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The homology of real subspace arrangements. (English) Zbl 1213.14102
Let $V$ be a vector space and let $G$ be a building set, i.e. a finite collection of subspaces of the dual ${V}^{*}$, whose elements are indecomposable. The open set $V-{\bigcup }_{{H}_{i}\in G}{H}_{i}^{\perp }$ has a natural emebdding in the product of the projective spaces $ℙ\left(V/{H}_{i}^{\perp }\right)$. The closure ${Y}_{G}$ of the image is the De Concini–Procesi model of $G$. The variety ${Y}_{G}$, in the case where $G$ is a braid arrangement, is connected with the real part of the closure of the moduli space ${\overline{M}}_{0,n}\left(ℝ\right)$ of marked rational curves. Starting with a combinatorial description of the homology of $V-{\bigcup }_{{H}_{i}\in G}{H}_{i}^{\perp }$, it is possible to characterize the homology of ${\overline{M}}_{0,n}\left(ℝ\right)$. The author performs a similar analysis when $G$ is a general building set. Using chains of blow down of real De Concini - Procesi models, the author obtains a description of the ring structure of the homology of ${Y}_{G}$. By using this method, the author also proves that the homology of ${\overline{M}}_{0,n}\left(ℝ\right)$ has no odd torsion.

##### MSC:
 14N20 Configurations and arrangements of linear subspaces 14F25 Classical real and complex cohomology