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On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold

Danica Kosanović

TL;DR

The paper analyzes the fundamental groups of spaces of framed embeddings $\mathrm{Emb}(\nu\mathbb{S}^1,X)$ into oriented 4-manifolds $X$, motivated by parametrised diffeomorphisms of 4-manifolds and the surgery maps that realize them. It develops a loop-space framework by relating framed immersions to loops in the frame bundle via Smale's theory, and then derives explicit exact sequences for $\pi_1$ in terms of Stiefel–Whitney data $w_2^s$ and $w_2^h$, and a Dax-type invariant $\mathsf{dax}^{whisk}_c$ that governs the immersion-to-embedding transition. The main contributions are the precise descriptions and splitting criteria for $\pi_1(\mathrm{Imm}(\nu\mathbb{S}^1,X);\nu c)$ and $\pi_1(\mathrm{Emb}(\nu\mathbb{S}^1,X);\nu c)$, expressed through invariant-driven exact sequences that depend on the spin structure of $X$ and the obstruction data encoded in $w_2^s$, $w_2^h$, and $w_2$. These results connect the geometry of framings and pseudo-isotopies with diffeomorphism theory in dimension four, and they provide a framework that could extend to higher dimensions with Bott-periodic adjustments. Overall, the work clarifies when framing twists and full rotations contribute nontrivially to the fundamental groups of embedding spaces and when they split, tying these algebraic features to classical topological invariants.

Abstract

Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of $S^1\times D^3$ in 4-manifolds. The majority of work goes into the case of framed immersed circles.

On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold

TL;DR

The paper analyzes the fundamental groups of spaces of framed embeddings into oriented 4-manifolds , motivated by parametrised diffeomorphisms of 4-manifolds and the surgery maps that realize them. It develops a loop-space framework by relating framed immersions to loops in the frame bundle via Smale's theory, and then derives explicit exact sequences for in terms of Stiefel–Whitney data and , and a Dax-type invariant that governs the immersion-to-embedding transition. The main contributions are the precise descriptions and splitting criteria for and , expressed through invariant-driven exact sequences that depend on the spin structure of and the obstruction data encoded in , , and . These results connect the geometry of framings and pseudo-isotopies with diffeomorphism theory in dimension four, and they provide a framework that could extend to higher dimensions with Bott-periodic adjustments. Overall, the work clarifies when framing twists and full rotations contribute nontrivially to the fundamental groups of embedding spaces and when they split, tying these algebraic features to classical topological invariants.

Abstract

Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of in 4-manifolds. The majority of work goes into the case of framed immersed circles.
Paper Structure (14 sections, 22 theorems, 30 equations)

This paper contains 14 sections, 22 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The key players
  4. Framed immersions
  5. Framed embeddings
  6. Immersions
  7. Free loop spaces
  8. Nonframed immersions
  9. Framed immersions
  10. Second Stiefel-Whitney classes
  11. The frame bundle
  12. Framed immersions
  13. The connecting maps
  14. Embeddings

Key Result

Theorem 1

For an oriented smooth 4-manifold $X$ and a framed embedded circle $\nu c\colon\nu\mathbb{S}^1=\mathbb{S}^1\times\mathbb{D}^3\hookrightarrow X$, we have the following exact sequences. Moreover, $\mathsf{rot}_{\nu c}$ splits if $w_2=0$ ($X$ spin). More generally, this splits whenever the extension $\mathbb{Z}/2\to\mathsf{Fix}_{\nu c}(\pi_1\mathsf{Fr}X)\twoheadrightarrow\mathsf{Fix}_c(\pi_1X)$ does

Theorems & Definitions (34)

  • Theorem 1: Corollary \ref{['cor:framed-imm']}
  • Theorem 2
  • Corollary 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 24 more