Nordic Topology Meeting 27 - 28 November 2008


A. Berglund: From Sullivan models to Koszul models
Abstract: Rational homotopy types are represented by Sullivan models - commutative cochain algebras (A,d) where, among other things, the underlying algebra A is free. An important feature is that rational homotopy invariants, like rational homotopy groups, are easily computed from the Sullivan model.

In this talk I will introduce a generalization of Sullivan models that I propose to call `Koszul models'. These are commutative cochain algebras (A,d) whose underlying algebra A is a, not necessarily free, Koszul algebra. Koszul models share the computational benefits of Sullivan models, but are sometimes more tractable. Examples to illustrate their usefulness are provided by moment-angle complexes associated to simplicial complexes. These spaces have infinitely generated Sullivan models but admit finitely generated Koszul models. If a space X admits a finitely generated Koszul model then a consequence is that the homotopy Lie algebra π*X) ⊗ Q is finitely generated and that the Poincaré series of the loop space homology H*X;Q) is rational.

B. Dundas: Two-vector bundles I & II
Abstract: If you cover a space by open sets, a continuous function can be given by defining the function on each open set and demanding that the values agree at intersection points. A vector bundle can be defined by defining transition isomorphisms on intersections and demanding that they agree on triple intersections. A two-vector bundle is a natural extrapolation of these ideas, where the isomorphisms occur at triple intersections and the cocycle condition on quadruple intersections. In fact, two-vector bundles is what you get if you - when defining vector bundles - replace the ring of complex numbers by the category of finite dimensional complex vector spaces with sum and tensor as operations. Otherwise said, they are to vector bundles what gerbes are to line bundles.

The strange fact is that this very naïve setup has homotopy theoretical content. The chromatic picture gives a hierarchy of cohomology theories according to how deep structure of stable homotopy theory the cohomology theory detects. Chromatic filtration zero, one and infinity have nice geometric interpretations through functions, vector bundles and bordisms, and the geometric origin is important for the analysis of key problems. Such geometric interpretations have been missing in higher finite filtrations.

Elliptic cohomology and topological modular are examples at chromatic filtration two, and Segal conjectured that there ought to be a geometric interpretation of elliptic cohomology through quantum field theories.

Recently it was shown that two-vector bundles give rise to a cohomology theory which is represented by the algebraic K-theory of topological K-theory ku. By results of Ausoni and Rognes K(ku) is of (a connected version of) chromatic filtration two. Hence we have a naturally defined geometric theory of the desired sort.

The connection to quantum field theories and the set-up of Stolz and Teichner is, however, still mysterious. There was a hope that an "integration of determinants through loops" construction would give a functor from two-vector bundles to quantum field theories, but this is unfortunately not the case.

Whereas commutative rings support determinants, this is not (in the most naïve sense) true for commutative ring spectra. In fact, neither the sphere spectrum nor topological K-theory support determinants. The latter is important for us since it rules out the conjectured integral functor to quantum field theories.

However the reason for its failure is very interesting: it stems from an observation that Rognes that Ausoni's calculations of K(ku) implies that the group of gerbes on the three dimensional sphere do not split off as a direct summand of the group of "virtual" two-vector bundles. This leaves one speculating about the geometry of two-vector bundles, even over very simple spaces.

Rationally, this problem vanishes, and Ausoni and Rognes have pushed through the program in this case: giving a virtual two-vector bundle on X is rationally the same as giving its virtual "dimesion bundle" and an "anomality bundle" on the free loop space of X.

Two-vector bundles is a theory in its infancy, and much is still left to explore. In particular the geometric and analytic aspects are so far largely terra incognita.

In my talk I will try to explain some of these ideas and results.

L. Fajstrup: Ditopology - why and how
Abstract: With motivation from problems arising in concurrent computing, spaces with a preferred (time)-direction are studied from various points of view. A direction on a topological space X is given by a preferred set of paths, the dipaths P(X) ⊂ XI, closed under concatenation, increasing reparametization, subpath and containing the constant paths. The motivating examples are geometric realizations of cubical sets, where dipaths in a cube In are paths increasing in each coordinate, and all other dipaths are concatenations of such. The pair (X,P(X)) is called a d-space. Ditopology is the study of topology respecting the direction. E.g., two dipaths γ1 and γ2 are dihomotopic, if there is a dihomotopy H : I × IX as in ordinary topology, but respecting the direction along paths. Two dipaths may be homotopic but not dihomotopic. To study the motivating cubical examples, fruitful approaches may include studying a larger category - all d-spaces or some subcategory - or one may take a more combinatorial viewpoint. Hence, in analogy with ordinary topology, we get combinatorial, categorical, algebraic methods complementing each other.

Recent results will be presented: Motivated by directed covering theory - important for applications - a new more convenient category for d-homotopy has been introduced and investigated (J.Roscicky and L.F.). The topology of the space of directed paths modulo reparametrizations, the space of traces of cubical complexes, has recently been described in terms closer to calculations (M.Raussen). A categorical framework for "components" of a (nice) d-space is about to be implemented (E.Goubault and E. Haucourt) for the purpose of static analysis of real life concurrent programs.

S. Lunĝe-Nilsen: On the homology of the Tate construction on THH
Abstract: For C = S1 or any of its closed p-subgroups, we study the homology of the C-Tate construction on THH(R). The goal is to compute the stable homotopy groups of the (closely related) C-fixed point spectra THH(R)C and further, the topological cyclic homology of R.

In the talk I will show how to approximate the C-Tate construction on THH(R) by a topological version of the algebraic Singer construction. As an application, I will report on work in progress that aims to calculate the (continuous) homology of the S1-Tate construction on THH(MU).

This work is joint with J.Rognes.

I. Madsen: Cobordism categories and their classifying spaces I & II
Abstract: I will discuss the proper definition of the cobordism categories, namely the cobordism categories of embedded manifolds, and their classifying spaces. The relationship between diffeomorphism groups and the cobordism category is next. For surfaces Harer stability gives a very tight relation which I will use to prove an integral Riemann-Roch theorem for bundles of Riemann surfaces on the one hand,and discuss the Atiyah-Bott moduli space of flat connections of SU(n)-bundles over Riemann surfaces on the other hand. Finally I will outline a new result due to Johannes Ebert: The 3-dimensional analogue of the Mumford conjecture is definitely false.
J. Rognes: Topological Hochschild homology of topological modular forms
Abstract: According to the red-shift conjecture, the mod (p, v1, v2) algebraic K-theory of the topological modular forms spectrum should be v3-periodic in high degrees. In particular, K(tmf) should represent the next kind of cohomology theory after rational cohomology, topological K-theory and elliptic cohomology, and the unit map SK(tmf) should detect v3-periodic families (=γ-families) in the stable homotopy groups of spheres. At p = 2, no such families are yet known. I will report on a program to compute K(tmf) by way of THH(tmf) and TC(tmf). The first step, computing THH(tmf) at p = 2, is well underway, and is joint work with Bob Bruner. The second step, of understanding the (approximate) S1-homotopy fixed points of THH(tmf), will be a joint project with Bruner and Sverre Lunĝe-Nielsen.
C. Schlichtkrull: Higher topological Hochschild homology of Thom spectra
Abstract: We analyze the higher topological Hochschild homology of Thom spectra in general and we explain its relationship to free torus mapping spaces. We also comment on the relationship to topological Andre-Quillen homology.
A. Stacey: Comparative smootheology
Abstract: Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject.

However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.

In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.

This is based partly on arxiv:0803.0611 [math.DG] arXiv:0802.2225 [math.DG].

D. Thurston: Bordered Heegaard Floer homology: A toy model
Abstract: Heegaard Floer homology is a homological invariant of 3-manifolds and knots whose Euler characteristic is the Alexander polynomial. It detects knot genus (or more generally the Thurston norm) and fibration, and has many other uses. There is an elegant combinatorial formulation of knot Heegaard Floer homology from grid diagrams. After reviewing this construction, we then use grid diagrams to motivate an extension of the theory to 3-manifolds with parametrized boundary.

This is joint work with Robert Lipshitz and Peter Ozsváth.

N. Wahl: Algebraic structure on the Hochschild homology of an algebra
Abstract: Costello and Kontsevich-Soibelman showed that the Hochschild homology of a Frobenius algebra is a homological conformal field theory, that is it admits an action of the homology of the moduli space of Riemann surfaces. In this talk, we will give a new point of view on this theorem, with a generalisation to other types of algebras.