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Manifolds and Disc-presheaves

Alexander Kupers

Abstract

This essay explains an approach to the study of smooth manifolds which compares them to presheaves on a category of discs, also known as embedding calculus. We highlight recent work that shows this approach has many desirable properties, as well as recent applications demonstrating its strength.

Manifolds and Disc-presheaves

Abstract

This essay explains an approach to the study of smooth manifolds which compares them to presheaves on a category of discs, also known as embedding calculus. We highlight recent work that shows this approach has many desirable properties, as well as recent applications demonstrating its strength.

Paper Structure

This paper contains 48 sections, 10 theorems, 46 equations, 1 figure.

Table of Contents

  1. Disc-presheaves
  2. A pointillist view of manifolds
  3. Embeddings of points
  4. Embedding of open discs
  5. Embeddings of open discs and maps between them
  6. Disc-presheaves
  7. Moduli spaces and structure spaces
  8. Desideratum \ref{['enum:des-prop']}: properties and structure of Disc-presheaves
  9. A field-theoretic structure
  10. Disjoint union and gluing by algebras and modules
  11. From bordisms to bimodules
  12. Tangential structures and smoothing theory
  13. Operadic foundations
  14. Disc-presheaves as right $E_d^\mathrm{o}$-modules
  15. The "particle" tangential structure
  16. ...and 33 more sections

Key Result

Theorem 4.1

Suppose that $M$ and $N$ are compact $d$-dimensional smooth manifolds, possibly with boundary. Then the comparison map $\rm{comp}$ is an equivalence if

Figures (1)

  • Figure 1: An element of $\mathrm{Emb}^\mathrm{o}(\ul{2} \times \mathbf{R}^d,M)$. The map to the framed ordered configuration space records the centres of the two open discs on the right, as well as the induced frame in the tangent spaces at those points.

Theorems & Definitions (24)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.7
  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Example 3.1
  • Remark 3.3
  • ...and 14 more