Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes

Author:
Luis Paris

Journal:
Trans. Amer. Math. Soc. **340** (1993), 149-178

MSC:
Primary 52B30; Secondary 32S25

DOI:
https://doi.org/10.1090/S0002-9947-1993-1148044-X

MathSciNet review:
1148044

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real vector space. An *arrangement of hyperplanes* in is a finite set of hyperplanes through the origin. A *chamber* of is a connected component of . The arrangement is called *simplicial* if and every chamber of is a simplicial cone. For an arrangement of hyperplanes in , we set

Let be an arrangement of hyperplanes of . Salvetti constructed a simplicial complex and proved that has the same homotopy type as . In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex and prove that there is a natural map which is the universal cover of . At the end, we use to give a new proof of Deligne's result: "if is a simplicial arrangement of hyperplanes, then is a space." Namely, we prove that is contractible if is a simplicial arrangement.

**[Br]**K. S. Brown,*Buildings*, Springer-Verlag, New York, 1989. MR**969123 (90e:20001)****[Co]**R. Cordovil,*On the homotopy of the Salvetti complexes determined by simplicial arrangements*, preprint.**[De]**P. Deligne,*Les immeubles des groupes de tresses généralisés*, Invent. Math.**17**(1972), 273-302. MR**0422673 (54:10659)****[FR]**M. Falk and R. Randell,*On the homotopy theory of arrangements*, Complex Analytic Singularities, Advanced Studies in Pure Math., 8, North-Holland, Amsterdam, 1987, pp. 101-124. MR**894288 (88f:32045)****[JT]**M. Jambu and H. Terao,*Free arrangements of hyperplanes and supersolvable lattices*, Adv. Math.**52**(1984), 248-258. MR**744859 (86c:32004)****[Lj]**E. S. Ljapin,*Semigroups*, Transl. Math. Monographs, vol. 3, Amer. Math. Soc., Providence, R.I., 1974. MR**0352302 (50:4789)****[LW]**A. T. Lundell and S. Weingram,*The topology of**complexes*, Van Nostrand Reinhold, New York, 1969. MR**0238319 (38:6595)****[Or]**P. Orlik,*Introduction to arrangements*, CBMS Regional Conf. Ser. in Math., no. 72, Amer. Math. Soc., Providence, R.I., 1989. MR**1006880 (90i:32018)****[Pa1]**L. Paris,*The covers of a complexified real arrangement of hyperplanes and their fundamental groups*, preprint. MR**1243871 (94i:52014)****[Pa2]**-,*The Deligne complex of a real arrangement of hyperplanes*, preprint.**[Pa3]**-,*Arrangements of hyperplanes with property*, preprint.**[Sa1]**M. Salvetti,*Topology of the complement of real hyperplanes in*, Invent. Math.**88**(1987), 603-618. MR**884802 (88k:32038)****[Sa2]**-,*On the homotopy theory of complexes associated to metrical-hemisphere complexes*, preprint.**[Te1]**H. Terao,*Arrangements of hyperplanes and their freeness*. I, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**27**(1980), 293-320. MR**586451 (84i:32016a)****[Te2]**-,*Modular elements of lattices and topological fibration*, Adv. Math.**62**(1986), 135-154. MR**865835 (88b:32032)****[We]**A. Weil,*Sur les théorèmes de De Rham*, Comment. Math. Helv.**26**(1952), 119-145. MR**0050280 (14:307b)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
52B30,
32S25

Retrieve articles in all journals with MSC: 52B30, 32S25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1148044-X

Article copyright:
© Copyright 1993
American Mathematical Society