Relative Non-Positive Immersion

A 2-complex $K$ has collapsing non-positive immersion if for every combinatorial immersion $X\to K$, where $X$ is finite, connected and does not allow collapses, either $\chi(X)\le 0$ or $X$ is point. This concept is due to Wise who also showed that this property implies local indicability of the fundamental group $\pi_1(K)$. In this paper we study a relative version of collapsing non-positive immersion that can be applied to 2-complex pairs $(L,K)$: The pair has relative collapsing non-positive immersion if for every combinatorial immersion $f\colon X\to L$, where $X$ is finite, connected and does not allow collapses, either $\chi(X)\le \chi(Y)$, where $Y$ is the essential part of the preimage $f^{-1}(K)$, or $X$ is a point. We show that under certain conditions a transitivity law holds: If $(L,K)$ has relative collapsing non-positive immersion and $K$ has collapsing non-positive immersion, then $L$ has collapsing non-positive immersion. This article is partly motivated by the following open question: Do reduced injective labeled oriented trees have collapsing non-positive immersion? We answer this question in the affirmative for certain important special cases.