Crossed modules, pictures, and 2-dimensional topology

Two-dimensional cell complexes form a remarkably rich class of objects; indeed, one can regard all of group theory as a sub-theory of 2-dimensional topology. One of the key invariants at our disposal is the second homotopy group, π2, and starting with the work of Whitehead and Reidemeister, a good theory of the structure of π2 has been developed. The aim of the talk is to describe this basic structure and explain the connection with group theory. The talk is in three sections. § 1 is a very brief crash course in basic homotopy theory. In § 2, we introduce crossed modules, and give two manifestations: relative π2 groups, and identities between the relations in a group presentation. In fact these two examples are essentially the same, and this will be one application of the ‘pictures’ developed in § 3, which provide combinatorial representatives for elements of π2.