Notre Dame Topology Seminar

Fall 2008 Schedule

October 2 at 1PM in H258: Tom Fiore, University of Chicago; A Thomason Model Structure on nFoldCat; pretalk 10-11 AM
October 9 at 10AM in H258: Stacy Hoehn; Parametrized Wall Finiteness and Siebenmann End Obstructions
October 16 at 10AM in H258: Stacy Hoehn; Constructing Family Versions of Geometric Invariants
November 25 at 10AM in HH125: Sunil Chebolu; Some applications of the Bloch-Kato conjecture
December 11 at 10AM in H258: Tony Elmendorf; Permutative categories, multicategories, and algebraic K-theory

Fall 2008 Abstracts

October 2, 2008: Tom Fiore

A Thomason Model Structure on nFoldCat

When are two categories the same? One possible notion of weak equivalence is an equivalence of categories, another is a functor whose nerve is a weak homotopy equivalence of simplicial sets. As is well known, these distinct notions of weak equivalence between categories have been encoded in model structures by Joyal-Tierney and Thomason. One can ask the same question for Ehresmann's internal categories in Cat: when are two double categories the same? Or, more generally, when are two n-fold categories the same? There are several reasonable notions of weak equivalence. One uses the nerve: an n-fold functor is a weak equivalence if and only if the diagonal of its nerve is a weak equivalence of simplicial sets. Our Theorem states that such n-fold functors form the weak equivalences of a model structure on nFoldCat, and that this model structure is Quillen equivalent to the standard structure on simplicial sets.

There are several reasons for interest in this topic. A double category is good context in which to embed two types of morphisms: ring homomorphisms and bimodules form the morphisms of a double category, as do cobordisms and diffeomorphisms. A double category is also a good place to treat base change. The Thomason model structure is of interest because of Quillen's Theorems A and B, as well as Thomason's result on certain nerves of categories as models for certain homotopy colimits.

This is joint work with Simona Paoli.

October 9, 2008: Stacy Hoehn

Parametrized Wall Finiteness and Siebenmann End Obstructions

Given a finitely dominated space X, the classical finiteness obstruction of Wall is an element in a certain group that vanishes if and only if X is homotopy equivalent to a finite CW complex. Similarly, given a non-compact manifold M that satisfies certain tameness properties near its ends, the classical end obstruction of Siebenmann is an element in a certain group that vanishes if and only if M can be completed to a compact manifold with boundary. Both of these classical obstructions can be parametrized to answer questions about families of spaces. We will discuss both of these parametrized obstructions and describe their relationships with one another.

October 16, 2008: Stacy Hoehn

Constructing Family Versions of Geometric Invariants

Many classical geometric invariants, like Wall's finiteness obstruction and the symmetric signature of certain spaces, have parametrized versions that can be used to answer questions about families of spaces. In the naive approach to constructing these family versions, problems may arise in guaranteeing compatibility among the different spaces in the family. We will describe how these problems can be avoided in certain situations by working on the level of categories instead of topological spaces.

November 25, 2008: Sunil Chebolu

Some applications of the Bloch-Kato conjecture

The Bloch-Kato conjecture claims that the reduced Milnor K-theory of a field F is isomorphic to the Galois cohomology of F. (I will explain these terms and the conjecture in detail in my talk.) This conjecture has been recently settled by Voevodsky and Rost. I will present some applications of this conjecture. In particular, I will show how this conjecture allows us to detect absolute Galois groups, and give cohomological characterisations of quadratic closures. This is joint work with Jan Minac.

December 11, 2008: Tony Elmendorf

Permutative categories, multicategories, and algebraic K-theory

One of the standard methods for producing algebraic K-theory spectra involves a construction that associates a spectrum to any permutative category. In one of his last papers, Robert Thomason showed that this construction produces all connective spectra, as well as all homotopy classes of maps between them, and claimed that he could produce a symmetric monoidal multiplicative structure on permutative categories that models the multiplicative structure of connective stable homotopy. In this talk I will describe joint work with Mike Mandell in which we give a symmetric monoidal category that is also bicomplete, which maps multiplicatively to connective spectra and in which permutative categories embed. The trick is to look at more general structures than permutative categories, called multicategories (also known as colored operads) and show that the construction of the associated spectrum extends, and actually has nicer properties than the construction used by Thomason and others.

2007-2008 Topology Seminar
Graduate Topology/Geometry Seminar
Notre Dame Department of Mathematics

Questions? Contact Kate Ponto