University of Chicago
Department of Mathematics

Fusion in stable homotopy theory

Kari Ragnarsson

In recent work with Radu Stancu we proved a bijection between saturated fusion systems on a finite p-group and certain idempotents in its p-localized double Burnside ring. The Segal conjecture relates the double Burnside ring to stable maps between classifying spaces, and so we can regard saturated fusion systems as objects in stable homotopy theory. In the talk I will show how, adopting this point of view, we can answer a long-standing question on stable splittings of classifying spaces, and give a (best possible?) generalization of the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. Time permitting, I will present a new, simplified model for p-local finite groups.

4:30 pm Tuesday, November 17, 2009

Algebraic Topology

Quillen cohomology of Pi-algebras

Martin Frankland

A Pi-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object A, one may ask if it can be realized topologically: Is there a space X such that pi_*(X) is isomorphic to A as a Pi-algebra, and if so, can we classify them? Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a Pi-algebra, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of Pi-algebras. What do these groups look like, and can we compute them? We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated Pi-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.

4:30 pm Tuesday, November 24, 2009

Algebraic Topology

Homotopical Properties of weakly globular models of homotopy types

Simona Paoli

Homotopy n-types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than n. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of n-fold categories internal to groups, also called catⁿ-groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of catⁿ-groups is a weak equivalence. In this talk we introduce a new model of connected n-types through a subcategory of catⁿ-groups, which we call weakly globular, for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model, and discuss the relevance of these structures for higher category theory. In the second part of this talk we discuss the use of the notion of weak globularity to deal with the non-path connected case in low dimension (n=2) and we illustrate an application. The second part of this talk is joint work with David Blanc.

University of ChicagoDepartment of Mathematics

University of Chicago
Department of Mathematics