  We develop some new topological tools to study maximal subgroups of free
idempotent generated semigroups. As an application, we show that the rank 1
component of the free idempotent generated semigroup of the biordered set of a
full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal
subgroup isomorphic to the multiplicative subgroup of Q.
