By Ciro Ciliberto, E. Laura Livorni, Andrew J. Sommese

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**Example text**

Group-like: For every two triples x ≤ y ≤ z, x ≤ y ≤ z of elements L, if two of the corresponding pairs of labelings are equal, so is the third. b a a b b a Figure 8. 11. An instructive example is one of the Garside structures that lead to the Artin group of type A2 . In this case (as for every ﬁnite-type Artin group) the poset is the weak order of the corresponding Coxeter Group, with the natural labeling by simple roots. We depict the poset with an equivalent labeling in Figure 8. For details on the weak order and the labeling see [8].

For every cover Gρ → G, we have Uρ Mρ . Uρ . For this, Proof. 4). In particular, (Δ(Gρ (F2 )), Δ(Gρ (F1 ))) is a NDR-pair and g is a closed coﬁbration. 6, obtaining a homotopy equivalence hocolimΔ(Gρ ) colimΔ(Gρ ). We are left with showing that the right-hand side is the complex Uρ . Indeed, every simplex is contained in (maybe more than) a Δ(Tρ (γ)). The maps of the diagram are inclusions, so colimΔ(Gρ ) = Δ(Tρ (γ)) γ∈Ob(Gρ ) ∼ 28 Emanuele Delucchi and we only have to check the identiﬁcations.

In the context of local system homology of arrangements, attention has been paid to the computation of the homology of cyclic covers of arrangement complements, as they generalize in many ways the Combinatorics of Arrangement Covers 33 Milnor ﬁbre (see the work of Cohen and Orlik [27] and, for a survey and the relevant bibliography, the paper by Suciu [74]). Therefore we want to point out that there exist spectral sequences that calculate the homology and cohomology of homotopy colimits of diagrams of spaces, thus oﬀering an alternative to the spectral sequence approach described by Denham [30] and later generalized by Papadima and Suciu [58].