The second homotopy group in terms of colorings of locally finite models   and new results on asphericity
  We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-connected complexes, and derive new results on asphericity for two-dimensional complexes and group presentations. 
Jonathan Ariel Barmak and Elias Gabriel Minian
math.AT math.GT