Families of Arcs in 4-Manifolds and Maps of Configuration Spaces
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911247978463232 |
|---|---|
| author | Sridhar-Shapiro, Shruthi |
| author_facet | Sridhar-Shapiro, Shruthi |
| contents | 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 $π_3\mathsf{Emb}_\partial(I,M)$ 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 $ω$-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 $π_3T_3\mathsf{Emb}_\partial(I,M)$ (however, we conjecture that it is non-trivial in $π_3T_4\mathsf{Emb}_\partial(I,M)$). In Chapter 5 we use the Bousfield-Kan spectral sequence for homotopy groups of cosimplicial spaces to show that the rational homotopy group $π^{\mathbb{Q}}_3\mathsf{Emb}_\partial(I,S^1 \times B^3)$ is $\mathbb{Q}$. This thesis extends work by Budney and Gabai which proves analogous results for $π_2\mathsf{Emb}_\partial(I,M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Families of Arcs in 4-Manifolds and Maps of Configuration Spaces Sridhar-Shapiro, Shruthi Geometric Topology Algebraic Topology 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 $π_3\mathsf{Emb}_\partial(I,M)$ 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 $ω$-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 $π_3T_3\mathsf{Emb}_\partial(I,M)$ (however, we conjecture that it is non-trivial in $π_3T_4\mathsf{Emb}_\partial(I,M)$). In Chapter 5 we use the Bousfield-Kan spectral sequence for homotopy groups of cosimplicial spaces to show that the rational homotopy group $π^{\mathbb{Q}}_3\mathsf{Emb}_\partial(I,S^1 \times B^3)$ is $\mathbb{Q}$. This thesis extends work by Budney and Gabai which proves analogous results for $π_2\mathsf{Emb}_\partial(I,M)$. |
| title | Families of Arcs in 4-Manifolds and Maps of Configuration Spaces |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2511.02111 |