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 Lie_{n}, and decipher the resulting algebraic structure on rational homotopy theory.
This is joint work with Brenda Johnson and Jack Morava.
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.
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.
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.
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 F_{b} ® E ® B, where b is a fixed point of the induced map [`(f)] on the base B, and F_{b} is the fibre over b. Then under orientability and commutativity conditions the formula [Fix[`(f)]^{x}_{*}; p_{*} (Fix f^{p(x)}_{*})] N(f) = N_{K} (f_{b}) ·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 Fixg^{x}_{*} denotes the subgroup of p_{1} (X,x) consisting of elements a with f_{*} (a) = a, finally N_{K} (f_{b}) denotes the mod K Nielsen number of the restriction f_{b} of f to the fibre with K the kernel of the induced map p_{1} (F_{b}) ® p_{1} (E).
An analogous formula [ Coin([`(f)]^{b}_{*}, [`(g)]^{b}_{*}); p_{*}( Coin(f_{*}^{x}, g^{x}_{b*}) ) ] N(f,g) = N_{K} (f_{b}, g_{b})·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.
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 manifold M.
References: K-theory of hyperbolic 3-manifolds, math.GT/0110227.
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.
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.
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 manifolds.
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.