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

TL;DR

This paper investigates the topological properties of intersections of subcomplexes in non-positively curved 2-complexes, proving injectivity of fundamental group maps and contractibility of intersection components.

Contribution

It establishes conditions under which the intersection of contractible subcomplexes in CAT(0) 2-complexes is also contractible, extending understanding of their topological structure.

Findings

The inclusion-induced map on fundamental groups is injective under certain conditions.

Components of intersections of contractible subcomplexes are contractible.

Provides new insights into the topology of non-positively curved 2-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.