The linking systems of the Solomon 2-local finite groups are simply connected

A p-local finite group is an algebraic structure which includes two categories, a fusion system and a linking system, which mimic the fusion and linking categories of a finite group over one of its Sylow subgroups. The p-completion of the geometric realization of the linking system is the classifying space of the finite group. In this paper, we study the geometric realization, without completion, of linking systems of certain exotic 2-local finite groups of which the existence was predicted by Solomon and Benson, and prove that they are all simply connected.