Conclusiones

In this section we return to the covering spaces of the Salvetti complex. The aim is to classify these covering spaces according to subgroups of the fundamental group. In Section3.4.1, using the Salvetti-type diagram models, we constructed covering spaces of the Salvetti complex. Now we prove that any connected topological covering space of the Salvetti complex corresponds to a Salvetti-type diagram model. The strategy we employ is a standard result in algebraic topology establishing correspondence between isomorphism classes of covering spaces and conjugacy classes of subgroups (see for example, [44, Theorem 1.38]).

Recall that in previous section we saw that given a cover ρ: Gρ → G of the ar-

rangement groupoid G, one can construct a simplicial complex Sρ which is a covering

space of the associated Salvetti complex Sal(A). In Remark 3.4.11 it was mentioned that the simplicial complexSρis indeed the barycentric subdivision of a CW-complex.

For notational simplicity we denote the CW-complex bySρand start by analyzing its

2-skeleton. The plan is to show that the fundamental group ofSρis isomorphic to the

object group of Gρ.

0-skeleton: The 0-cells of Sρ correspond bijectively to the objects of Gρ and can be

written as [ρ(v), v] wherev _∈Ob(Gρ) andρ(v)∈C(A).

1-skeleton: There is a 1-cell between two vertices [ρ(v1), v1],[ρ(v2), v2] if and only if the corresponding chambersρ(v1), ρ(v2) share a codimension 1 face F1. In fact, there is one 1-cell [F1_{, v}

1] directed from [ρ(v1), v1] to [ρ(v2), v2] and another 1-cell [F1, v2] directed from [ρ(v2), v2] to [ρ(v1), v1].

77 The Fundamental Group

2-skeleton: For a codimension 2 face F2 _{and an object v ∈ Ob(G}

ρ) (such that

ρ(v) _≺ F2_{) there is one 2-cell [F}2_{, v] in S}

ρ. The vertices of [F2, v] are indexed by

the chambers whose closures containF2_{. To such a chamber C, there corresponds an} object in the covering groupoid given by t(ρ(v) → C, F2₎<v>_{. Hence the vertex set}

of [F2_{, v] is given by {[ρ(v}

i), vi] | ρ(vi) ≺ F2, i = 1, . . . ,2k}. A 1-cell [F1, vi] in the

boundary of [F2_{, v] corresponds to codimension 1 face F}1 _{such that F}1 _{≺ F}2_{. The} vertex v1 is such that the chamber ρ(vi) is on the same side as ρ(v) with respect to

F1_{. Moreover, the cell is oriented to havev}

i as its start vertex.

Remark 3.5.21. Attachment of higher dimensional cells is analogous to the construc- tion of Salvetti complex. Hence it can be shown that these covering complexes Sρ

have the structure of a MH-complex. However we will not prove it here.

The next step in the classification is to show that the fundamental group of Sρ is

isomorphic to the object group ofGρ. We compare the relations given by 2-cells of Sρ

and the relations that defineGρ.

Recall that the equivalence relation in the arrangement groupoid G is generated by identifying two paths (in the arrangement graph) with same end points and form the boundary of a 2-cell (of Sal(A)). As a result, the identity in an object group is an equivalence class of loops based at that object that can be written as γγ0−1_{, where}

γ, γ0 _{are paths bounding a 2-cell. For a given groupoid cover ρ: G}

ρ →GletGρ denote

the graph underlying Gρ. The morphisms ofGρare equivalence classes of paths in the

graphGρ. This equivalence relation is the one induced byρ. To be precise, two paths

˜

α,β˜ in Gρ are identified if and only if they have same end points and ρ(˜α) ∼ ρ( ˜β)

(where ˜α,β˜ are lifts of α, β respectively). Hence the relations in Gρ are generated by

the loops of the form ˜γ1γ˜2−1 such that [γ1] = [γ2] (in Gρ), denote this set by Σρ. In a

nutshell we have the following:

π(Gρ)∼=π1(Gρ)/Σρ

On the other hand let σρ denote the smallest normal subgroup of π1((Sρ)1) (1- skeleton) generated by the relations imposed by the boundary of every 2-cell. Then:

π1(Sρ)∼=π1((Sρ)1)/σρ

Theorem 3.5.22. With the notation as above we have following isomorphism

π(Gρ)∼=π1(Sρ)

Proof. The proof is clear once we use the above discussion, the ‘moves’ defining the arrangement groupoid and the Corollary 3.5.6.

Remark 3.5.23. We note that the above theorem was first proved in the context of cov- ers of the hyperplane complement, in Delucchi’s thesis [23, Lemma 4.2.2, Proposition 4.2.3], the proof is more direct and transparent.

Now we are in the position to state the classification of covering spaces of the Sal- vetti complex. The proofs of following statements essentially involve diagram chasing and are fairly straightforward (see [23, Section 4.5] for the original arguments). Fi- nally, we recall that the covering spaces Sρ are homotopy colimits of diagrams Dρ

(defined over F∗_{) of posets, see Definition 3.4.9.}

Theorem 3.5.24. LetAbe a submanifold arrangement in a l-manifoldX, letSal(A)

denote the associated Salvetti complex and G denote the arrangement groupoid. For any topological coverp: S _→Sal(A), there exists a cover of the arrangement groupoid

ρ: Gρ → G such that the homotopy colimit of the associated diagram of spaces Dρ is isomorphic to S as a covering space of Sal(A)

Proof. Let H denote the isomorphic image of p∗(π1(S)) inside π(G(A)) (under the isomorphism in Theorem 3.5.22). Applying Theorem 2.3.6 we see that there is a covering groupoid ρ: Gρ →G(A) such that π(Gρ)∼=H. Let Sρ denote the homotopy

colimit of the diagram of spaces defined usingGρ. Again appealing to Theorem3.5.22

it implies that π1(Sρ) =∼ H. Letιρ denote the inclusion of Gρ (the graph underlying

Gρ) into Sρ as its 1-skeleton. Since Gρ is equivalent to the fundamental groupoid of

Sρ the following diagram commutes.

π(Gρ) ∼ = −−−→ π1(Sρ) ρ∗   y   y(Λρ)∗ π(G) −−−→∼= π1(Sal(A))

Since (Λρ)∗(Sρ)∼=H we have that S and Sρ are isomorphic as covering spaces.

Following corollaries are immediate.

Corollary 3.5.25. Any cover p: S →Sal(A) of the Salvetti complex can be written as the order complex of a poset.

Corollary 3.5.26. Let ρˆ: Gρˆ → G denote the universal cover of the arrangement

groupoid. Then Sρˆ, the homotopy colimit of the associated diagram of spaces (posets)

79 Higher Homotopy Groups