Topology / Topologie
(Org: Keith Johnson, Dalhousie University and/et Renzo Piccinini, University of Milan)
- KRISTINE BAUER, University of Calgary, Calgary, AB T2N 1N4
The identity functor and rational homotopy theory
Goodwillie's calculus of functors is a way of making homotopy functors
more tractable by providing approximations to these functors. A good
analogy is the Taylor series approximation of a function.
Goodwillie's approximations play the role of the closest finite degree
polynomial functor to the given functor.
Instead of approximating very complicated functors by simpler ones,
the calculus can be used in the opposite way to associate rich
information to seemingly simple functors. For example, the Goowillie
tower of the identity functor from spaces to spaces has incredibly
complex structure. The homogeneous degree n approximations, computed
by Johnson and further studied by Arone and Mahowald, provide a
filtration between unstable and stable homotopy theory.
We explore certain operad actions appearing in the Goodwillie tower of
the identity functor. We relate these to the operad Lien, and
decipher the resulting algebraic structure on rational homotopy
This is joint work with Brenda Johnson and Jack Morava.
- PETER BOOTH, Memorial University, St John's, Newfoundland, Canada
On the Classification of Fibrations whose Fibres are Products
of Eilenberg-MacLane Spaces
Suitably defined Mapping Spaces will be shown to act as Classifying
Spaces for these fibrations. Computational results will then be
derived from the aforementioned theorem.
- CARLOS BROTO, Universitat Autònoma de Barcelona
The theory of p-local finite groups: a link between
homotopy theory and group theory
The theory of p-local finite groups, recently introduced in joint
work with Ran Levi and Bob Oliver has its origins in the p-local
properties of finite groups and their classifying spaces. A p-local
finite group consists of a finite p-group S together with two
categories, F and L, of which the first one encodes "conjugacy"
relations among the subgroups of S and the second one contains just
enough information in order to associate a classifying space. This is
a p-complete space that shares many of the same homotopy theoretic
properties of p-completed classifying spaces of finite groups. We
will give an overview of the theory.
- DAVIDE FERRARIO, Università di Milano-Bicocca
A cellular (co)homology theory of fibred spaces
It is possible at one time to generalize the notion of fiber bundle,
stratified space (Thom-Mather) and cellular G-space by introducing
the idea of fibred space with fibres controlled by a suitable
structure category F. This unified approach allows to define a
generalized homology and cohomology theory with local coefficients, a
natural notion of homotopy and a topological Atiyah-Hirzebruch
K-theory. Classical theorems for CW-complexes hold in this more
general setting, like Blakers-Massey theorem, Whitehead theorems,
obstruction theory, Hurewicz homomorphism, Wall finiteness theorem,
Whitehead torsion, principal bundle theorem and pull-back theorem.
- PHIL HEATH, Memorial University
Groupoids and Nielsen product/sum theorems
In the interaction between Nielsen theory and Fibre spaces, there are
a number of product formulas (addition) formulas. For example let f
be a fibre preserving self map of a fibration Fb ® E ® B, where
b is a fixed point of the induced map [`(f)] on the base B,
and Fb is the fibre over b. Then under orientability and
commutativity conditions the formula [Fix[`(f)]x*; p* (Fix
fp(x)*)] N(f) = NK (fb) ·N([`(f)]) holds.
Here for a self map g : X ® X, N(g) denotes the Nielsen
number of g, and if x is a fixed point, the symbol Fixgx*
denotes the subgroup of p1 (X,x) consisting of elements a
with f* (a) = a, finally NK (fb) denotes the mod K
Nielsen number of the restriction fb of f to the fibre with K
the kernel of the induced map p1 (Fb) ® p1 (E).
An analogous formula [ Coin([`(f)]b*, [`(g)]b*); p*( Coin(f*x, gxb*) ) ] N(f,g) = NK (fb, gb)·N([`(f)], [`(g)]), holds in the context of coincidences. In
this talk we indicate how these, and other formulas, follow from the
theory of fibrations of groupoids.
- KEITH JOHNSON, Dalhousie
- IGOR NIKOLAEV, University of Calgary
On number fields associated to hyperbolic 3-manifolds
We consider 3-dimensional manifolds M, which are fibre bundles over
the circle with surface fibre S and pseudo-Anosov monodromy f.
The action of j fixes a pair of foliations on surface S.
There exists a natural notion of "slope" of foliation, and such a
slope is always an algebraic number q Î Q(Öd). The aim
of our talk is to show that the number field K = Q(Öd) absorbs
critical data on geometry, topology and combinatorics of the
References: K-theory of hyperbolic 3-manifolds, math.GT/0110227.
- DOUG RAVENEL, Department of Mathematics, University of Rochester,
Rochester, NY 14627
Using Abelian varieties to construct cohomology theories
In elliptic cohomology one uses the 1-dimensional formal group law
associated with a family of elliptic curves to construct a cohomology
theory. This FGL can have height at most 2. It would be desirable to
have naturally occuring 1-dimensioanl FGLs of larger heights.
Associated to a curve of genus g is an Abelian variety with a
g-dimensional FGL. We will describe a family of curves for which
this FGL has a 1-dimensional summand.
- LAURA SCULL, University of British Columbia
Formal spaces are those whose rational homotopy type is completely
determined by their cohomology; this has proved a very useful concept.
I will discuss adapting this idea to the equivariant case, and compare
several alternate definitions of equivariant formality.
- AFEWORK SOLOMAN, Memorial
- STEPHEN THERIAULT, University of Aberdeen, Aberdeen, United Kingdom
The H-structure of low rank torsion free H-spaces
Start with a fixed prime p and a space X of t odd dimensional
cells, where t < p-1. After localizing at p, Cooke, Harper, and
Zabrodsky constructed a finite H-space Y with the property that
the mod-p homology of Y is generated as an exterior Hopf algebra
by the reduced mod-p homology of X. Cohen and Neisendorfer, and
later Selick and Wu, reproduced this result with different
constructions. We use the latter approaches to show that Y is
homotopy associative and homotopy commutative if X is a suspension
and t < p-2. Interesting examples include low rank mod-p Stiefel
- PETER ZVENGROWSKI, The University of Calgary, Dept. of Mathematics and
Statistics, Calgary, Alberta T2N 1N4
The Order of Real Line Bundles
It is shown that for any real line bundle x over an arbitrary
topological space X such that nx admits r ³ 1 independent
sections, there is a power of 2 that is a natural upper bound on the
order of [x], as an element of the real K-theory KO(X). The
relation to calculations of the (complex) K-theory of the projective
Stiefel manifolds by various authors will be explained, and
applications to classifying spaces, the Alexandrov line, Stiefel
manifolds, and projective Stiefel manifolds will be sketched.