Consequences

In this section we explain the connection between the construction described in Section 2.2 and the one described in Section 2.4. Namely we will prove (Theorem 2.5.1) that these are two different ways of constructing the same object. This will allow us to prove the main results of this chapter, namely Theorem 2.5.4, stating that any two maximal sphere systems can be represented in standard form; and Theorem 2.5.6, stating that standard form is in some sense unique. Recall that we are still supposing that the two sphere systems do not have any sphere in common.

Let us start with some remark. Given a manifoldMg and two sphere systems

Σ1, Σ2 in standard form we have two ways of constructing a dual square complex.

The first way consists of constructing the square complex directly from the 3-manifold, as described in Section 2.2.

The second way uses the procedure described in Section 2.4 in the following way. LetT1 be the tree dual to Mf_g and Σf₁ and T₂ be the tree dual to Mf_g and Σf₂

(here I use the notation introduced in Section 2.1.1). Both trees are endowed with an action by the free group Fg, coming from the action of the group on Mf_g. The

productT1×T2 is therefore endowed with the diagonal action defined in Section 2.4.

The Gromov boundary ofT1 can be identified to the Gromov boundary of T2 since

they both can be identified to the Gromov boundary of the group Fg and to the

construct a square complexC(T1, T2). The quotient ofC(T1, T2) under the diagonal

action ofFg is a compact locally CAT(0) square complex. Throughout this Section

we will denote the complexC(T1, T2)/Fg by ∆(T1, T2).

We prove that these two constructions produce the same outcome, i.e::

Theorem 2.5.1. If Mg is the connected sum of g copies of S2×S1 and Σ1,Σ2 are

two embedded maximal sphere systems in standard form, then the square complex associated to Mf_g, Σf₁ and Σf₂ is isomorphic to the core C(T₁, T₂), where T₁ is the 3-valent tree dual to Mf_g andΣf₁ and T₂ is the 3-valent tree dual to Mf_g and Σf₂.

Proof. Since no ambiguity can occur, for the remainder of the proof we will denote by ∆ the square complex associated toe Mf_g, Σf₁ and Σf₂. Note that ∆ is endowede

with a free properly discontinuous action of the free groupFg, induced by the action

ofFg on the manifold Mf_g.

Let T1 and T2 be the trees defined above and, as usual, denote the core as

C(T1, T2). As mentioned above, T1 and T2 are endowed with free properly discon-

tinuous actions of the free group Fg, and the core is endowed with the diagonal

action.

The proof will consist of three steps: we first prove that ∆ can be identifiede

to a subcomplex of the productT1×T2; then we prove that∆ is actually containede

inC(T1, T2); eventually we prove thatC(T1, T2) is contained in∆.e

In order to show the first step, we first prove the existence of two equivariant projectionsp1 :∆e →T₁ andp₂ :∆e →T₂. We start defining these projections on the

0-skeleton of∆, then we extend them to the 1-skeleton and finally to the 2-skeleton.e

Let v be a vertex in ∆, this vertex represents a 3-piece ofe Mf_g, i. e. a

complementary component ofΣf₁∪Σf₂ inMf_g. In particular this 3-piece is contained

in a uniquely determined component ofMf_g \Σf₁, which is represented by a vertex

v1 inT1, and in a uniquely determined component of Mf_g\Σf₂, which is represented

by a vertex v2 inT2. Setp1(v) =v1 andp2(v) =v2.

Now we extent these projections to the 1-skeleton of ∆. Lete e be an edge

in∆. The edgee erepresents a 2-piece. This 2-piece belongs to a sphere σ₁ in Σf₁ if

eis a black edge and to a sphere σ2 in Σf₂ ife is a red edge. Suppose for example

thateis a red edge. The red 2-piece it represents lies entirely in a component P1 of f

Mg\Σf₁ and is contained in the boundary of two adjacent components ofMf_g\Σf₂.

We setp1(e) to be the vertex representing P1 andp2(e) to be the edge representing

σ2. The same holds for any black edge.

Note now that, if s is a square in ∆, then the two black (resp. red) edgese

belonging to the same sphere. Therefore the projectionsp1 andp2 naturally extend

to the 2-skeleton of∆.e

The projections p1 and p2 are clearly equivariant under the action of the

groupFg.

The existence of the projectionsp1 and p2 implies that there exists a surjec-

tive map of∆ onto a subcomplex of the producte T₁×T₂. To conclude the first step

we only need to show that this map is also an injection.

To see why this map is injective, note that standard form implies that a sphereσ1inΣf₁and a sphereσ₂inΣf₂intersect at most once. Therefore, if we denote

bye1 the edge inT1 corresponding to σ1 and bye2 the edge inT2 corresponding to

σ2, there is at most one square s in∆ satisfyinge p₁(s) = e₁ and p₁(s) = e₂. This

concludes the proof of the first step. In the remainder of the proof we will consider

e

∆ as a subcomplex of the productT1×T2.

We show now the second step, i. e. we show that that∆ is contained in thee

core C(T1, T2). Since each edge in ∆ bounds a square, in order to prove thate ∆ ise

contained inC(T1, T2), it is sufficient to prove that each square of∆ is contained ine

C(T1, T2).

Letsbe a square in∆. The two black (horizontal) edges ofe sproject through

p1 onto an edge e1 in T1 and the two red (vertical) edges of s project through p2

onto an edgee2 inT2. The fact thats is a square in ∆ means that the spherese σ₁

andσ2 represented bye1 ande2 intersect, and therefore, by Lemma 2.1.10, the two

partitions induced byσ1andσ2on the boundary ofMf_gare not nested. Consequently

the partitions induced by the edgese1 and e2 on the boundary ofT1 andT2 are not

nested and therefores=e1×e2 is a square inC(T1, T2).

To finish the proof of Theorem 2.5.1, we only need to prove thatC(T1, T2)

is contained in∆. As above it is sufficient to prove that each square ofe C(T₁, T₂) is

contained in∆.e

Let s =e1×e2 be a square in C(T1, T2). The edge e1 represents a sphere

in Σf₁, call this sphere σ₁; the edge e₂ represents a sphere in Σf₂, call this sphere

σ2. The fact that sis contained in the core means that the partitions induced by

the edgese1 and e2 on the boundary of T1 and T2 are not nested; this implies that

the partitions induced by the spheresσ1 andσ2 on the space of ends ofMf_g are not

nested. Therefore, by Lemma 2.1.10, the two spheres σ1 and σ2 intersect in Mf_g

and their intersection consists of exactly one circle. This means that there are two 2-pieces ofσ1and two 2-pieces ofσ2 inMf_g, all four of them intersecting in a 1-piece,

Remark 2.5.2. Note that Theorem 2.5.1 and Proposition 2.4.11 immediately imply Lemma 2.2.7 and Lemma 2.2.8, whose proof we had omitted in Section 2.2.

It is easy to note that, if we see ∆(Mf_g,Σf₁,Σf₁) as a subcomplex ofT₁×T₂,

then theFg-action on this square complex induced by theFg-action on the manifold f

Mg coincides with the diagonal action of Fg on the productT1×T2. Therefore an

immediate consequence of Theorem 2.5.1 is the following:

Corollary 2.5.3.Under the hypothesis of Theorem 2.5.1 the square complex∆(Mg,Σ1,Σ2)

is isomorphic to the square complex ∆(T1, T2).

We have shown so far that the construction described in Section 2.2 and the one described in Section 2.4 produce the same result.

Note anyway that the construction described in Section 2.4 is much more general. Above all, we can perform this construction starting just with a three- manifoldMg and two embedded maximal sphere systems which do not contain any

sphere in common. We do not need the two sphere systems to be in minimal or standard form.

Note also that, if we have two trees T1, T2 with no edge in common, we

construct the core C(T1, T2) and quotient it by the diagonal action of the group

Fg, then we obtain a square complex ∆(T1, T2) satisfying all the properties 1-6

described in Section 2.4, and therefore we can associate to it a 3-manifoldMg with

two maximal sphere systems,QRand QB, in standard form.

Summarising, if we have a manifoldMg and two embedded maximal sphere

systems, not necessarily in standard form, then we can associate to each system a dual tree with a group action. We can construct the core of the two trees. Then, using the method described in Section 2.3 we can build a manifold Mg with two

sphere systems in standard form.

The remarks I have just made are the main ingredients for the proof of existence of standard form. In fact, we are now ready to prove the following:

Theorem 2.5.4. Let Mg be the connected sum of g copies of S2×S1 and let Σ1,

Σ2 be two embedded maximal sphere systems such that no sphere inΣ1 is homotopic

to a sphere inΣ2. Then there exist maximal sphere systems Σ01, Σ02 such that Σ0i is

homotopic toΣi for i= 1,2, and Σ01, Σ02 are in standard form.

Before starting the proof, to avoid confusion, we clarify some terminology. Given two infinite trivalent trees T and T0 endowed with (free, properly discon- tinuous and cocompact) actions by the group Fg and therefore with a boundary

simplicial isomorphism ϕ : T → T0 such that for each edge e in T its image ϕ(e) induces the same partition ase. We are now ready to prove Theorem 2.5.4.

Proof. (of Theorem 2.5.4) Let Σ1, Σ2 be two sphere systems in Mg satisfying the

hypothesis of the theorem. As usual denote by Mf_g the universal cover of M_g and

by Σf₁ and Σf₂ the entire lifts of Σ₁ and Σ₂. Let T₁ be the tree dual to Mf_g and Σf₁

and letT2 be the tree dual toMf_g andΣf₂. BothT₁ andT₂ are endowed with a (free,

properly discontinuous and cocompact) Fg-action. Let C(T1, T2) be the core of T1

andT2.

Now, applying the procedure explained in Section 2.3 to the square complex C(T1, T2) we can construct a simply connected three manifold Mf0, with two embed-

ded maximal sphere systems, Qf_R and gQ_B, in standard form with respect to each

other. The manifoldMf0 is abstractly homeomorphic to Mf_g.

By construction, C(T1, T2) is the square complex dual to the manifold Mf0

and the two sphere systemsQf_RandgQ_B, therefore, by theorem 2.5.1, it is isomorphic

to the core of the two trees associated toMf0 andgQ_B, and toMf0 andQf_R. The trees

associated toMf0 and gQ_B and to Mf0 and Qf_R are the two projections of the square

complexC(T1, T2), namely T1 and T2. Therefore, without loss of generality, up to

permuting the labels QR and QB, we can suppose that T1 is the tree associated to f

M0 _and

g

QB, andT2 is the tree associated toMf0 and Qf_R.

The space of ends ofMf0 can be identified to the space of ends ofMf_g, since

they both can be identified to the boundaries ofT1 andT2. Moreover, since the tree

dual toMf_g and Σf₁ is the same as the tree dual toMf0 and gQ_B (they both coincide

with the treeT1), then for each sphereσ inΣf₁ there is a sphere ingQ_B inducing the

same partition as σ and for each sphere s in Qg_B there is a sphere in Σf₁ inducing

the same partition ass. The same holds forΣf₂ and Qf_R.

We can choose a homeomorphism H : Mf0 → Mf_g which is consistent with

the identification on the space of ends and such that, for each spheresinQf_R∪gQ_B,

the partition induced by H(s) is the same as the partition induced by s. Denote H(gQ_B) byΣf0

1 andH(Qf_R) by Σf0 2.

The systemsΣf0₁ andΣf0₂ are maximal and are in standard form with respect

to each other, since they are homeomorphic image of two maximal sphere systems in standard form. Moreover, for each sphere in Σf₁ (resp. Σf₂) there is a sphere in f

Σ0₁ (resp. Σf0₂) inducing the same partition and vice versa. Therefore, for i = 1,2

the sphere systemfΣ_i is homotopic in Mf_g to the sphere systemfΣ0_i.

Let Σ₁0 and Σ0₂ in Mg be the projections of Σf0₁ and Σf0₂ through the covering

respect to each other and moreover for i= 1,2 the sphere system Σ0_i is homotopic inMg to the sphere system Σi.

As an immediate consequence of Theorem 2.5.4 we can show something we had mentioned without proof in Section 2.1.2. Namely:

Remark 2.5.5. An immediate consequence of Theorem 2.5.4 is that, as promised in Section 2.1.2, two maximal sphere systems not containing any sphere in common can always be homotoped to be in strong minimal form with respect to each other. In other words a strong minimal form always exists for two maximal sphere sys- tems containing no spheres in common. This implies that the three definitions of minimality given in Section 2.1.2 are all equivalent.

To summarise what we have done, basically, in the proof of Theorem 2.5.4, we have shown a constructive way to find a standard form for two maximal sphere systems inMg. Namely, given two embedded maximal sphere systems Σ1 and Σ2 in

Mg which do not contain any sphere in common; letT1 be the tree dual toMf_g and f

Σ1 and letT2be the tree dual toMf_gandΣf₂. LetC(T₁, T₂) be the core ofT₁×T₂and

let ∆(T1, T2) be the quotient of C(T1, T2) by the diagonal action of Fg. Applying

the procedure explained in Section 2.3 to ∆(T1, T2) we construct a 3-manifold M

homeomorphic toMg, with two embedded maximal sphere systems,QRand QB, in

standard form with respect to each other. Note that this construction is defined up to twists around spheres inQR and QB. Note also that the construction depends

only on the homotopy class of the systems Σ1 and Σ2.

We show now that a standard form for two maximal sphere systems is “in some sense”unique. More precisely:

Theorem 2.5.6. Let (Σ1,Σ2), (Σ01,Σ02) be two pairs of embedded maximal sphere

systems in Mg. Suppose that both pairs of sphere systems are in standard form

and do not contain any sphere in common (i.e. that no sphere inΣ1 (resp. Σ01) is

homotopic to a sphere in Σ2 (resp. Σ02)). Suppose also that Σi is homotopic to Σ0i

for i= 1,2.

Then there exists a homeomorphism F : Mg → Mg such that F(Σi) = Σ0i

for i= 1,2. The homeomorphism F induces the identity (up to conjugacy) on the fundamental group ofMg.

The proof of Theorem 2.5.6 is based on Lemma 2.3.13 and on the following lemma.

Lemma 2.5.7. For g ≥3, let F :Mg →Mg be a self-homeomorphism of Mg. Let

Σbe a maximal sphere system embedded inMg. Suppose that for each sphere σ inΣ

the imageF(σ)is homotopic toσ. Then the induced homomorphismF∗:π1(Mg)→

π1(Mg) is an inner automorphism of the free groupFg.

Proof. Denote as usual as Mf_g the universal cover ofM_g, and denote the full lift of

Σ byΣ. The manifolde Mf_g is endowed with an action by the free group F_g and the

quotient of Mf_g by this action is the manifold M_g. In order to prove Lemma 2.5.7

we will show that a liftFe of the homeomorphismF is equivariant under this group

action.

First note that the action ofFg onMf_g induces an action ofF_gon the space of

ends. This action on the space of ends determines the action onMf_g up to homotopy;

in fact, since each component of Mf_g \Σ is a 3-holed 3-sphere, then the action ofe

Fg on Mf_g is determined by the action of F_g on Σ; and the action ofe F_g on Σ ise

determined up to homotopy by the action ofFg on the space of ends ofMf_g.

Now we exhibit a particular lift Fe. Note that, since each component of f

Mg\Σ is a 3-holed 3-sphere, then the mape Feis determined, up to homotopy, by its

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