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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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