Algebraic and Geometric Topology / Topologie algébrique et
(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
- 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
- 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
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
- 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
- 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
- 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 .
- I. Nikolaev, K-theory of hyperbolic 3-manifolds.
- 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
- 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
- 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.
- HAL SADOFSKY, Oregon
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)
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
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.