The first relative k-invariant

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 $h\text{\hspace{0.17em}⁣}:Y_0\to Y_1$, 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 ${\pi }_2$.