# Seminars & Events for Algebraic Topology Seminar

##### Spanier-Whitehead $K$-duality

Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. We consider a noncommutative version, termed Spanier-Whitehead $K$-duality, which is defined on the category of $C^*$-algebras whose $K$-theory is finitely generated and that satisfy the UCT, with morphisms the Kasparov $KK$-groups. Examples from foliations, hyperbolic dynamics, and other highly non-commutative $C^*$-algebras illustrate the truly new phenomena encountered. There are many open questions associated with relaxing the assumptions on the algebras. For example, does the Calkin algebra have a Spanier-Whitehead $K$-dual? This is joint work with Jerry

Kaminker.

##### The Grothendieck -Teichmuller lie algebra and homotopy equivalences of configuration spaces

The Grothendieck - Teichmuller Lie algebra grt is a Lie algebra, over the rational numbers Q, which is clearly very interesting and equally clearly not very well-understood. It crops up in many different areas of mathematics.

In this talk I will explain how the Lie algebra grt is related to the space of homotopy equivalences of the configuration spaces F(k, R^n) and F(k, S^n). Here F(k,X) is the space of k distinct ordered points in the topological space X. Much of the talk will be devoted to explaining, in as concrete a way as possible, the background needed for the statements of the results. I will also try to explain one or two of the key ideas in the proofs.

##### Topology and Combinatorics of 'unavoidable' complexes and the family tree of the Van Kampen-Flores theorem

Special classes of simplicial complexes (chessboard, `unavoidable’, threshold, `simple games’, etc.) play a central role in applications of algebraic topology in discrete geometry and combinatorics. As an illustration we outline the proof of a (so far) the most general theorem of Van Kampren-Flores type (arXiv:1502.05290 [math.CO],** **Theorem 1.2), confirming a conjecture of Blagojevic, Frick, and Ziegler (Tverberg plus constraints, Bull. London Math. Soc., 46 (2014) 953-967). The results presented in the lecture are a joint work with Dusko Jojic (University of Banja Luka) and Sinisa Vrecica (University of Belgrade).

##### A geometric model of twisted differential K-theory

A twisted vector bundle is a weaker notion of an ordinary vector bundle whose cocycle condition is off by a U(1)-valued Cech 2-cocycle (the cycle data of a U(1)-gerbe) called a topological twist. We will introduce a geometric model of a differential extension of twisted complex K-theory using twisted vector bundles with connection as cycles and U(1)-gerbes with connection as differential twists. Here a U(1)-gerbe with connection is a total degree 2 cocycle in the Cech-de Rham double complex. We will give an introduction to the Chern-Weil theory of twisted vector bundles, define a twisted differential K-theory, and introduce a hexagon diagram of twisted differential K-theory.

##### Spaces of commuting elements in Lie groups

Spaces of group homomorphisms $Hom(\pi,G)$ from a discrete group to a Lie group have been studied in various contexts. We study the space of pairwise commuting $n$-tuples, i.e. $\pi$ is free abelian, in a compact and connected Lie group G, from the topological viewpoint. We will describe a way to stabilize spaces of homomorphisms by introducing an infinite dimensional topological space, reminiscent of a Stiefel variety, that assembles the spaces of commuting tuples into a single space. Hilbert-Poincare series will be also described, in addition to other properties.

##### An Algorithm for Hecke Operators

Hecke operators act on the cohomology of locally symmetric spaces for SL(n,R), and the Hecke eigenvalues are important in number theory and automorphic forms. When n = 2, the case of classical modular forms, modular symbols (due to Manin) give an algorithm for computing the Hecke operators. Ash and Rudolph extended the algorithm to all n, but only in the top non-vanishing degree of the cohomology--the virtual cohomological dimension, or vcd. Gunnells found an algorithm for all n, but only in degree one less than the vcd. This talk is on work in progress on an algorithm that computes the Hecke operators in all degrees. This is joint work with Robert MacPherson.

##### Symmetrized topological complexity of the circle

We prove that it is impossible to have two continuous rules telling how to move between any two points on the circle in such a way that the path from Q to P is the reverse of the path from P to Q. It is easy to see that this can be done with three such rules.

##### Topology of Character Varieties

Character varieties parametrize conjugacy classes of representations of discrete groups into algebraic groups. I'll discuss some recent result on the topology of these varieties. In particular, I'll explain how studying their fundamental groups leads to information about centralizers in Lie groups. This is joint with with Biswas, Lawton, and Florentino.

##### On some symplectic aspects of moduli stack of Chen connections

The study of the Poisson geometry of the Teichmuller space and the moduli space of local systems gave rise to the discovery of the Goldman bracket of curves on an oriented surface which in turn led Chas and Sullivan to discover string topology operations on chains on the free loop space of an arbitrary oriented manifold. Their string topology operations also generalized the Turaev cobracket which did not come from a Poisson geometric origin, and the search for the geometric meaning of all string topology operations continues. In this direction, I will discuss some Poisson geometric aspects of the moduli stack of Z-graded Chen connections and how in the large N-limit an additional relevant structure appears (N=dimension of the fibre).

##### Topology of the space of metrics of positive scalar curvature

**This is a joint Algebraic Topology / Topology seminar. **We use recent results on the moduli spaces of manifolds, relevant index and surgery theory to study the index-difference map from the space ${\mathcal R}iem^+(W^d)$ of psc-metrics to the space $\Omega^{d+1}KO$ representing the real $K$-theory. In particular, we show that the index map induces nontrivial homomorphism in homotopy groups $\pi_k {\mathcal R}iem^+(W^d) \to \pi_k \Omega^{d+1}KO$ once the target groups $\pi_k \Omega^{d+1}KO= KO_{k+d+1}$ are not trivial. This work is joint with J. Ebert and O. Randall-Williams. In this talk, I also plan to discuss some recent applications of those results.

##### Stable moduli space of high-dimensional handlebodies

**This is a joint Topology / Algebraic Topology seminar. **We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^n)^{\natural g}$, i.e. the classifying space $\mathrm{BDiff} (D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near an embedded disk $D^{2n} \subset \partial (D^{n+1}\times S^n)^{\natural g}$. We prove that there is a natural map $$\colim_{g\to\infty}\mathrm{BDiff}((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \;\longrightarrow \; Q_{0}BO(2n+1)\langle n \rangle_{+}$$ which induces an isomorphism in integral homology when $n\geq 4$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. This work is joint with N.Perlmutter. I also plan to discuss some recent results related to this work.