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Embedding, compression, and the relative Hopf invariant

John R. Klein

Abstract

We establish Poincaré embedding results in the relative setting, generalizing previously known results in the absolute case. Our primary motivation comes from applications to non-simply connected Poincaré surgery, which will be developed in a forthcoming paper. Along the way, we introduce a new tool: the relative Hopf invariant in the equivariant setting.

Embedding, compression, and the relative Hopf invariant

Abstract

We establish Poincaré embedding results in the relative setting, generalizing previously known results in the absolute case. Our primary motivation comes from applications to non-simply connected Poincaré surgery, which will be developed in a forthcoming paper. Along the way, we introduce a new tool: the relative Hopf invariant in the equivariant setting.
Paper Structure (20 sections, 12 theorems, 116 equations, 3 figures)

This paper contains 20 sections, 12 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relative Poincaré embeddings
  3. The space of Poincaré embeddings
  4. Some fiberwise notions
  5. The Compression Theorem
  6. Poincaré immersions and the fiberwise collapse
  7. Applications
  8. The relative Hopf invariant
  9. Preliminaries
  10. Model categories of spaces
  11. Poincaré duality spaces
  12. The Hopf invariant
  13. Properties
  14. The coaction
  15. The relative Hopf invariant
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that homotopy spine dimension of $P$ is $\le p$, with $p \le d-3$ and $3p+4\le 2d$. If $\nu$ is trivial, then up to concordance, the relative embedding $P\times J \to M \times I$ compresses to a relative embedding $P\to M$.

Figures (3)

  • Figure 1: A depiction of a Poincaré embedding of $P$ in $M$ relative to $\partial_0 P$, where $\partial_1 M$ is the the unlabeled part of the boundary.
  • Figure 2: Depiction of a relative Poincaré embedding of $P\times J$ in $M \times I$. The ruled portion of $P\times J$ represents the embedding of $\partial_0 P\times J$ in $\partial (M\times I)$.
  • Figure 3: A smooth manifold immersion that schematically depicts the Poincaré immersion $f\cup \phi$ appearing in the proof of Theorem \ref{['bigthm:handle']}. The bottom rectangle is an immersion $\phi\colon \partial_0 H {\times} I \to \partial M {\times} I$ which restricts to an embedding $\partial_0 H {\times} \partial I \to \partial M {\times} \partial I$. The top rectangle is an immersion $f\colon H \to M$. The immersions coincide at the embedding $\phi_0\colon \partial_0 H {\times} \{0\} \to \partial M {\times} \{0\}$. In the figure, $\partial M \times \{0\}$ is the edge where the rectangles meet and $\partial M \times \{1\}$ is the bottom edge of the lower rectangle.

Theorems & Definitions (47)

  • Definition 1.1: GK
  • Remark 1.2
  • Definition 1.3: GK
  • Remark 1.5
  • Remark 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1: Compression Theorem
  • Remark 1.10
  • ...and 37 more