# TTT70: Manchester

School of Mathematics, Alan Turing Building, University of Manchester.

Supported by the London Mathematical Society and The Royal Society/Russian Foundation for Basic Research.

Informal meetings were held in the Frank Adams seminar room during the period 4–6 November. The primary purpose was for Manchester PhD students to discuss ongoing research work with their Moscow counterparts.

## Date

2–3 November 2009

## Speakers

### Taras Panov (Moscow State University)

Cohomological rigidity of quasitoric manifolds and simple polytopes

Cohomological rigidity of quasitoric manifolds and simple polytopes

A family of (quasi)toric manifolds is said to be cohomologically rigid if the manifolds within the family are distinguished up to homeomorphism by their cohomology rings. Although it seems unlikely that the whole class of quasitoric manifolds is cohomologically rigid, no counterexamples are yet known. In other words, it is unknown if an isomorphism between the cohomology rings of two quasitoric manifolds implies the existence of a homeomorphism.

Moreover (and quite surprisingly), cohomological rigidity holds for some particular families, such as Bott towers of height up to 3, or topologically trivial Bott towers of arbitrary height. There is also a related combinatorial concept of cohomological rigidity for simple polytopes: a polytope P is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of any (quasi)toric manifold over P.

Not all polytopes are cohomologically rigid, but the rigidity may be established for certain important families, and the arguments involve some nice combinatorial commutative algebra. We shall discuss several results on cohomological rigidity for manifolds and polytopes, and suggest some open problems.

### Gery Debongnie (Louvain-la-Neuve and Manchester)

On the rational homotopy type of subspace arrangements

On the rational homotopy type of subspace arrangements

We shall explore different properties of the complement spaces of subspace arrangements, from the viewpoint of rational homotopy theory. A rational model will be described, from which we deduce several results. For example, we give a complete description of coordinate subspace arrangements whose complement space is a product of spheres.

### Andrey Kustarev (Moscow State University)

Almost complex quasitoric manifolds

Almost complex quasitoric manifolds

We prove that there exists a T^n-invariant almost complex structure on any quasitoric manifold M^{2n} with positive omniorientation.

### Nickolay Erohovets (Moscow State University)

On the Buchstaber invariant of simple polytopes

On the Buchstaber invariant of simple polytopes

The Buchstaber invariant s(P) of a simple polytope P is the maximal dimension of a subtorus of T^m that acts freely on the moment angle complex Z_P; it is not difficult to see that s(P) cannot exceed m-n, where P has m facets and dimension n.

We will show that s(P)=1 if and only if P is a simplex, and that for any k \geq 2, there exists a P with m-n=k and s(P)=2. We will also relate s(P) to chromatic numbers of P, flips on P, flag polytopes, and f-vectors, and compute s(P) for m=n+3 in terms of bigraded Betti numbers.

### Alexander Gaifullin (Moscow State University)

Sets of links of vertices of simplicial and cubic manifolds

Sets of links of vertices of simplicial and cubic manifolds

To each oriented (simplicial or cubic) closed combinatorial manifold one may assign the set (with repetitions) of isomorphism classes of links of its vertices. The resulting transformation L is the main object of our talk, and we pose a problem on its inversion: for a given set Y_1,Y_2,... ,Y_k of oriented (n-1)-dimensional combinatorial spheres, is there an oriented (simplicial or cubic) n-dimensional combinatorial manifold K whose set of links of vertices coincides up to isomorphism with the given set Y_1,Y_2,...,Y_k?

It is easy to obtain a balancing condition, which is necessary for the existence of such a manifold K; that is, for a set of isomorphism classes of combinatorial spheres to belong to the image of L. We shall give an explicit construction providing that each balanced set of isomorphism classes of combinatorial spheres is in the image of L after passing to a multiple set and adding several pairs of the form (Z,-Z), where -Z is the sphere Z with the orientation reversed.

We shall also discuss the relationship of this problem with Steenrod's problem on realization of cycles and the problem of finding local combinatorial formulae for the rational Pontryagin classes of triangulated manifolds.

### Dmitri Gugnin (Moscow State University)

The graded theory of Frobenius n-homomorphisms, and topological applications

The graded theory of Frobenius n-homomorphisms, and topological applications

Building on work of Buchstaber and Rees, we introduce an algebraic theory of graded Frobenius n-homomorphisms, and describe two topological applications. The first of these involves the theory of Dold-Smith branched coverings and their associated transfer maps, which are closely related to actions of finite groups on topological spaces.

The second application concerns the notion of an $n$-Hopf algebra for an arbitrary graded commutative algebra. We show that such a structure exists on the rational cohomology ring of any n-valued topological group, and deduce results on the nonexistence of n-valued multiplications for new series of manifolds, where n=2 and 3.

### Ivan V Arzhantsev (Moscow State University)

Homogeneous toric varieties

Homogeneous toric varieties

This is joint work with Sergey Gaifullin. We classify all toric varieties with a transitive action of a semisimple algebraic group. It turns out that such toric varieties lie between a product of punctured affine spaces and a product of projective spaces. The result is based on Cox's realization of a toric variety.

### Natalia Dobrinskaya (Vrije University, Amsterdam)

Loops on quasitoric manifolds

Loops on quasitoric manifolds

This is joint work with Nigel Ray. We construct combinatorial models for loop spaces of (quasi)-toric manifolds, using piecewise geodesics with certain very special properties. In particular, all breaks of these geodesics lie in the union of the singular strata and one free orbit. We also discuss related results concerning free loops on quasitoric manifolds, and some homology calculations.

## Further information

The meeting was supported by the London Mathematical Society and The Royal Society/Russian Foundation for Basic Research.