Families of Arcs in 4-Manifolds and Maps of Configuration Spaces

TL;DR

This thesis constructs and analyzes 3-parameter families of embedded arcs in 4-manifolds, using embedding calculus and diagrammatic frameworks to study their homotopy properties and rational homotopy groups.

Contribution

It introduces a new diagrammatic approach to study families of arcs and demonstrates their triviality in certain Taylor approximations, extending previous work to higher homotopy groups.

Findings

G(p,q,r) is trivial in  T_3mbedding space

The rational homotopy group ^{}  of embeddings in S^1 imes B^3 is 

Extends results from  by Budney and Gabai to higher homotopy groups.

Abstract

In this thesis we construct 3-parameter families $G(p,q,r)$ of embedded arcs with fixed boundary in a 4-manifold. We then analyze these elements of $\pi_3\mathsf{Emb}_\partial(I,M)$ using embedding calculus by studying the induced map from the embedding space to ``Taylor approximations" $T_k\mathsf{Emb}_\partial(I,M)$. We develop a diagrammatic framework inspired by cubical $\omega$-groupoids to depict $G(p,q,r)$ and related homotopies. We use this framework extensively in Chapter 4 to show explicitly that $G(p,q,r)$ is trivial in $\pi_3T_3\mathsf{Emb}_\partial(I,M)$ (however, we conjecture that it is non-trivial in $\pi_3T_4\mathsf{Emb}_\partial(I,M)$). In Chapter 5 we use the Bousfield-Kan spectral sequence for homotopy groups of cosimplicial spaces to show that the rational homotopy group $\pi^{\mathbb{Q}}_3\mathsf{Emb}_\partial(I,S^1 \times B^3)$ is $\mathbb{Q}$. This thesis extends work by Budney and Gabai which proves analogous results for $\pi_2\mathsf{Emb}_\partial(I,M)$.