Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q > 1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Related stable splittings for spaces of commuting elements in a compact Lie group will also be discussed.

There are two primary theories of "curve counting" on a Calabi-Yau threefold, Donaldson-Thomas theory and Gromov-Witten theory. Both theories extend to Calabi-Yau orbifolds. A three dimensional Calabi-Yau orbifold X has a canonical Calabi-Yau resolution Y. The four curve counting theories DT(X), DT(Y), GW(X), and GW(Y) are expected to be equivalent. We give an overview of these theories and conjectural equivalences.

An old theorem of C. T. C. Wall's states that every 3-manifold admits a smooth embedding in the 5-sphere. So if M is a closed 3-manifold different from the 3-sphere, the lowest-dimensional sphere that it embeds in is either 4 or 5. This talk will explore what is known about which closed 3-manifolds admit smooth embeddings into the 4-sphere.

In the last decade, 3-manifold topology has seen the first major applications of the ideas developed 20 years ago by the late Andreas Floer, most notably through the use of the Heegaard-Floer homology defined by Ozsváth and Szabó. Through recent work of Kronheimer and Mrowka, many of these advances can be recovered through a version of instanton Floer homology. In this talk we aim to look at some consequences for SU(2)-representation varieties of knots in S³. In particular we shall explain how SU(2)-representation varieties of 2-fold branched covers detect the unknot and give a simple proof of a classical result of Scharlemann which asserts that unknotting number 1 knots are prime.

A tangle is a pair (B,A), where B is the 3-sphere with the interiors of a finite number of 3-balls removed, and A is a disjoint union of properly embedded arcs in B such that A meets each component of ¶B in four points. The Hexatangle is a certain tangle having six boundary components and a projection into the plane with no crossings. By filling the boundary components of a tangle with rational tangles we get knots and links in the 3-sphere. In a previous work we determined all the integral fillings on the hexatangle that produce the trivial knot. Now we consider arbitrary rational fillings of the hexatangle, and have a conjecture which says exactly when we can get the trivial knot. We show some partial results about this conjecture. The double branched cover of the hexatangle is a certain hyperbolic link L of six components in S³. Our problem is equivalent to determining all Dehn surgeries on L that produce the 3-sphere. This link is interesting, for infinitely many hyperbolic knots which have exceptional surgeries are obtained by performing surgery on 5 components of L, and then a solution of the conjecture will lead to a listing of all such knots that are obtained from L.

This is joint work with Lorena Armas-Sanabria.

I will speak on new developments in moment angle complexes and toric topology.

Topological complexity was introduced by M. Farber as a way to measure intrinsic hardness in motion planning algorithms. It has been shown that this invariant captures the immersion dimension of projective spaces. I will describe the corresponding relationship between symmetric topological complexity and embedding dimension.

This is joint work with Peter Landweber.

A subset U of a closed topological n-manifold M is S¹-contractible (in M) if there exist maps f : U ® S¹, a: S¹ ® M such that the inclusion of U into M is homotopic to af.

M has S¹-category £ k if it can be covered by k open subsets which are S¹-contractible.

Previously we have determined, for n £ 3, the n-manifolds with S¹-category 2.

Theorem 1 If n > 3 and Mⁿ has S¹-category 2 then Mⁿ is an n-sphere or an S^n-1-bundle over S¹.

With J. C. Gómez-Larrañaga and W. Heil.

Suppose that H is an algebraic group which is defined over a field k, and let L be the algebraic closure of k. The canonical stalk for the etale topology on k induces a simplicial set map from the classifying space B(H-tors) of the groupoid of H-torsors (a.k.a. principal H-bundles) to the space BH(L). The homotopy fibres of this map are groupoids of pointed torsors, suitably defined. These fibres can be analyzed with cocycle techniques: their path components are representations of the absolute Galois groupoid in H, and each path component is contractible. The arguments for these results are simple, and applications will be displayed.

In this talk I will report on my joint work with Ana Gonzalez and Carlos Segovia on the study of a generalization of Frobenius Algebras motivated by String Topology.

I will explain of the relation of such algebras with topological field theories, and to orbifolds.

The complex flag manifold Fl_n (C) is the space of all nested sequences

An outstanding conjecture by M. Gromov asserts that a one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface. Motivated by this conjecture, we study the existence of immersions of closed surfaces in 2-dimensional CW-complex. Our results provide sufficient conditions for the existence of such immersions and imply the existence of surface subgroups in a particular class of hyperbolic groups.

Joint work with M. Forester and N. Brady.

We find the horofunction boundary of the (2n+1)-dimensional Heisenberg group with the Koranyi metric and show that it is homeomorphic to a 2n-dimensional disk. We also show that the Busemann points correspond to the (2n-1)-sphere boundary of this disk and that the compactified Heisenberg group is homeomorphic to a (2n+1)-dimensional sphere. As an application, we find all isometries of the Koranyi metric.

This is joint work with Tom Klein.

We will indicate how to compute the algebraic K-theory groups of virtually free groups using the following facts:

(1) these are hyperbolic groups;
(2) they satisfy the Farrell-Jones Conjecture for K-theory.

A group is left-orderable if there is a strict total ordering < of its elements so that y < z implies xy < xz. If a group G is left-orderable, the set LO(G) of all left orderings has a natural topology, introduced by Sikora. I will outline the current state-of-the-art in understanding the space of orderings, and some of the applications of this theory.

For a topological space X, cat(X) is one less than the least number of open sets in X that it takes to cover X. Ganea conjectured that for any X, cat(X ×Sⁿ) = cat(X)+1. In 1997 Iwase constructed a series of counterexamples using the instability of certain Hopf invariants. The least dimension of the counterexamples was ten. On the other hand results of Vandembrouq imply that no simply connected counterexamples exist with dimension less that or equal to five. We construct a counter-example of dimension seven. Instead of unstable Hopf invariants, the example comes from a Hopf invariant stably factoring through another map. We will also discuss the remaining question of counterexamples of dimension six.

This is joint work with Hugo Rodriguez.