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

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

2010-09-16

arxiv.org/abs/1009.3089v4

Mathematics

Metric Geometry

Small corrections in v2, v3, v4. To appear in Advances in Geometry

Scientific paper

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.

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