• Corpus ID: 56100668

Introduction to the manifold calculus of Goodwillie-Weiss ⁄

@article{Munson2010IntroductionTT,
  title={Introduction to the manifold calculus of Goodwillie-Weiss ⁄},
  author={Brian A. Munson},
  journal={arXiv: Algebraic Topology},
  year={2010}
}
  • B. Munson
  • Published 11 May 2010
  • Mathematics
  • arXiv: Algebraic Topology
We present an introduction to the manifold calculus of functors, due to Goodwillie and Weiss. Our perspective focuses on the role the derivatives of a functor F play in this theory, and the analogies with ordinary calculus. We survey the construction of polynomial functors, the classification of homogeneous functors, and results regarding convergence of the Taylor tower. We sprinkle examples throughout, and pay special attention to spaces of smooth embeddings. 
View PDF on arXiv
11 Citations
Background Citations
2

Figures from this paper

11 Citations

Citation Type

Citation Type

Low stages of the Taylor tower for r-immersions
We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of r-immersions, which are immersions without r-fold self-intersections. We describe the first r…
Convergence of the Taylor tower for embeddings in $\mathbb R^n$
Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the…
Toward Finite Models for the Stages of the Taylor Tower for Embeddings of the 2-Sphere
  • Adisa Bolić, Franjo Sarcevic, I. Volić
  • Mathematics
    Advanced Technologies, Systems, and Applications IV -Proceedings of the International Symposium on Innovative and Interdisciplinary Applications of Advanced Technologies (IAT 2019)
  • 2019
We provide the beginning of the construction of a finite model for the stages of the Taylor tower for embeddings of the 2-sphere in a smooth manifold. We show how these stages can be described as…
A streamlined proof of the convergence of the Taylor tower for embeddings in ${\mathbb R}^n$
Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the…
Configuration space integrals and the topology of knot and link spaces
This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type…
A Homotopy-Theoretic Approach to the Topological Tverberg Conjecture
Let r ≥ 2 and d ≥ 1 be integers, let N = (d + 1)(r − 1), and let ∆ denote a standard N -simplex. The Topological Tverberg Conjecture states that any continuous map f : ∆ → R has r-fold…
Little disks operads and configuration spaces
Speaker’s abstract. Operads are objects that govern categories of algebras. Initially introduced in the sixties to study iterated loop spaces, they have proved useful in several areas of mathematics.…
Polynomial functors in manifold calculus
Cubical Homotopy Theory
Preface Part I. Cubical Diagrams: 1. Preliminaries 2. 1-cubes: homotopy fibers and cofibers 3. 2-cubes: homotopy pullbacks and pushouts 4. 2-cubes: the Blakers-Massey Theorems 5. n-cubes: generalized…
An application of the h-principle to manifold calculus
  • Apurv Nakade
  • Mathematics
    Journal of Homotopy and Related Structures
  • 2020
Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this…
...
1
2
...

References

SHOWING 1-10 OF 27 REFERENCES
Finite type knot invariants and the calculus of functors
  • I. Volić
  • Mathematics
    Compositio Mathematica
  • 2006
We associate a Taylor tower supplied by the calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along…
A Haefliger style description of the embedding calculus tower
Configuration space integrals and Taylor towers for spaces of knots
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…
Spaces of smooth embeddings, disjunction and surgery
We describe progress in the theory of smooth embeddings over more than 50 years, starting with Whitney’s embedding theorem, continuing with the generalized Whitney tricks of Haefliger and Dax, early…
Embeddings in the 3/4 range
A manifold calculus approach to link maps and the linking number
We study the space of link maps Link(P1,...Pk;N), which is the space of maps P1 ` � � � ` Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors…
Coformality and rational homotopy groups of spaces of long knots
We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in R-d for d >= 4 collapses at the E 2 page. The main ingredients in the proof…
A one-dimensional embedding complex
A stable range description of the space of link maps
We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions,…
...
1
2
3
...