 for any protocol   and any input simplex  , the subcomplex   is a manifold. We saw that any such protocol cannot solve  -set agreement for  . In this chapter, we investigate another important topological property of the complex  : having no "holes" in dimensions   and below, a property called  - _connectivity_. We will see that if every   is  -connected, then   cannot solve  -set agreement. We will see later that there are natural models of computation for which protocol complexes are not manifolds, but they are  -connected for some  . We will also use this notion of connectivity in later chapters to characterize when protocols exist for certain tasks.  ##