University of Chicago Algebraic Topology Seminar

4:30 pm   Tuesday, October 14, 2008   in Eckhart 203

Algebraic Topology: A cup product for intersection cochains
by Jim McClure (Purdue University)


This is joint work with Greg Friedman. There has been very little discussion of intersection cohomology in the literature. One reason for this is that there is no Alexander-Whitney diagonal map for intersection chains, so there's no obvious way to define the cup product except as the Poincare dual of the intersection pairing. We show nevertheless that there is a diagonal map on intersection chains which can serve as a replacement for the Alexander-Whitney map, and therefore there is a cup product on intersection cochains. Applications include a multiplicative de Rham theorem for intersection cohomology and a construction of the symmetric signature for singular spaces. A key ingredient in our work is Greg Friedman's intersection Kunneth theorem.


4:30 pm   Tuesday, October 21, 2008   in Eckhart 203

Algebraic Topology: Correlators and actions from moduli spaces
by Ralph Kaufmann (Purdue University)


We show how to associate multilinear maps (correlators) to certain types of Ribbon graphs. Under suitable assumptions these maps can be dualized to give operadic actions on the Hochschild co-chains of an associative algebra (which again has to satisfy some conditions, e.g. be Frobenius). These actions generalize the actions of chains of little discs, framed little discs and string topology to moduli space and its Penner/Kontsevich compactification. Lastly we show that one can stabilize the situation and obtain an operad that contains E_{infty} operad. Now this operad can be used to detect infinite loop spaces. A striking feature is that the usual filtration in terms of dimension is replaced be a filtration in genus of the graphs. The operation on the Hochschild chains stabilizes if and only if the algebra is semi-simple an normalized. This observation provides a link to Gromov-Witten theory.


4:30 pm   Tuesday, October 28, 2008   in Eckhart 203

Algebraic Topology: Some computations of Bridgeland's stability conditions
by Rina Anno (University of Chicago)


The notion of a stability condition on a general triangulated category, introduced by Bridgeland, was inspired by physics (if a category is a category of A- or B-branes in some sigma-model, a stability condition describes stable branes with a given central charge). From the mathematical point of view, the space of all stability conditions may be called a "moduli space" of t-structures, because it has a structure of a smooth manifold, and one can associate a t-structure to every stability condition. The spaces of stability conditions (or their certain connected components, or certain subspaces) were described for a number of triangulated categories. I will briefly review Bridgeland's argument for derived categories of coherent sheaves on minimal resolutions of Kleinian singularities, and proceed with a computation for derived categories of coherent sheaves on the total space of the cotangent bundle to a flag variety (by R. Bezrukavnikov and myself).


4:30 pm   Tuesday, November 4, 2008   in Eckhart 203

Algebraic Topology: Obstructions to homotopy sections of curves over number fields
by Kirsten Wickelgren (Stanford University)


Grothendieck's anabelian conjectures say that hyperbolic algebraic curves over number fields should be K(pi,1)'s in algebraic geometry; it follows that conjecturally the rational points on a hyperbolic algebraic curve cut out by equations with rational coefficients are the homotopy fixed points of the action of the absolute Galois group of Q on the curve. We use obstructions first introduced by Jordan Ellenberg which are constructed from the lower central series of the fundamental group to study these homotopy fixed points. We will relate these obstructions to Massey products, and explicitly compute mod 2 versions of the first and second for P^1 - {0,1,infty}.


4:30 pm   Tuesday, November 11, 2008   in Eckhart 203

Algebraic Topology: Homeomorphisms of manifolds and algebraic K-theory
by Lars Hesselholt (MIT)


The celebrated work of Waldhausen, Weiss-Williams, Igusa, and Farrell-Jones shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed negatively curved Riemannian manifold can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead space of the circle and the canonical involution on these groups. In this talk, I will use the cyclotomic trace of Bokstedt-Hsiang-Madsen and a theorem of Dundas to evaluate the latter homotopy groups in low degrees. 


4:30 pm   Tuesday, November 18, 2008   in Eckhart 203

Algebraic Topology: Units of ring spectra, Thom spectra, and a twisted Umkehr map
by Matt Ando (UIUC)


Let R be an associative ring spectrum. I shall describe several constructions of the R-module Thom spectrum associated to a map f : X -> BGL_1(R). The space BGL_1(R) classifies the twists of R-theory, and to a fibration of manifolds g : Y -> X I shall associate an Umkehr map g! from the (f,g)-twisted R-theory of Y to the f-twisted R-theory of X. In the case of K-theory, this twisted Umkehr map appears in the study of D-brane charge. I shall review this story, and then discuss the analogous construction for TMF .


4:30 pm   Tuesday, November 25, 2008   in Eckhart 203

Algebraic Topology: Completion of G-spectra and stable maps between classifying spaces
by Kári Ragnarsson (Depaul University)


For a finite group G, completion at the augmentation ideal I(G) of the Burnside ring A(G) features in cohomological completion theorems such as the Atiyah--Segal completion theorem and Carlsson's theorem (formerly the Segal Conjecture). Completion of G-spectra at ideals of the Burnside ring was introduced by Greenlees--May to obtain topological versions of these theorems, letting the completion take place at the level of spectra rather than cohomology groups. When G is a p-group, May--McClure showed that I(G)-adic completion coincides with p-completion (modulo some basepoint issues) but in general I(G)-adic completion seems very complicated. In this talk I will describe how, for the purposes of I(G)-completion, one may replace a G-spectrum E by a related G-spectrum E' whose isotropy groups are all groups of prime power order. The I(G)-adic completion of the latter is often more approachable, admitting a description as a colimit of completions at primes. As an application I will give a simplified description of the function spectrum F(BG,BK) for a finite group and a compact Lie group K.


4:30 pm   Tuesday, December 2, 2008   in Eckhart 203

Algebraic Topology: Presentations of monoidal model categories
by Samuel Isaacson (Harvard)


Let C be a model category. In a 2003 paper, Dan Dugger showed that if C is combinatorial, it can be realized as a left Bousfield localization of simplicial presheaves on some small site. I'll describe a variation of this theorem: by replacing simplicial sets with another model for the homotopy category, we can produce a presentation for C when C is symmetric monoidal that retains the monoidal structure of C as the convolution product.