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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).

57R19 Algebraic topology on manifolds and differential topology
55Q05 Homotopy groups, general; sets of homotopy classes
55P10 Homotopy equivalences in algebraic topology
Full Text: DOI EuDML
[1] Brieskorn, E.: Sur les groupes de tresses. Sém. Bourbaki (1971/72); Lect. Notes Math.,317, 21-44 (1973) · Zbl 0277.55003
[2] Deligne, P.: Les immeubles des groupes de tresses g?éraligés. Invent. Math.17, 273-302 (1972) · Zbl 0238.20034
[3] Cartier, P.: Arrangements d’hyperplans: un chapitre de géométrie combinatoire. Sém. Bourbaki (1980/81). Lect. Notes Math., vol. 901. Berlin-Heidelberg-New York: Springer, 1977, pp. 1-22
[4] Hendriks, H.: Hyperplane complements of large type. Invent. Math.79, 375-381 (1985) · Zbl 0564.57016
[5] Lyndon, R.C., Schupps, P.E.: Combinatorial group theory. Ergebnisse der Mathematik. Berlin-Heidelberg-New York: Springer 1977
[6] Zavlasky, T.: Facing up to arrangements.: face-count formulas for partitiones of space by hyperplanes. Mem. Am. Math. Soc.154 (1975)
[7] Randell, R.: The fundamental group of the complement of a union of complex hyperplanes. Invent. Math.69, 103-108 (1982) · Zbl 0505.14017
[8] Randell, R.: The fundamental group oof the complement of a union of complex hyperplanes: correction. Invent. Math.80, 467-468 (1985) · Zbl 0596.14014
[9] Hattori, A.: Topology of ? N minus a finite number of affine hyperplanes in general position. J. Fac. Sci., Univ. Tokio, Sect. IA22, 205-219 (1975) · Zbl 0306.55011
[10] Fox, R.H., Neuwirth, L.: The braid groups. Scand. Math.10, 119-126 (1962) · Zbl 0117.41101
[11] Viet Dung, N.: The fundamental groups of the space of regular orbits of the affine Weyl groups. Topology22, 425-435 (1983) · Zbl 0524.57015
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