82 research outputs found
Calculus of the embedding functor and spaces of knots
We give an overview of how calculus of the embedding functor can be used for
the study of long knots and summarize various results connecting the calculus
approach to the rational homotopy type of spaces of long knots, collapse of the
Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite
type knot invariants, etc. Some open questions and conjectures of interest are
given throughout.Comment: A survey prepared for the AIM Workshop On Moduli Spaces of Knot
Interpolation categories for homology theories
For a homological functor from a triangulated category to an abelian category
satisfying some technical assumptions we construct a tower of interpolation
categories. These are categories over which the functor factorizes and which
capture more and more information according to the injective dimension of the
images of the functor. The categories are obtained by proving the existence of
truncated versions of resolution or -model structures. Examples of
functors fitting in our framework are given by every generalized homology
theory represented by a ring spectrum satisfying the Adams-Atiyah condition.
The constructions are closely related to the modified Adams spectral sequence
and give a very conceptual approach to the associated moduli problem and
obstruction theory. As application we establish an isomorphism between certain
E(n)-local Picard groups and some Ext-groups.Comment: 40 pages, corrected version of second part of the replaced version,
first part will appear sepparately as "Truncated resolution model
structures", to appear in JPA
Towers of MU-algebras and the generalized Hopkins-Miller theorem
Our results are of three types. First we describe a general procedure of
adjoining polynomial variables to -ring spectra whose coefficient
rings satisfy certain restrictions.A host of examples of such spectra is
provided by killing a regular ideal in the coefficient ring of MU, the complex
cobordism spectrum. Second, we show that the algebraic procedure of adjoining
roots of unity carries over in the topological context for such spectra. Third,
we use the developed technology to compute the homotopy types of spaces of
strictly multiplicative maps between suitable K(n)-localizations of such
spectra. This generalizes the famous Hopkins-Miller theorem and gives
strengthened versions of various splitting theorems
Completed representation ring spectra of nilpotent groups
In this paper, we examine the `derived completion' of the representation ring
of a pro-p group G_p^ with respect to an augmentation ideal. This completion is
no longer a ring: it is a spectrum with the structure of a module spectrum over
the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In
order to explain the origin of some of these higher homotopy classes, we define
a deformation representation ring functor R[-] from groups to ring spectra, and
show that the map R[G_p^] --> R[G] becomes an equivalence after completion when
G is finitely generated nilpotent. As an application, we compute the derived
completion of the representation ring of the simplest nontrivial case, the
p-adic Heisenberg group.Comment: This is the version published by Algebraic & Geometric Topology on 26
February 200
Cosimplicial models for spaces of links
We study the spaces of string links and homotopy string links in an arbitrary manifold using multivariable manifold calculus of functors. We construct multi-cosimplicial models for both spaces and deduce certain convergence properties of the associated Bousfield-Kan homotopy and cohomology spectral sequences when the ambient manifold is a Euclidean space of dimension four or more
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