Georgia Institute of Technology College of Sciences

Combinatorial homotopy theory, or $A$-theory, is a homotopy theory of simplicial complexes which is known to have far-reaching applications. In the graph case, it coincides with a notion of homotopy first introduced by Maurer to study matroid basis graphs. In the language of $A$-theory, Maurer's celebrated homotopy theorem states that matroid basis graphs have trivial fundamental group. We ask whether this result can be strengthened and make progress toward showing that matroid basis graphs are $A$-contractible. We look at this problem through the lens of Malle's ``gangster problem," which formulates $A$-contractibility of graphs in terms of gangsters travelling between towns.