Intersections of subcomplexes in non-positively curved 2-dimensional complexes

Abstract

Let $X$ be a contractible $2$-complex which is a union of two contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\cap Z$ contractible as well? In this note, we prove that the inclusion-induced map $\pi _{1}(Y\cap Z)\rightarrow \pi _{1}(Z)$ is injective if $Y$ is $\pi _{1}$-injective subcomplex in a locally CAT(0) 2-complex $X$. In particular, each component in the intersection of two contractible subcomplexes in a CAT(0) 2-complex is contractible.