Algebraic and Geometric Topology / Topologie algébrique et géométrique
(Org: Laura Scull, Peter Zvengrowski and/et George Peschke)

KRISTINE BAXTER BAUER, University of Western Ontario, London, Ontario  N6A 5B7
Towards classifying functors from spaces to spaces

This is work in progress with Greg Arone, Dan Christensen and Dan Isaksen.

In the past, topologists have been successful in classifying degree n functors from the category of topological spaces to spectra (e.g. Dwyer-Rezk, McCarthy). However, understanding the situation for functors from the category of topological spaces to itself remains elusive. We consider another classification of functors from spaces to spectra and explain how this leads to a classification of functors from spaces to spaces.

DAVID BLANC, University of Haifa, Haifa, Israel
Rectifying homotopy commutative diagrams of spaces

A problem which arises in many contexts in homotopy theory is the following: can we replace a diagram of topological spaces and maps which commutes only up to homotopy by a strict diagram of topological spaces? This comes up, for instance, when we try to determine when an H-space is actually (homotopy equivalent to) a topological group. We shall give a survey on two basic approaches to the problem-one geometric, in terms of higher homotopy operations, and the other algebraic, in terms of cohomology groups-and try to explain how they are related.

PETER BOOTH, Memorial University, St. John's, Newfoundland
On the use of non-standard mapping spaces in homotopy theory

Some topological constructions are basic to Homotopy Theory. Thus Product Spaces, Topological Sums, Mapping Spaces with the Compact-Open Topology, Pullback Spaces, Adjunction Spaces, CW-Complexes and Topological Joins are part of the standard toolkit of practitioners in the area. Certain other basic constructions-the various types of Fibred Mapping Spaces-are easily defined and can be applied in a wide variety of situations. Yet they have been used by only a relatively small number of topologists.

In this talk I will review some of these latter constructions and indicate how they can be applied to numerous topics.

RYAN BUDNEY, University of Rochester, Rochester, New York
Geometric coincidences on knots

I will describe how a coefficient of the Alexander-Conway polynomial of a knot is a count of the number of straight lines that intersect the knot in four points. Perhaps counting intersections of knots with other families of algebraic varieties might give rise to topological invariants? Progress on this question will be described.

EDDY CAMPBELL, Queen's University, Jeffery Hall, Kingston, Ontario  K7L 3N6
Modular invariant theory

Suppose a group G has a representation V over a field F. Then G acts as a group of algebra automorphisms on the coordinate ring F[V]. The algebra of functions left fixed by G, the invariant ring, is denoted F[V]G. Invariant theory asks for the strucure of or generators for F[V]G. The most obvious connection to algebraic topology is that H0(G,F[V]) = F[V]G.

Modular invariant theory is the study of the case in which the order |G| of the finite group G is divisible by the characteristic p of the field F. I will survey recent results with an emphasis on p-groups.

RALPH COHEN, Stanford University, Stanford, California
Graphs, loop spaces, and Morse theory

In this lecture I will describe certain ways in which spaces of graphs can be used to parameterize (co)homology operations. I will first discuss Morse theory on a compact manifold, in which a moduli space of "graph flows" can be used to describe classical operations such as cup products, Steenrod squares, and Stiefel-Whitney classes. We then apply these ideas to loop spaces of manifolds, and show how one can describe "string topology" operations Morse theoretically, using ribbon graphs and the corresponding moduli spaces of graph flows.

OCTAVIAN CORNEA, University of Montreal
The Serre spectral sequence and Lagrangian intersections

One of the central problems in symplectic topology is finding invariants associated to generic pairs (L0,L1) of Lagrangian submanifolds of some fixed symplectic manifold. The most famous and now classical such example is Floer homology. In this talk which presents work in collaboration with J.-F.Barraud from Lille I will discuss a more refined such invariant. It consists of a spectral sequence which coincides with the Serre spectral sequence of the path-loop fibration of base L0 when L1 is isotopic to L0 by a hamiltonian isotopy. This opens the way to defining interesting symplectic invariants whose homotopical content is richer than just homology.

DIARMUID CROWLEY, Department of Mathematics, Penn State University, University Park, State College, Pennsylvania  16802, USA
Classifying bordisms

Let M be an n-dimensional manifold, n > 4. We classify certain minimal bordisms based on M using quadratic forms. As a corollary, we deduce new classifications for certain classes manifolds and their mapping class groups.

PO HU, Wayne State University
Some aspects of conformal field theory

I will talk about some algebraic structures which arise in defining closed and open conformal field theory; in genus 0, one encounters operad algebras and modules (in a generalized sense). In higher genus, more complicated structures appear. I will define these structures and give some applications.

IGOR KRIZ, University of Michigan, Michigan, USA
D-brane cohomology and elliptic cohomology

I will discuss a mathematically rigorous formalism for axiomatizing both closed and open conformal field theories. Related topics include a candidate for a "higher mode algebra" whose category classifies stable D-branes, and a 2-vector space approach to modular functors which makes a connection between an additive theory for elliptic cohomology, conformal field theories, and Rognes' K-theory of K-theory.

GAUNCE LEWIS, Syracuse University, Syracuse, New York, USA
Localizing Mackey functors at Mackey functor prime ideals of the Burnside ring

Localizing a Mackey functor M at Mackey functor prime ideals (rather than ordinary prime ideals) of the Burnside ring allows one to isolate more cleanly the contributions of individual subgroups of the ambient group G to M. This talk is devoted to a discussion of this localization process and to the applications of this process to equivariant stable homotopy theory. The techniques used for this localization process are borrowed from the theory of noncommutative rings.

PETER MAY, University of Chicago
Parametrized equivariant homotopy theory

Modern model theoretic foundations for parametrized homotopy theory, whether equivariant or not, will be explained. The main theme is how to obtain derived functors on homotopy categories with good algebraic properties when the usual methods of Quillen adjunction fail hopelessly.

ANDREW NICAS, McMaster University, Hamilton, Ontario  L8S 4K1
Trace and duality in symmetric monoidal categories

Traces taking values in suitable "Hochschild complexes" are defined in a general context and applied to various categories of chain complexes, simplicial abelian groups, and symmetric spectra. Topological applications to parametrized fixed point theory are given.

IGOR NIKOLAEV, University of Calgary, Calgary, Alberta
Noncommutative geometry of 3-manifolds

Let M be surface bundle over the circle with a pseudo-Anosov monodromy f. Thurston showed that M is a hyperbolic 3-manifold. One can associate to M an ordered abelian group E (a.k.a "dimension group") coming from a f-invariant geodesic lamination. We prove that E is dimension group of a stationary type. Intrinsically, group E is described by its endomorphism ring (Handelman). We calculate EndE @ OK, where OK is the ring of integers of a quadratic number field K=Q(Öd).

Theorem 1. Let M be hyperbolic manifold, which has minimal volume in its commensurability class. Then VolM=2loge/Öd, where eis the fundamental unit of K.

Theorem 2. Let hK be the class number of field K. Then Gromov-Thurston map M® VolM has degree hK in the point M. Both theorems are proved by the methods of noncommutative geometry [1].

## References

[1]
I. Nikolaev, K-theory of hyperbolic 3-manifolds. math.GT/0110227.

DUANE RANDALL, New Orleans
On tangent 4-fields with finite singularities

This project represents joint collaboration with Professors Maria Herminia de Mello, Nancy Cardim and Mario Olivero da Silva. Let u be any tangent 4-field with finite singularities on a closed connected smooth manifold M of dimension n > 8. Whenever the index of u is independent of u, we determine indu in terms of generators given by James and Nomura for the homotopy of the Stiefel manifold of orthonormal 4-frames in Euclidean n-space. Applications are given to the total spaces of differentiable fiber bundles of closed manifolds.

DALE ROLFSEN, University of British Columbia, Vancouver, British Columbia
Virtually orderable 3-manifold groups

A group is orderable if there exists a linear ordering of its elements which is invariant under multiplication on both sides. It has been conjectured that the fundamental group of any closed 3-manifold has a finite-index subgroup which is orderable. I will discuss the connection between this and other famous conjectures in 3-manifold theory. Moreover, I will show that in any closed orientable 3-manifold one can find a link whose complement has orderable fundamental group.

YULI RUDYAK, University of Florida, Gainesville, Florida  32611, USA
Toward the general theory of affine linking numbers

(joint with Vladimir Chernov)

Let N1, N2, M be smooth manifolds such that dimN1 +dimN2+1 = dimM and let fi, i=1,2, be smooth mappings of Ni to M such that Áf1ÇÁf2=Æ. The classical linking number lk(f1,f2) is defined only when f1*[N1]=f2*[N2]=0 Î H* (M).

Affine linking number a is the generalization of the classical invariant to the case of nonzero-homologous f1* [N1],f2*[N2]. Recently we have constructed the first examples of a-invariants of nonzero-homologous spheres in the spherical tangent bundle of a manifold and showed that alk is intimately related to the causality relation of wave fronts on manifolds.

In this paper we develop the general theory of a-invariants in the case of nonzero-homologous f1*[N1] and f2*[N2]. We show that alk is a universal Goussarov-Vassiliev invariant of order £ 1. In case of f1* [N1]=f2*[N2]=0 Î H*(M) the alk-invariant appears to be a splitting of the classical linking number into a collection of independent invariants.

To construct alk we introduce a new pairing on the bordism groups of space of mappings of N1 and N2 into M. For the case N1=N2=S1 this pairing can be regarded as an analog of the string-homology pairing constructed by Chas and Sullivan, and it is a generalization of the Goldman Lie bracket.

Morava K-theory and inverse limits

We discuss a spectral sequence converging to the Morava K-theory of the homotopy inverse limit of a tower (of Morava K-theory local) spectra.

The E2 term of this spectral sequence is, in a certain sense, the local cohomology of the Morava stabilizer group with certain coefficients derived from applying Morava K-theory to the spectra in the tower.

We then give examples where this spectral sequence can be used to calculate Morava K-theory of a homotopy inverse limit. One such example is an infinite product of spectra which have a global bound on the size of their Morava K-theory.

PARAMESWARAN SANKARAN, Institute of Mathematcal Sciences, CIT Campus, Taramani, Chennai  600113, India
A coincidence theorem for holomorphic maps to G/P

Let G denote a complex semi simple algebraic group and P Ì G a maximal parabolic subgroup so that G/P is a homogeneous variety. Let f,g: M® G/P be any two holomorphic maps where M is a compact connected complex manifold. Assume that at least one of f, g is surjective. Then we show that f and g have a coincidence: f(x)=g(x) for some x Î M. We shall show that if M is a generalized Hopf manifold over G/P with at least one of f, g non-constant then f and g have a coincidence.

DON STANLEY, Ottawa
The rational homotopy type of a blowup

Suppose f: V® W is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well know in both complex and symplectic geometry that one can then construct a manifold [(W)\tilde] which is the blowup of W along V. Assume that dimW ³ 2dimV +3 and that H1(f) is injective. We construct an algebraic model of the rational homotopy type of the blow-up [(W)\tilde] from an algebraic model of the embedding and the Chern classes of the normal bundle. This implies that if the space W is simply connected then the rational homotopy type of [(W)\tilde] depends only on the rational homotopy class of f and on the Chern classes of the normal bundle. This model can be used to prove that certain Symplectic manifolds have no Kähler structure.