On the local structure and the homology of CAT(κ) spaces and euclidean buildings

Abstract

Abstract We prove that every open subset of a euclidean building is a finite-dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our approach to this result is the following: the space of directions ∑ oX of a CAT(κ) space X is homotopy equivalent to a small punctured disk Bɛ (X, o) – o. The second ingredient is the local homology sheaf of X. Along the way, we prove some results about the local structure of CAT(κ)-spaces which may be of independent interest.