# Seminars & Events for Algebraic Topology Seminar

##### Projective product spaces

A projective product space is a space obtained from a product of spheres by modding out by the antipodal action in all factors. We discuss cohomology, splittings, span, parallelizability, and immersion dimension of these spaces.

##### On the KO-theory of toric spaces

Central in toric geometry and topology are several important spaces which include moment-angle complexes, the Davis-Januszkiewicz space and toric manifolds. In any complex-oriented cohomology theory, the cohomology rings of many of these spaces have elegant descriptions in terms of the underlying combinatorics. For KO-theory however the situation is more complex. Even so, a surprising amount of the structure does survive from the complex-oriented case. A report of joint work in progress with: Luis Astey, Martin Bendersky, Fred Cohen, Don Davis, Matthias Franz, Sam Gitler, Mark Mahowald, Nigel Ray and Reg Wood.

##### Spaces of knots and configuration spaces

The structure of the space of 'long knots in \R3' was developed in a series of papers by R. Budney with applications in joint work with Budney. An introduction to this space together with homological properties, and their connections to configuration spaces will be developed. For example, the singular homology of the space of knots will be considered.

##### Some results in toric topology

##### Homotopy theory and spaces of representations

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 $G$. Homotopy properties of the $B(q,G)$ will be described for finite groups, and cohomology calculations provided for compact Lie groups. Recent results on understanding both the number and stable homotopy type of the components of related spaces of representations will also be discussed.

##### Codes, arithmetic and topology

##### Periodicity and Duality

This talk will produce periodic families of Poincare duality spaces, giving a partial answer to a problem posed in the proceedings of the 1982 Northwestern homotopy theory conference. The ideas will also relate James Periodicity to the four-fold periodicity of the surgery obstruction groups.

##### Equivariant Computations and the Gap Theorem

I'll show how elementary computations with equivariant chain complexes and homology can be used to prove the vanishing of certain homotopy groups. Together with the detection theorem, this shows that the group in which the Kervaire classes would be detected is the zero group.

##### Motivic invariants of the rational numbers

We report on joint work in progress with Kyle Orsmby. Via local-to-global techniques and Adams spectral sequence computations we access motivic invariants such as K-theory, cobordism and stable stems of the rational numbers.

##### Pointed torsors and Galois groups

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 (aka. principal $H$-bundles) to the space $BH(L)$. The homotopy fibres of this map are groupoids of pointed torsors. These fibres are analyzed with cocycle techniques, and their path components are categorical representations of the absolute Galois groupoid (suitably defined) in $H$. Analogous results hold for finite etale sites: pointed torsors in that context are classified by continuous morphisms defined on a Grothendieck fundamental groupoid.

##### On the equivariant K-theory of toric varieties

A recent computation of Bahri, Franz and Ray identified the equivariant cohomology of weighted projective space with a ring of piecewise polynomials. This talk will report on joint work of Ray and the speaker, on recent developments regarding the equivariant topological K-theory of toric varieties. In particular, we relate the zeroth equivariant K-theory to an appropriate ring of Laurent polynomials, and consider extensions to the odd equivariant K-groups for a restricted class of varieties.

##### Topological complexity, Euclidean embeddings of $RP(n)$, and the cohomology of configuration spaces of pairs of distinct points in $RP(n)$

This will be a report on joint work with Jesus Gonzalez, about topics related to topological complexity (TC), introduced by Michael Farber in 2003 as a numerical measure of the complexity of robot motion planning problems. TC of real projective space $RP(n)$ coincides with the Euclidean immersion dimension of $RP(n)$ for $n$ different from 1, 3 and 7. For symmetric TC of $RP(n)$, there is a close relation to the Euclidean embedding dimension of $RP(n)$. Our current study of symmetric TC involves configuration spaces of pairs of distinct points in $RP(n)$ and has led to a calculation of their integral cohomology groups.

##### Equivariant formality and beyond

A space with a torus action is called "equivariantly formal" if its equivariant cohomology (say, with rational coefficients) is free over the polynomial ring $H^*(BT)$. Many interesting spaces fall into this class. A nice feature of them is that their equivariant cohomology can be easily computed from the fixed point set and the one-dimensional orbits. It turns out that this last property holds for more spaces than just the equivariant formal ones. We will characterise these spaces and present some examples. We will also discuss equivariant formality for cohomology with integer coefficients.