How highly connected can an orbifold be?

Abstract

On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that $n$-connected $n$-orbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions, we prove slightly better bounds. We obtain sharp results up to dimension 5.