The first relative k-invariant

TL;DR

This paper introduces a relative k-invariant for pairs of spaces, serving as an obstruction measure for extending sections and maps in homotopy theory, with applications to 4-manifold classification.

Contribution

It defines the relative k-invariant for CW pairs and demonstrates its role as a complete obstruction for extending maps and sections in homotopy theory.

Findings

Relative k-invariant characterizes obstructions to sections.

Provides a complete obstruction criterion for extending maps.

Applicable to homotopy classification of 4-manifolds.

Abstract

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 $B\pi_1(X)\to P_2(X)$ 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 $\pi_1$ and $\pi_2$.