On the local structure and the homology of CAT$(\kappa)$ spaces and euclidean buildings

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 $\Sigma_oX$ of a CAT$(\kappa)$ space $X$ is homotopy quivalent to a small punctured disk $B_\eps(X,o)\setminus o$. The second ingredient is the local homology sheaf of $X$. Along the way, we prove some results about the local structure of CAT$(\kappa)$-spaces which may be of independent interest.