Fields Institute Program on Homotopy Theory and its Applications University of Western Ontario London, Ontario September, 2003

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Abstracts for all conference lectures

Alejandro Adem, Periodicity, Fixity and Group Actions

One of the basic problems in group cohomology is to relate properties such as periodicity to group actions. In this lecture we will discuss how homotopy theory can be applied to this problem. It turns out that periodic methods can also be applied to non-periodic situations, and in particular we will discuss recent work on constructing finite group actions on products of spheres. We will also use the notion of fixity to produce natural examples of smooth actions.

John Baez, Euler Characteristic versus Homotopy Cardinality

Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups. There are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined. However, the two quantities have many of the same properties. In many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series, and the two then agree. We give examples, describe applications of homotopy cardinality to a topological version of Joyal's theory of "species", and beg the audience to find some unifying concept which has both Euler characteristic and homotopy cardinality as special cases.

Paul Baum, K-Theory for group C^* algebras

Let A be a Banach algebra (i.e. a complete normed algebra over the complex numbers). Associated to A is a spectrum whose homotopy groups are the (topological) K-theory groups of A. If G is a topological group which is locally compact, Hausdorff, and second countable, then the reduced C^* algebra of G is a Banach algebra and thus its K-theory groups are defined. How can these K-theory groups be calculated? This talk states the conjecture of P.Baum and A.Connes which proposes an answer to this question. Note that G could be a discrete (countable) group. At the present time no counter-examples to the conjecture are known.The conjecture has been proved for certain classes of groups including amenable groups, a-t-amenable groups, and hyperbolic groups. When true the conjecture has among its corollaries the Novikov conjecture, the stable Gromov-Lawson-Rosenberg conjecture and the Kadison-Kaplansky conjecture.

Daniel Biss, On the stable homology of Aut(F_n)

The stable automorphism group of the free group can be thought of as a 1-dimensional analogue to the stable mapping class group, or as the first (possibly) nontrivial step in an infinite filtration of the algebraic K-theory of a point; it also has connections with Kontsevich's graph homology, and, consequently, with finite type invariants of knots and 3-manifolds. I will make some comments about the homology of this group.

Gunnar Carlsson, Algebraic Topology as Applicable Mathematics

I will discuss various attempts to use algebraic topology as a tool for studying data sets and for recognizing familiar shapes, by locating and classifying geometrically interesting features. We will also discuss some of the algorithmic problems arising in this approach, which has lead to some new ideas and algorithms for computing homology.

Wojtek Chacholski, Global properties of fibrations of topological spaces

I will describe one way of quantifying how complicated fibrations of topological spaces are. This will be done by comparing the operations of taking fibrations (twisted products) and loop spaces with the more basic operations of taking wedges and push-outs. I will show that adding twisted products coincide with the stabilization process of topological spaces. I will then use the classification of the thick subcategories of the stable category and the classification of unstable Bousfield classes to describe the complexity of fibrations. I will also describe the generic fiber of this stabilization process.

Denis-Charles Cisinski, Pursuing models for homotopy types

This talk is a presentation of the realization of a part of Grothendieck's program in Pursuing Stacks to lay the foundations of homotopy theory. Here, the main point is to understand to which extent the homotopy theory of simplicial sets can be generalized to an arbitrary category of (pre)sheaves. Following ideas of Grothendieck, we give a reasonable answer to this question and obtain in this way new models for (eventually equivariant) homotopy types in terms of Quillen model category structures, with an emphasis on cohomological invariants.

Jesper Grodal, A finite loop space not rationally equivalent to a compact Lie group

(Joint work with K. Andersen, T. Bauer, E. Pedersen)

I will exhibit a concrete finite loop space which is not rationally equivalent to any compact Lie group, hence providing a counterexample to a classical conjecture. Our finite loop space has rank 66. We however also verify that that this cannot happen for loop spaces of rank 65 or less, hence extending the classically known bound of 5.

Lars Hesselholt, Motivic complexes and topological Hochschild homology

This is joint work with T. Geisser. The motivic complexes Z(q) form the E₁-term of a spectral sequence for the algebraic K-theory of smooth scheme over a field. By analogy, we introduce complexes W(q) that we conjecture form the E₁-term of a spectral sequence for the p-typical curves on the K-theory of a smooth scheme over a field.

Finnur Larusson, Model structures in complex analysis

The new notion of a simplicial site, due to Toen and Vezzosi, with their generalization of the homotopy theory of simplicial presheaves on ordinary, discrete sites to prestacks on simplicial sites, provides a framework for applying homotopical algebra in analysis. We illustrate this by results to the effect that holomorphic maps satisfy certain versions of the Oka Principle if and only if they are fibrations in suitable model structures on the category of prestacks on the site of Stein manifolds.

Ib Madsen, Cohomology of some moduli spaces.

I will consider the moduli space of Riemann surfaces with a continuous map to some fixed background space Y. This is analoguos to the much studied moduli space of Riemann surfaces with a holomorphic map to Y in case Y is a symplectic manifold.

Peter May, Parametrized equivariant stable homotopy theory

We explain the structure and some applications of the stable homotopy category of G-spectra parametrized over a base G-space. The power and interest of the theory come from base change functors relating different base spaces. To the best of my knowledge, there has not previously been a full study of any such situation in a model categorical framework, and the theory gives a striking illustration of both the power and the intrinsic limitations of model category theory. Pullback base change functors have both left and right adjoints. It turns out that, in any interesting model structure, it is impossible for both of these adjunctions to be Quillen adjunctions. In fact, it is nontrivial to show that both point-set level adjunctions descend to adjunctions relating the derived homotopy categories. The same will be true in analogous situations in algebraic geometry. While the solutions to these and to related problems with smash products and duality are special to our topological context, they might serve as a guide to anyone interested in modernizing the analogous foundational frameworks that appear in algebraic geometry.

This is joint work with Johann Sigurdsson.

Ieke Moerdijk, Categories of embeddings and classifying spaces for foliations

I will present a generalisation of an old theorem of Graeme Segal, and show that the classifying space of a foliation (an infinite dimensional non-Hausdorff space) is weakly homotopy equivalent to the classifying space of a discrete category. I will give two applications, namely a geometric construction of characteristic classes of foliations, and a classification of gerbes over leaf spaces.

Jack Morava, Thom prospectra for loop group representations

There is reason to believe that a good theory of an Atiyah dual LM^-TLM for a free loopspace would fit very nicely with the homotopy-theoretic approach to conformal field theory under development by Chas-Sullivan, Madsen-Tillmann-Weiss, R. Cohen et al, W. Dwyer, J. Klein, A. Voronov, and others. However, the fact that the tangent bundle of a loopspace is not defined by a positive-energy representation presents real technical difficulties. This talk is a report on recent work with Nitu Kitchloo, based on the use of a topological affine Tits building, which promises to resolve these problems elegantly.

Fabien Morel, On cohomology operations in unramified Milnor K-theory

In this talk we will address the problem of constructing and computing cohomology operations in the cohomology theory on smooth schemes with coefficents in unramified mod p Milnor K-theory. We will give some examples and applications, related to the Bloch-Kato conjecture and to the Milnor conjecture on quadratic forms.

Vic Snaith, The homotopy category of perfect complexes applied to arithmetic-algebraic geometry

When G is abelian and l is a prime we show how elements of the relative K-group K₀(Z_p[G], Q_p) give rise to annihilator/Fitting ideal relations of certain associated Z[G]-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which G is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates-Sinnott and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions. When these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.

Neil Strickland, Arithmetic equivariant elliptic cohomology

Bertrand Toen, Some applications of homotopical algebraic geometry

In algebraic geometry, geometric objects (e.g. schemes, algebraic stacks, ...) are obtained by gluing together commutative rings. Homotopical algebraic geometry is a more general theory (developed in collaboration with Vezzosi) for which geometric objects are obtained by gluing together homotopy commutative ring-like objects (e.g. commutative differential graded algebras, E-infinity-ring spectra, symmetric monoidal categories, ...). In this talk I will first recall briefly the general formalism, and then present some of its applications to the construction of certain moduli spaces in algebraic geometry and algebraic topology.

Thomas Wenger, Derived deformation theory patterns and mixed Tate motives