Arrangements, local systems and singularities: CIMPA Summer by Fouad El Zein, Alexander I. Suciu, Meral Tosun, Muhammed

By Fouad El Zein, Alexander I. Suciu, Meral Tosun, Muhammed Uludag, Sergey Yuzvinsky

This quantity includes the Lecture Notes of the CIMPA/TUBITAK summer time university preparations, neighborhood structures and Singularities held at Galatasaray college, Istanbul in the course of June 2007. the quantity is meant for a wide viewers in natural arithmetic, together with researchers and graduate scholars operating in algebraic geometry, singularity idea, topology and similar fields. The reader will discover a number of open difficulties concerning preparations, neighborhood structures and singularities proposed through the teachers on the finish of the school.

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Charney, M. W. Davis; Finite K(π, 1)’s for Artin groups. Prospects in topology (Princeton, NJ, 1994), 110–124, Ann. of Math. Stud. 138, Princeton Univ. Press, Princeton, NJ, 1995. [24] R. Charney, D. Peifer; The K(π, 1)-conjecture for the affine braid groups. Comment. Math. Helv. 78 (2003), no. 3, 584–600. [25] R. Charney, J. Meier, K. Whittlesey; Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105 (2004), 171–188. [26] D. C. Cohen, P. Orlik; Arrangements and local systems.

This concludes the proof. We end by proving the announced proposition. 11. Let Θ → Γ(A) be a cover of oriented systems, ρ : Gρ → G the associated cover of the arrangement groupoid, and Dρ the corresponding diagram. Then NΘ = NvF F ∈ max(F ) v∈Ob(Gρ ) is a covering by open, contractible subsets with empty or contractible intersections. Moreover, the nerve of this covering is the poset PlimDρ . Proof. After the above preparations, we only have to show that NvF11 ∩ NvF22 is not empty if and only if it equals Nv˜F1 ∧F2 , where v˜ represents the path obtained by concatenating any representative of v1 with the lift (ρ(v1 ) → ρ(v1 )F1 ∧F2 ) v1 (where the wedge is taken in F ).

Indeed, N (v) ∩ N (w) = M (ρ(v)) ∩ M (ρ(w)) ∩ (Z(v, w) + iV ). 3. Combinatorial stratifications and diagram models We now draw the link between Paris’ construction and the diagram models. For this, we describe a stratification of NΘ by contractible subsets with contractible intersections, and show that its nerve is isomorphic to PlimDρ . 21]) this proves directly hocolimΔ(Dρ ) NΘ . 8. In comparing the two constructions one has to bear in mind that a vertex of an oriented system corresponds to an object of the corresponding covering groupoid, which is an equivalence class of paths in the arrangement graph.

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