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We describe the second homotopy group of any CW-complex $K$ by analyzing theuniversal cover of a locally finite model of $K$ using the notion of$G$-coloring of a partially ordered set. As applications we prove ageneralization of the Hurewicz theorem, which relates the homotopy and homologyof non-necessarily simply-connected complexes, and derive new results onasphericity for two-dimensional complexes and group presentations.

The second homotopy group in terms of colorings of locally finite models and new results on asphericity