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Topology of the complement of real hyperplanes in $${\mathbb C}^ N$$. (English) Zbl 0594.57009
Let $$Y={\mathbb{C}}^ N\setminus \cup_{i}M_ i$$, where $$\{M_ i\}$$ is a locally finite family of affine hyperplanes with real equation. The problem of determining the homotopy type of Y is considered. In the first part, a regular cellular complex $$X\subset Y$$ and a homotopy equivalence between X and Y are constructed. The real part Q of X is the dual to the subdivision of $${\mathbb{R}}^ N$$ induced by the hyperplanes, and over each cell of Q, dual to the facet $$F^ j$$ of codimension j, there are as many j-cells in X as the chambers of $${\mathbb{R}}^ N$$ which contain $$F^ j$$ in their closure.
In the second part $$\pi_ 1(X)=\pi_ 1(Y)$$ is studied. It is proved that two ”minimal positive” paths in the 1-skeleton of X which have the same ends are homotopic. An algorithm is found for determining presentations of $$\pi_ 1(X)$$, which contains the Van Kampen method as a particular case.
Finally, a known result is re-proved using the complex X, namely if the hyperplanes of the subdivision are in general position then X (hence Y) is not a K($$\pi$$,1).

##### MSC:
 57R19 Algebraic topology on manifolds and differential topology 55Q05 Homotopy groups, general; sets of homotopy classes 55P10 Homotopy equivalences in algebraic topology
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