We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove, that the set-valued algebraic theory obtained by applying π0 is the theory of ordinary n-nilpotent groups. Finally ...

We prove results exploring the relationship between fundamental group and second Betti number of minimal symplectic fillings lens spaces. These unify generalize several disparate facts appearing in literature. The Fibonacci numbers make a cameo appearance.