Gabriel Minian: New methods for studying asphericity of 2-complexes and group presentations

A path-connected space is called aspherical if its homotopy groups are vanish in dimensions greater than or equal to 2. For 2-complexes (i.e. CW-complexes of dimension 2) this is equivalent to requiring  that the second homotopy group of the space vanishes. In this talk I will
discuss new methods for studying asphericity of compact 2-complexes and group presentations. These methods were recently introduced in a joint paper with J. Barmak, and they are based on a description of the second homotopy group of a regular compact CW-complex in terms of group-colorings of the poset of its cells.

In the first part of my talk I will recall some classical problems and results on asphericity (including Whitehead asphericity conjecture) and the relationship between finite presentations of groups (up to extended Nielsen transformations) and compact 2-complexes (up to 3-deformations).

In the second part of the talk I will show how to describe the second homotopy group of a 2-complex in terms of colorings of the poset of
its cells and use this description to derive new results on asphericity of group presentations.