$ determined by the cohomology ring of $$G$$, while the Chen ranks of $$G$$ are the lower central series ranks of its maximal metabelian quotient. I will talk about the first resonance varieties and the Chen ranks of a class of pure braid-like groups, including the pure virtual braid groups, the pure welded braid groups, and certain of their subgroups. As an application, I will use these invariants to test the formality properties and decide various isomorphism problems for such groups. This talk is about joint work with Alex Suciu.