Plan de estrategias

We use in this subsection the same notation as in Section 2.3.

Again we start with a complex ∆ satisfying the properties stated in the previous subsection and we want to construct a 3-manifold M∆ associated to ∆.

Again we can prove that this manifold is the connected sum ofg copies ofS2×S1 with two embedded maximal sphere systems in standard form, possibly containing spheres in common.

Let ∆ be a square complex satisfying the following properties:

-1) ∆ is connected and is the union of finite V-H square complexes and finite trees; the edges belonging to these trees will be called “joining edges”.

-2) ∆ is locally CAT(0)

-3) The fundamental group of ∆ is the free groupFg of rankg.

-4) All the vertex links in ∆ are of the types A1-A9 drawn in Figure 2.1 if the vertex is not contained in a joining edge and of the three types drawn in Figure 2.8 if the vertex is contained in a joining edge.

-5) All the hyperplanes in ∆ are either points or finite trees.

-6) If we denote by ∆ the universal cover of ∆, then there are two surjective pro-e

jections p1 : ∆e → T₁ and p₂ : ∆e → T₂, where T₁ and T₂ are infinite three-valent

trees.

Note that these properties are mostly the same as the six properties stated in Section 2.3 apart from property 4 and property 1.

We construct the topological spaceM∆using the same method as in Section

2.3.

Again we associate a circle to each square in ∆ and call these circles 1-pieces. As for edges, ifeis not a joining edge (i. e. it bounds at least a square) then we associate to e a 2-piece as described in Section 2.3. Ife is a joining edge than the 2-piece associated toeis a 2-sphere.

We call the 2-pieces associated to vertical edges the “red”2-pieces and the 2-pieces associated to horizontal edges the “black”2-pieces. We consider the spheres associ- ated to the joining edges as both black and red.

As for 3-pieces, if v is a vertex not contained in any joining edge, than we associate to v a handlebody as explained in Figure 2.3. If v is a vertex contained in a joining edge than the 3-piece associated to v will be a holed handlebody as described in Figure 2.9.

holed solid torus

3-holed 3-sphere

associated 3-piece

vertex link

2-holed 3-sphere

Figure 2.9: How to associate to a vertex containing a joining edge, a 3-piece with its boundary pattern. In all of the three cases above, instead of drawing the actual 3-piece, I draw the complement of the 3-piece in S3. Spheres coloured with pale blue are the ones belonging to both systems Σ1 and Σ2.

Again we take the 1-skeleton C1 to be the disjoint union of 1-pieces.

We glue the 2-pieces with boundary components to the 1-pieces as explained in Section 2.3, we denote the space obtained in this way as C₂0. Note that if the complex ∆ contains joining edges thanC₂0 is not connected. LetC2 be the disjoint

union ofC₂0 and the spheres associated to joining edges.

Again we “fill”C2 by gluing the 3-pieces, and we obtain in this way the space

M∆. Again we denote asQR the union of red 2-pieces and asQB the union of black

2-pieces. The pieces associated to joining edges will belong to both QR and QB.

The goal is to prove thatM∆ is the connected sum of g copies of S2×S1.

In order to reach this goal it is sufficient to prove that all Lemmas stated in Section 2.3 also hold in this general case.

Lemma 2.3.2 clearly holds in the general case also. The proof works in the same way even if we allow the possibility that a 2-piece might be a 2-sphere.

Lemma 2.3.3 holds in the general case also, in fact it obviously holds for the 2-pieces associated to joining edges.

Lemma 2.3.5 holds, again we can check it case by case for the three new cases.

Lemma 2.3.6 holds again, because it depends only on Lemmas 2.3.3 and 2.3.5.

As in Section 2.3, let∆ be the universal cover of ∆. Again we can constructe

M

e

∆ from ∆.e

Lemma 2.3.7 holds again, since the proof works in the same way. Lemma 2.3.8 holds in this case too. The proof works in the same way. Lemma 2.3.9 also holds true in the general case.

Lemma 2.3.10 also holds since it depends on Lemma 2.3.7 and Lemma 2.3.8. Also the proof of Lemma 2.3.11 works in the same way, therefore Lemma 2.3.11 also holds.

Therefore the space M∆ is also in the general case the connected sum of g

copies of S2×S1, with two embedded maximal sphere systems: QR and QB, in

standard form with respect to each other, and the square complex ∆.is the complex associated to Mg,QRand QB.

Also Remark 2.3.12, and Lemma 2.3.13 hold in the general case too; the proofs work in the same way.