E-theory for C[0,1]-algebras with finitely many singular points

We study the E-theory group $E_{[0,1]}(A,B)$ for a class of C*-algebras over the unit interval with finitely many singular points, called elementary $C[0,1]$-algebras. We use results on E-theory over non-Hausdorff spaces to describe $E_{[0,1]}(A,B)$ where $A$ is a sky-scraper algebra. Then we compute $E_{[0,1]}(A,B)$ for two elementary $C[0,1]$-algebras in the case where the fibers $A(x)$ and $B(y)$ of $A$ and $B$ are such that $E^1(A(x),B(y)) = 0$ for all $x,y\in [0,1]$. This result applies whenever the fibers satisfy the UCT, their $K_0$-groups are torsion-free and their $K_1$-groups are zero. In that case we show that $E_{[0,1]}(A,B)$ is isomorphic to $\text{Hom}(\mathbb{K}_0(A), \mathbb{K}_0(B))$, the group of morphisms of the K-theory sheaves of $A$ and $B$. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.