in the case when the topological information is encoded by bundles which are supported away

from the boundary. As a first application, we establish a ``long neck principle'' for a compact

Riemannian spin n-manifold with boundary X, stating that if Scal(X)≥n(n-1) and there is a nonzero degree map

into the sphere f: X → S

boundary of X is at most π/n. This answers, in the spin setting, a question recently asked by

Gromov. As a second application, we consider a Riemannian manifold X obtained by removing

k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if Scal(X)>σ>0 and Y

satisfies a certain condition expressed in terms of higher index theory, then the radius of a

geodesic collar neighborhood of ∂X is at most π[(n-1)(nσ)]

n-manifold V diffeomorphic to Nx[-1,1], with N a closed spin manifold with nonvanishing

Rosenberg index. In this case, we show that if Scal(X)≥σ>0, then the distance between the boundary

components of V is at most π[(n-1)(nσ)]