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

TL;DR

This paper investigates the local topological and homological properties of CAT() spaces and Euclidean buildings, establishing their homotopy types, local structures, and rigidity of homeomorphisms.

Contribution

It proves that open subsets of Euclidean buildings are finite dimensional ANRs and demonstrates the homotopy equivalence of the space of directions to a punctured disk, also establishing rigidity results.

Findings

Open subsets of Euclidean buildings are finite dimensional ANRs.

The space of directions in CAT() spaces is homotopy equivalent to a punctured disk.

Results on local structure of CAT()-spaces may be of independent interest.

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.