The first relative k-invariant
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914106385104896 |
|---|---|
| author | Conway, Anthony Kasprowski, Daniel |
| author_facet | Conway, Anthony Kasprowski, Daniel |
| contents | Motivated by work on the homotopy classification of $4$-manifolds with boundary, we define a relative $k$-invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair $(X,Y)$ with Postnikov $2$-type $X \to P_2(X)$, the relative $k$-invariant is the obstruction to the existence of a section $Bπ_1(X)\to P_2(X)$ extending $Y \hookrightarrow X \to P_2(X)$. Given CW pairs $(X_0,Y_0)$ and $(X_1,Y_1)$, as well as a map $h \colon Y_0 \to Y_1$, we also prove that relative $k$-invariants provide a complete obstruction to constructing a map $X_0^{(3)} \cup Y_0 \to X_1$ that extends $h$ and induces given isomorphisms on $π_1$ and $π_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18796 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The first relative k-invariant Conway, Anthony Kasprowski, Daniel Geometric Topology Motivated by work on the homotopy classification of $4$-manifolds with boundary, we define a relative $k$-invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair $(X,Y)$ with Postnikov $2$-type $X \to P_2(X)$, the relative $k$-invariant is the obstruction to the existence of a section $Bπ_1(X)\to P_2(X)$ extending $Y \hookrightarrow X \to P_2(X)$. Given CW pairs $(X_0,Y_0)$ and $(X_1,Y_1)$, as well as a map $h \colon Y_0 \to Y_1$, we also prove that relative $k$-invariants provide a complete obstruction to constructing a map $X_0^{(3)} \cup Y_0 \to X_1$ that extends $h$ and induces given isomorphisms on $π_1$ and $π_2$. |
| title | The first relative k-invariant |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2510.18796 |