Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Embedding calculus and smooth structures

Ben Knudsen, Alexander Kupers

TL;DR

It is proved that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro, and it is shown that embeding calculus does distinguish certain exotic spheres in higher dimensions.

Abstract

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro. In contrast, we show that embedding calculus does distinguish certain exotic spheres in higher dimensions. As a technical tool of independent interest, we prove an isotopy extension theorem for the limit of the embedding calculus tower, which we use to investigate several further examples.

Embedding calculus and smooth structures

TL;DR

It is proved that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro, and it is shown that embeding calculus does distinguish certain exotic spheres in higher dimensions.

Abstract

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro. In contrast, we show that embedding calculus does distinguish certain exotic spheres in higher dimensions. As a technical tool of independent interest, we prove an isotopy extension theorem for the limit of the embedding calculus tower, which we use to investigate several further examples.

Paper Structure

This paper contains 32 sections, 40 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Embedding calculus
  4. Calculus for manifolds with boundary
  5. A simplicial category of topological manifolds
  6. Microbundles
  7. Topological tangency
  8. Comparing tangent bundles
  9. Formally smooth manifolds
  10. Simplicial categories of Riemannian and formally smooth manifolds
  11. Smooth embeddings of Euclidean spaces
  12. Topological embeddings of Euclidean spaces
  13. Formally smooth embeddings of Euclidean spaces
  14. Consequences for embedding calculus
  15. Embedding calculus in dimension 4
  16. ...and 17 more sections

Key Result

Theorem 1

Let $M$ and $M'$ be smooth, simply connected, compact $4$-manifolds. If $M$ and $M'$ are homeomorphic, then for any $k \geq 0$ we have

Figures (1)

  • Figure 1: Several subspaces of $D^3$ which appear in Section \ref{['Sec:codim 2 convergence']}.

Theorems & Definitions (105)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Definition 2.2: Boavida--Weiss
  • Theorem 2.3: Goodwillie--Klein--Weiss
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 95 more