Irmak Luis Elmas

Paris

Injective homomorphisms of mapping class groups of non-orientable surfaces

Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0, and let M(N) be the mapping class group of N . We show that, if G is a finite index subgroup of M(N) and ϕ : G → M(N) is an injective homomorphism, then there exists f 0 ∈ M(N) such that ϕ(g) = f 0 gf -1 0 for all g ∈ G . We deduce that the abstract commensurator of M(N) coincides with M(N).

57N05

Introduction

Let N be a connected and compact surface that can be non-orientable and have boundary but whose Euler characteristic is negative. We denote by Homeo(N) the group of homeomorphisms of N . The mapping class group of N , denoted by M(N), is the group of isotopy classes of elements of Homeo(N). Note that it is usually assumed that the elements of Homeo(N) preserve the orientation when N is orientable. However, the present paper deals with non-orientable surfaces and we want to keep the same definition for all surfaces, so, for us, the elements of Homeo(N) can reverse the orientation even if N is orientable. On the other hand, we do not assume that the elements of Homeo(N) pointwise fix the boundary. In particular, a Dehn twist along a circle isotopic to a boundary component is trivial. If N is not connected, the mapping class group of N , denoted by M(N), has the same definition. Although our results concern connected surfaces, we shall use in some places mapping class groups of non-connected surfaces.

The purpose of this paper is to prove the following.

Theorem 1.1 Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0, let G be a finite index subgroup of M(N), and let ϕ : G → M(N) be an injective homomorphism. Then there exists f 0 ∈ M(N) such that ϕ(g) = f 0 gf -1 0 for all g ∈ G .

The same result is known for orientable surfaces (see Irmak [START_REF] Irmak | Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups[END_REF][START_REF] Irmak | Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups. II[END_REF][START_REF] Irmak | Complexes of nonseparating curves and mapping class groups[END_REF], Behrstock-Margalit [START_REF] Behrstock | Curve complexes and finite index subgroups of mapping class groups[END_REF], Bell-Margalit [START_REF] Bell | Injections of Artin groups[END_REF]). A first consequence of Theorem 1.1 is as follows. This corollary was proved in Atalan-Szepietowski [START_REF] Atalan | Outer automorphisms of mapping class groups of nonorientable surfaces[END_REF][START_REF] Atalan | Automorphisms of the mapping class group of a nonorientable surface[END_REF] for compact, connected, nonorientable surfaces of genus ρ ≥ 7. Our contribution is just an extension to compact, connected, non-orientable surfaces of genus 5 and 6.

Corollary 1.2 Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0. Then Aut(M(N)) ∼ = M(N) and Out(M(N)) ∼ = {1}.

Note that the ideas of Atalan-Szepietowski [START_REF] Atalan | Outer automorphisms of mapping class groups of nonorientable surfaces[END_REF][START_REF] Atalan | Automorphisms of the mapping class group of a nonorientable surface[END_REF] cannot be extended to the more flexible framework of Theorem 1.1 because they use rather precise relations in M(N) that do not necessarily hold in a finite index subgroup.

Let G be a group. We denote by Com(G) the set of triples (A, B, ϕ), where A and B are finite index subgroups of G and ϕ : A → B is an isomorphism. Let ∼ be the equivalence relation on Com(G) defined as follows. We set (A 1 , B 1 , ϕ 1 ) ∼ (A 2 , B 2 , ϕ 2 ) if there exists a finite index subgroup C of A 1 ∩ A 2 such that ϕ 1 (g) = ϕ 2 (g) for all g ∈ C. The (abstract) commensurator of G is defined to be the quotient Com(G) = Com(G)/ ∼. This is a group. A more general consequence of Theorem 1.1 is the following.

Corollary 1.3 Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0. Then Com(M(N)) ∼ = M(N).

Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0. We denote by C(N) the complex of curves of N and by T (N) the subcomplex of C(N) formed by the isotopy classes of two-sided circles. For α, β ∈ C(N) we denote by i(α, β) the intersection index of α and β and, for α ∈ T (N), we denote t α the Dehn twist along α. A map λ : T (N) → T (N) is called a super-injective simplicial map if the condition i(λ(α), λ(β)) = 0 is equivalent to the condition i(α, β) = 0 for all α, β ∈ T (N).

The proof of Theorem 1.1 is based on two other theorems. The first one, proved in Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF]Theorem 1.1], says that, if λ : T (N) → T (N) is a super-injective simplicial map, then there exists f 0 ∈ M(N) such that λ(α) = f 0 (α) for all α ∈ T (N). Let T (N) be the subfamily of T (N) formed by the isotopy classes of circles that do not bound one-holed Klein bottles. The second theorem is the aim of Section 3 of this paper (see Theorem 3.1). It says that, if G is a finite index subgroup of M(N) and ϕ : G → M(N) is an injective homomorphism, then there exists a super-injective simplicial map λ : T (N) → T (N) which sends a non-trivial power of t α to a non-trivial power of t λ(α) for all α ∈ T (N).

The proof of Theorem 3.1 uses several results on reduction classes of elements of M(N) and on maximal abelian subgroups of M(N). These results are classical and widely used in the theory of mapping class groups of orientable surfaces, but they are more or less incomplete in the literature for mapping class groups of non-orientable surfaces. So, the aim of Section 2 is to recall and complete what is known on non-orientable case.

In order to prove Theorem 3.1 we first prove an algebraic characterization of some Dehn twists up to roots and powers (see Proposition 3.3). This characterization seems interesting by itself and we believe that it could be used for other purposes. Note also that this characterization is independent from related works of Atalan [START_REF] Atalan | Outer automorphisms of mapping class groups of nonorientable surfaces[END_REF][START_REF] Atalan | An algebraic characterization of a Dehn twist for nonorientable surfaces[END_REF] and Atalan-Szepietowski [START_REF] Atalan | Automorphisms of the mapping class group of a nonorientable surface[END_REF].

The final proofs of Theorem 1.1 and Corollary 1.3 are given in Section 4. The proof of Corollary 1.2 is left to the reader.

Preliminaries

Canonical reduction systems

From now on N = N ρ,n denotes compact, connected, non-orientable surface of genus ρ ≥ 5 with n ≥ 0 boundary components. A circle in N is an embedding a : S 1 → N \ ∂N . This is assumed to be non-oriented. It is called generic if it does not bound a disk, it does not bound a Möbius band, and it is not isotopic to a boundary component. The isotopy class of a circle a is denoted by [a]. A circle a is called two-sided (resp. one-sided) if a regular neighborhood of a is an annulus (resp. a Möbius band). We denote by C(N) the set of isotopy classes of generic circles, and by T (N) the subset of C(N) of isotopy classes of generic two-sided circles.

The intersection index of two classes

α, β ∈ C(N) is i(α, β) = min{|a ∩ b| | a ∈ α and b ∈ β}. The set C(N)
is endowed with a structure of simplicial complex, where a finite set A is a simplex if i(α, β) = 0 for all α, β ∈ A. This simplicial complex is called the curve complex of N . Note that M(N) acts naturally on C(N), and this action is a simplicial action. Note also that T (N) is invariant under this action.

Let A = {α 1 , . . . , α r } be a simplex of C(N). Let M A (N) denote the stabilizer of A in M(N). Choose pairwise disjoint representatives a i ∈ α i , i ∈ {1, . . . , r}, and denote by N A the natural compactification of N \ (∪ r i=1 a i ). The groups M A (N) and M(N A ) are linked by a homomorphism Λ A : M A (N) → M(N A ), called the reduction homomorphism along A, defined as follows. Let f ∈ M A (N). Choose a representative F ∈ Homeo(N) of f such that F({a 1 , . . . , a r }) = {a 1 , . . . , a r }. The restriction of F to N \ (∪ r i=1 a i ) extends in a unique way to a homeomorphism F ∈ Homeo(N A ). Then Λ A (f ) is the element of M(N A ) represented by F .

We denote by t α the Dehn twist along a class α ∈ T (N). This is defined up to a power of ±1, since its definition depends on the choice of an orientation in a regular neighborhhood of a representative of α, but this lack of precision will not affect the rest of the paper. If A is a simplex of C(N), we set A T = A ∩ T (N), and we denote by

Z A the subgroup of M(N) generated by {t α | α ∈ A T }.
The following is a classical result for mapping class groups of orientable surfaces. It is due to Stukow [START_REF] Stukow | Dehn twists on nonorientable surfaces[END_REF][START_REF] Stukow | Commensurability of geometric subgroups of mapping class groups[END_REF] for non-orientable surfaces.

Proposition 2.1 (Stukow [START_REF] Stukow | Dehn twists on nonorientable surfaces[END_REF][START_REF] Stukow | Commensurability of geometric subgroups of mapping class groups[END_REF]) Let A be a simplex of C(N). Then Z A is the kernel of Λ A and it is a free abelian group of rank |A T |.

Let f ∈ M(N). We say that f is pseudo-Anosov if C(N) = ∅ and f n (α) = α for all α ∈ C(N) and all n ∈ Z \ {0}. We say that f is periodic if it is of finite order. We say that f is reducible if there is a non-empty simplex A of C(N) such that f (A) = A. In that case, A is called a reduction system for f and an element of A is called a reduction class for f . An element f ∈ M(N) can be both periodic and reducible, but it cannot be both pseudo-Anosov and periodic, and it cannot be both pseudo-Anosov and reducible.

Let f ∈ M(N) be a reducible element and let A be a reduction system for f . Let N 1 , . . . , N be the connected components of

N A . Choose n ∈ Z \ {0} such that f n (N i ) = N i for all i and denote by Λ i,A (f n ) ∈ M(N i ) the restriction of Λ A (f n ) to N i .
We say that A is an adequate reduction system for f if, for all i ∈ {1, . . . , }, Λ i,A (f n ) is either periodic or pseudo-Anosov. This definition does not depend on the choice of n. The following was proved by Thurston [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF] for orientable surfaces and then was extended to non-orientable surfaces by Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF].

Theorem 2.2 (Thurston [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF], Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF]) A mapping class f ∈ M(N) is either pseudo-Anosov, periodic or reducible. Moreover, if it is reducible, then it admits an adequate reduction system.

Let f ∈ M(N).

A reduction class α for f is an essential reduction class for f if for each β ∈ C(N) such that i(α, β) = 0 and for each integer m ∈ Z \ {0}, the classes f m (β) and β are distinct. We denote by S(f ) the set of essential reduction classes for f . Note that S(f ) = ∅ if and only if f is either pseudo-Anosov or periodic. The following was proved by Birman-Lubotzky-McCarthy [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF] for orientable surfaces and by Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF] and Kuno [START_REF] Kuno | Abelian subgroups of the mapping class groups for non-orientable surfaces[END_REF] for non-orientable surfaces.

Theorem 2.3 (Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF], Kuno [START_REF] Kuno | Abelian subgroups of the mapping class groups for non-orientable surfaces[END_REF]) Let f ∈ M(N) be a non-periodic reducible mapping class. Then S(f ) = ∅, S(f ) is an adequate reduction system for f , and every adequate reduction system for f contains S(f ). In other words, S(f ) is the unique minimal adequate reduction system for f . Remark The proof of Theorem 2.3 for orientable surfaces given in Birman-Lubotzky-McCarthy [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF] does not extend to non-orientable surfaces, since Lemma 2.4 in [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF] is false for non-orientable surfaces and this lemma is crucial in the proof. The proof for non-orientable surfaces is partially made in Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF] using oriented double coverings, and it is easy to get the whole result by these means. Kuno's approach [START_REF] Kuno | Abelian subgroups of the mapping class groups for non-orientable surfaces[END_REF] is different in the sense that she replaces Lemma 2.4 of [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF] by another lemma of the same type.

If f ∈ M(N) is a non-periodic reducible mapping class, then the set S(f ) is called the essential reduction system for f . Recall that, if f is either periodic or pseudo-Anosov, then S(f ) = ∅. The following lemma follows from the definition of S(f ).

Lemma 2.4 Let f ∈ M(N). Then (1) S(f n ) = S(f ) for all n ∈ Z \ {0}, (2) S(gfg -1 ) = g(S(f )) for all g ∈ M(N).
Moreover, we will often use the following. Lemma 2.5 Let A = {α 1 , . . . , α p } be a simplex of T (N), and, for every i ∈ {1, . . . , p}, let

k i ∈ Z \ {0}. Set g = t k 1 α 1 • • • t kp αp . Then S(g) = A.
Proof We have g(α i ) = α i for all i ∈ {1, . . . , p}, hence A is a reduction system for g. The reduction Λ A (g) of g along A is the identity, hence A is an adequate reduction system. Let i ∈ {1, . . . , p}. Set A i = A \ {α i }. Note that A i is also a reduction system for g. We choose disjoint representatives a j ∈ α j , j ∈ {1, . . . , p}, we denote by N Ai the natural compactification of N \ (∪ j =i a j ), and we denote by N the component of N Ai that contains a i . The restriction of Λ Ai (g) to N is t ki ai , which is neither periodic, nor pseudo-Anosov. Hence, A i is not an adequate reduction system. So, A is the minimal adequate reduction system for g, that is, A = S(g).

Abelian subgroups

Let A be a simplex of C(N) and let H be an abelian subgroup of M(N). We say that A is a reduction system for H if A is a reduction system for every element of H . Similarly, we say that A is an adequate reduction system for H if A is an adequate reduction system for every element of H . On the other hand, we set S(H) = f ∈H S(f ). The following can be proved exactly in the same way as Lemma 3.1 in Birman-Lubotzky-McCarthy [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF].

Lemma 2.6 Let H be an abelian subgroup of M(N). Then S(H) is an adequate reduction system for H and every adequate reduction system for H contains S(H).

The rank of N is defined by rk(N) = 3 2 ρ + n -3 if ρ is even and by rk(N)

= 1 2 (3ρ -1) + n -3 if ρ is odd. Let A be a simplex of C(N).
As ever, we set

A T = A ∩ T (N). For k ∈ Z \ {0}, we denote by Z A [k] the subgroup of M(N) generated by {t k α | α ∈ A T }. By Proposition 2.1, Z A [k] is a free abelian group of rank |A T |.
The rank of an abelian group H is denoted by rk(H).

Kuno [START_REF] Kuno | Abelian subgroups of the mapping class groups for non-orientable surfaces[END_REF] proved that the maximal rank of an abelian subgroup of M(N) is precisely rk(N). Her proof is largely inspired by Birman-Lubotzky-McCarthy [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF]. In the present paper we need the following more precise statement whose proof is also largely inspired by Birman-Lubotzky-McCarthy [START_REF] J S Birman | Abelian and solvable subgroups of the mapping class groups[END_REF].

Proposition 2.7 Let H be an abelian subgroup of M(N). Then rk(H) ≤ rk(N). Moreover, if rk(H) = rk(N), then the following claims hold.

(1) There exists a simplex A in T (N) such that |A| = rk(N) and S(H)∩T (N) ⊂ A.

(2) There exists

k ≥ 1 such that Z S(H) [k] ⊂ H . (3) None of the connected components of N S(H) is homeomorphic to N 2,1 .
Remark There are classes of circles that are not included in any simplex of T (N) of cardinality rk(N). These classes will be considered in Section 3. In particular, Part (1) of the above proposition is not immediate.

The rest of this subsection is dedicated to the proof of Proposition 2.7.

Let A be a simplex of C(N). We say that A is a pants decomposition if N A is a disjoint union of pairs of pants. It is easily seen that any simplex of C(N) is included in a pants decomposition. The following lemma is partially proved in Irmak [ Remark There are simplices in T (N) of cardinality rk(N). Their construction is left to the reader.

For ρ, n ≥ 0, we denote by S ρ,n the orientable surface of genus ρ and n boundary components. The first part of the following lemma is well-known. The second part is proved in Scharlemann [START_REF] Scharlemann | The complex of curves on nonorientable surfaces[END_REF] and Stuko [START_REF] Stukow | Dehn twists on nonorientable surfaces[END_REF]. The third part is easy to prove. Lemma 2.10 (1) The mapping class groups M(S 0,3 ) and M(N 1,2 ) are finite.

(2) The set T (N 2,1 ) is reduced to a unique element, β , and M(N 2,1 ) Z Z/2Z, where the copy of Z is generated by t β .

(3) If M is a connected surface, with negative Euler characteristic, different from S 0,3 and N 1,2 , then T (M) is non-empty.

The following lemma is well-known for orientable surfaces (see Fathi-Laudenbach-Poénaru [8, Thm. III, Exp. 12] and Ivanov [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF]Lem. 8.13]). It can be easily proved for a non-orientable surface N from the fact the the lift of a pseudo-Anosov element of M(N) to the mapping class group of the orientable double-covering of N is a pseudo-Anosov element (see Wu [START_REF] Ying | Canonical reducing curves of surface homeomorphism[END_REF]).

β i ∈ T (N i ). We set B = {β 1 , . . . , β } and A = S T B . Then A is a simplex of T (N)
, hence, by Corollary 2.9, we have |A| ≤ rk(N), and therefore

rk(H) = rk(H ∩ Z S ) + rk(Λ S (H)) ≤ |S T | + |B| = |A| ≤ rk(N) . Assume that rk(H) = rk(N). Then |A| = rk(N) and S T = S(H) ∩ T (N) ⊂ A. Moreover, H ∩ Z S is of finite index in Z S , hence there exists k ≥ 1 such that Z S [k] ⊂ H ∩ Z S .
Finally, for every i ∈ {1, . . . , }, there exists f ∈ Λ S (H) ∩ Ker ϕ such that the restriction of f to N i is pseudo-Anosov. By Lemma 2.10, this implies that N i cannot be homeomorphic to N 2,1 .

3 Super-injective simplicial maps

Let B be a subset of C(N). A map λ : B → B is called a super-injective simplicial map if the condition i(λ(α), λ(β)) = 0 is equivalent to the condition i(α, β) = 0 for all α, β ∈ B . It is shown in Irmak-Paris [13, Lemma 2.
2] that a super-injective simplicial map on T (N) is always injective. We will see that the same is true for the set B = T 0 (N) defined below (see Lemma 3.6).

Let α ∈ T (N). We say that α bounds a Klein bottle if N α has two connected components and one of them, N , is a one-holed Klein bottle. Recall that T (N ) is a singleton {α 0 } (see Lemma 2.10 ( 2)). The class α 0 is then called the interior class of α. The aim of this section is to prove the following theorem. This together with Theorem 1.1 of Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF] are the main ingredients for the proof of Theorem 1.1. We say that a class α ∈ T (N) is separating (resp. non-separating) if N α has two connected components (resp. has one connected component). We denote by T 1 (N) the set of separating classes α ∈ T (N) such that both components of N α are non-orientable of odd genus. We denote by T 2 (N) the set of separating classes α ∈ T (N) that bound Klein bottles. And we denote by T 0 (N) the complement of

T 1 (N) ∪ T 2 (N) in T (N).
The rest of the section is devoted to the proof of Theorem 3.1. In Subsection 3.1 we prove the restriction of Theorem 3.1 to T 0 (N) (see Proposition 3.4). In Subsection 3.2 we prove several combinatorial properties of super-injective simplicial maps T 0 (N) → T 0 (N). Theorem 3.1 is proved in Subsection 3.3.

Injective homomorphisms and super-injective simplicial maps of T 0 (N)

We define the rank of an orientable surface S ρ,n to be rk(S ρ,n

) = 3ρ + n -3. It is well- known that rk(S ρ,n ) is the maximal cardinality of a simplex of T (S ρ,n ) (= C(S ρ,n )).
By Corollary 2.9 the same is true for non-orientable surfaces.

Lemma 3.2 Let α ∈ T (N).

(1) We have α ∈ T 1 (N) if and only if there is no simplex A in T (N) of cardinality rk(N) containing α.

(2) We have α ∈ T 2 (N) if and only if there exists a simplex A in T (N) of cardinality rk(N) containing α and there exists a class β ∈ T (N), different from α, such that every simplex A in T (N) of cardinality rk(N) containing α also contains β .

(3) We have α ∈ T 0 (N) if and only if there are two simplices A and

A in T (N) of cardinality rk(N) such that A ∩ A = {α}. Proof Suppose that α is non-separating. It is easily seen that rk(N) = rk(N α ) + 1. Take two disjoint simplices A 1 and A 1 in T (N α ) of cardinality rk(N α ) and set A = A 1 ∪ {α} and A = A 1 ∪ {α}. Then |A| = |A | = rk(N) and A ∩ A = {α}.
Suppose that α is separating. Let N 1 and N 2 be the connected components of

N α . It is easily seen that rk(N) = rk(N 1 ) + rk(N 2 ) + 2 if α ∈ T 1 (N) and rk(N) = rk(N 1 ) + rk(N 2 ) + 1 otherwise. Suppose that α ∈ T 1 (N). Let A be a simplex of T (N) containing α. Set A 1 = A ∩ T (N 1 ) and A 2 = A ∩ T (N 2 ). We have A = A 1 A 2 {α}, hence |A| = |A 1 | + |A 2 | + 1 ≤ rk(N 1 ) + rk(N 2 ) + 1 = rk(N) -1 < rk(N) .
Suppose that α ∈ T 2 (N). Then one of the connected components of N α , say N 1 , is a one-holed Klein bottle. By Lemma 2.10 (2), T (N 1 ) contains a unique element,

β . Let A 2 be a simplex of T (N 2 ) of cardinality rk(N 2 ). Set A = A 2 ∪ {α, β}. Then A contains α and |A| = rk(N). Let A be a simplex of T (N) of cardinality rk(N) containing α. Set A 1 = A ∩ T (N 1 ) and A 2 = A ∩ T (N 2 ). Since A = A 1 A 2 {α} and |A 2 | ≤ rk(N 2 ) = rk(N) -2, we have A 1 = ∅, hence A 1 = {β}, and therefore β ∈ A.
Suppose that α ∈ T 0 (N). It is easily seen that there exist two disjoint simplices A 1 and A 1 in T (N 1 ) of cardinality rk(N 1 ). Similarly, one can find two disjoint simplices

A 2 and A 2 in T (N 2 ) of cardinality rk(N 2 ). Set A = A 1 ∪ A 2 ∪ {α} and A = A 1 ∪ A 2 ∪ {α}. Then A and A are two simplices of T (N) of cardinality rk(N) such that A ∩ A = {α}. For α ∈ T (N) we set R(α) = {f ∈ M(N) | there exists ∈ Z such that f ∈ t α \ {1}}.
Note that, by Lemma 2.4 and Lemma 2.5, we have S(f ) = {α} for all f ∈ R(α). We turn now to prove an algebraic characterization of the Dehn twists along the elements of T 0 (N), up to roots and powers. This result and its proof are independent from the characterizations given in Atalan [START_REF] Atalan | Outer automorphisms of mapping class groups of nonorientable surfaces[END_REF][START_REF] Atalan | An algebraic characterization of a Dehn twist for nonorientable surfaces[END_REF] and Atalan-Szepietowski [START_REF] Atalan | Automorphisms of the mapping class group of a nonorientable surface[END_REF], and they are interesting by themselves.

Proposition 3.3 (1) Let f be an element of infinite order in M(N). If there exist two abelian subgroups H and H in M(N) of rank rk(N) such that H ∩H = f , then there exists α ∈ T 0 (N) such that f ∈ R(α).

(2) Let G be a finite index subgroup of M(N), let α ∈ T 0 (N), and let k ∈ Z \ {0} such that t k α ∈ G . Then there exist two abelian subgroups H and

H in G of rank rk(N) such that H ∩ H = t k α .
Proof We take an element f ∈ M(N) of infinite order such that there exist two abelian subgroups H and H of rank rk(N) satisfying

H ∩ H = f . First we prove that S(H) ∩ T (N) = ∅. Suppose that S(H) ∩ T (N) = ∅.
) ∩ T (N) = ∅. Suppose that S(f ) ∩ T (N) = ∅.
Then all the elements of S(f ) are one-sided, N S(f ) is connected, and the reduction homomorphism

Λ S(f ) : M S(f ) (N) → M(N S(f ) ) is injective. Since S(f
) is an adequate reduction system for f , the element Λ S(f ) (f ) is either periodic or pseudo-Anosov. We know by the above that S(H) ∩ T (N) is non-empty, hence we can choose a class α ∈ S(H) ∩ T (N). We have α ∈ S(f ), since it is two-sided, hence α lies in T (N S(f ) ) and it is a reduction class for

Λ S(f ) (f ). So, Λ S(f ) (f ) is periodic,
A = {α 1 , . . . , α r } and A = {α 1 , . . . , α r } in T (N) of cardinality r = rk(N) such that A ∩ A = {α}. We assume that α = α 1 = α 1 . Since G is of finite index in M(N), there exists ≥ 1 such that t αi , t α i ∈ G for all i ∈ {2, . . . ,
H ∩ H = t k α . Let g ∈ H ∩ H . Since g ∈ H , there exist u 1 , u 2 , . . . , u r ∈ Z such that g = t ku 1 α t u 2 α 2 • • • t ur αr . Similarly, there exist v 1 , v 2 , . . . , v r ∈ Z such that g = t kv 1 α t v 2 α 2 • • • t vr α r
. By Lemma 2.5 we have S(g)

= {α i | u i = 0} = {α i | v i = 0}. Since A ∩ A = {α}, we have u i = v i = 0 for all i ∈ {2, . . . , r}, hence g = t ku 1 α ∈ t k α .
Proposition 3.4 Let G be a finite index subgroup of M(N) and let ϕ : G → M(N) be an injective homomorphism. Then there exits a super-injective simplicial map λ : T 0 (N) → T 0 (N) satisfying the following. Let α ∈ T 0 (N). There exist k, ∈ Z\{0}

such that t k α ∈ G and ϕ(t k α ) = t λ(α) . Proof Let α ∈ T 0 (N). Choose u ∈ Z \ {0} such that t u α ∈ G . By Proposition 3.3 there exist two abelian subgroups H, H of G of rank rk(N) such that H ∩ H = t u
α . The subgroups ϕ(H) and ϕ(H ) are abelian groups of rank rk(N) and ϕ(H) ∩ ϕ(H ) = ϕ(t u α ) , hence, by Proposition 3.3, there exists

β ∈ T 0 (N) such that ϕ(t u α ) ∈ R(β). There exist v, ∈ Z \ {0} such that ϕ(t uv α ) = ϕ(t u α ) v = t β . Then we set k = uv and λ(α) = β .
Let α, α ∈ T 0 (N) and let k, k ∈ Z \ {0}. By Stukow [START_REF] Stukow | Dehn twists on nonorientable surfaces[END_REF] we have that, if t k α = t k α , then α = α , and we have

t k α t k α = t k α t k α if and only if i(α, α ) = 0. We assume that k, k are such that ϕ(t k α ) = t λ(α) and ϕ(t k α ) = t λ(α ) for some , ∈ Z \ {0}. Then i(λ(α), λ(α )) = 0 ⇔ t λ(α) t λ(α ) = t λ(α ) t λ(α) ⇔ t k α t k α = t k α t k α ⇔ i(α, α ) = 0 . So, λ is a super-injective simplicial map.
3.2 Super-injective simplicial maps of T 0 (N)

In this subsection λ : T 0 (N) → T 0 (N) denotes a given super-injective simplicial map. Most of the results proved here are proved in Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF] for super-injective simplicial maps of T (N). However, we use in Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF] configurations of circles containing elements of T 2 (N) or T 1 (N), and, on the other hand, here we use results such as Lemma 3.7 that are false for super-injective simplicial maps of T (N). So, it is not easy to deduce the results of this subsection from Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF] without losing the reader, hence we give independent proofs from Irmak-Paris [START_REF] Irmak | Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces[END_REF] up to an exception (Lemma 3.11).

The following lemma is easy to show and its proof is left to the reader. It will be often used afterwards, hence it is important to keep it in mind. Lemma 3.5 Let A be a simplex of T (N) of cardinality rk(N). If ρ is even, then all the connected components of N A are homeomorphic to S 0,3 . If ρ is odd, then one of the connected components of N A is homeomorphic to N 1,2 and the other components are homeomorphic to S 0,3 .

Lemma 3.6

The map λ : T 0 (N) → T 0 (N) is injective.

Proof Let α, β be two distinct elements of T 0 (N). If i(α, β) = 0, then i(λ(α), λ(β)) = 0, hence λ(α) = λ(β). Suppose that i(α, β) = 0. It is easily shown that there exists γ ∈ T 0 (N) such that i(α, γ) = 0 and i(β, γ) = 0. Then i(λ(α), λ(γ)) = 0 and i(λ(β), λ(γ)) = 0, hence λ(α) = λ(β). Lemma 3.7 Let A = {α 1 , . . . , α r } be a simplex of T 0 (N) of cardinality r = rk(N). There exists β ∈ T 0 (N) such that i(α 1 , β) = 0 and i(α i , β) = 0 for all i ∈ {2, . . . , r}.

Proof We choose pairwise disjoint representatives a i ∈ α i , i ∈ {1, . . . , r}. We denote by N A the natural compactification of N \ (∪ r i=1 a i ) and by π A : N A → N the gluing map. We denote by a 1 and a 1 the two components of π -1 A (a 1 ), by P the connected component of N A containing a 1 , and by P the connected component containing a 1 .

Suppose that P = P . Set P = π A (P ) = π A (P ). Then P is homeomorphic to S 1,1 or N 2,1 and there exists i ∈ {2, . . . , r} such that a i = ∂P (say a 2 = ∂P). Since α 2 ∈ T 0 (N), P cannot be homeomorphic to N 2,1 , hence P is homeomorphic to S 1,1 . Let β ∈ T (N) be the class represented by the circle b drawn in Figure 3.1 (i). Then β ∈ T 0 (N), i(α 1 , β) = 1, and i(α i , β) = 0 for all i ∈ {2, . . . , r}. Suppose that P = P . By Lemma 3.5, either P and P are both homeomorphic to S 0,3 , or one is homeomorphic to S 0,3 and the other is homeomorphic to N 1,2 . The surfaces P and P are drawn in Let A = {α 1 , . . . , α r } be a simplex of T 0 (N). As before we denote by π A : N A → N the gluing map. We say that α 1 and α 2 are adjacent with respect to A if there exists a connected component P of N A and two boundary components a 1 and a 2 of P such that α 1 = [π A (a 1 )] and

α 2 = [π A (a 2 )].
Lemma 3.8 Let A = {α 1 , . . . , α r } be a simplex of T 0 (N) of cardinality r = rk(N).

If α 1 and α 2 are adjacent with respect to A then there exists β ∈ T 0 (N) such that i(α 1 , β) = 0, i(α 2 , β) = 0, and i(α i , β) = 0 for all i ∈ {3, . . . , r}.

Proof We choose pairwise disjoint representatives a i ∈ α i , i ∈ {1, . . . , r}. We denote by N A the natural compactification of N \ (∪ r i=1 a i ) and by π A : N A → N the gluing map. We denote by a i and a i the two components of π -1 A (a i ) for all i. We can assume that a 1 and a 2 are boundary components of the same connected component P 0 of N A . On the other hand, we denote by P 1 (resp. P 2 ) the connected component of N A containing a 1 (resp. a 2 ).

Suppose first that P 0 = P 1 . Set P = π A (P 0 ). Then P is homeomorphic to S 1,1 or N 2,1 , and a 2 is the boundary component of P. But α 2 ∈ T 0 (N), hence P cannot be homeomorphic to N 2,1 , thus P is homeomorphic to S 1,1 . There exists a subsurface K of N homeomorphic to S 1,2 endowed with the configuration of circles drawn in Lemma 3.9 Let A be a simplex of T 0 (N) of cardinality rk(N) and let α 1 , α 2 ∈ A.

β ∈ T 0 (N), i(α 1 , β) = 1, i(α 2 , β) = 2, and i(α i , β) = 0 for all i ∈ {3, . . . , r}. a 1 a 2 b a 1 a 2 b a 1 a 2 (i) (ii) (iii) b a 1 a 2 b c (iv) (v)
Then α 1 is adjacent to α 2 with respect to A if and only if λ(α 1 ) is adjacent to λ(α 2 ) with respect to λ(A).

Proof Set A = {α 1 , α 2 , α 3 , . . . , α r }. Suppose that α 1 is not adjacent to α 2 with respect to A. By Lemma 3.7 there exist β 1 , β 2 ∈ T 0 (N) such that i(β 1 , α 1 ) = 0, i(β 1 , α i ) = 0 for all i ∈ {2, 3, . . . , r}, i(β 2 , α 2 ) = 0, and i(β 2 , α i ) = 0 for all i ∈ {1, 3, . . . , r}. Since α 1 is not adjacent to α 2 with respect to A we also have i(β 1 , β 2 ) = 0. Since λ is a super-injective simplicial map, we have i(λ(β 1 ), λ(α 1 )) = 0, i(λ(β 1 ), λ(α i )) = 0 for all i ∈ {2, 3, . . . , r}, i(λ(β 2 ), λ(α 2 )) = 0, i(λ(β 2 ), λ(α i )) = 0 for all i ∈ {1, 3, . . . , r}, and i(λ(β 1 ), λ(β 2 )) = 0. This is possible only if λ(α 1 ) is not adjacent to λ(α 2 ) with respect to λ(A).

Suppose that α 1 is adjacent to α 2 with respect to A. By Lemma 3.8 there exists β ∈ T 0 (N) such that i(β, α 1 ) = 0, i(β, α 2 ) = 0, and i(β, α i ) = 0 for all i ∈ {3, . . . , r}. Since λ is a super-injective simplicial map, we have i(λ(β), λ(α 1 )) = 0, i(λ(β), λ(α 2 )) = 0, and i(λ(β), λ(α i )) = 0 for all i ∈ {3, . . . , r}. This is possible only if λ(α 1 ) is adjacent to λ(α 2 ) with respect to λ(A).

Let P = {α 1 , α 2 , α 3 } be a simplex of T 0 (N) of cardinality 3. We choose pairwise disjoint representatives a 1 , a 2 , a 3 of α 1 , α 2 , α 3 , respectively. We say that P is a 3simpants if there exists a subsurface P of N homeomorphic to S 0,3 whose boundary components are a 1 , a 2 , a 3 . Let P = {α 1 , α 2 } be a simplex of T 0 (N) of cardinality 2.

We choose disjoint representatives a 1 , a 2 of α 1 , α 2 , respectively. We say that P is a 2-simpants if there exists a subsurface P of N homeomorphic to S 0,3 whose boundary components are a 1 , a 2 and a boundary component of N . We say that P is a simpskirt if there exists a subsurface P of N homeomorphic to N 1,2 whose boundary components are a 1 , a 2 . Let β ∈ T 0 (N). We say that β bounds a torus if N β has two connected components, one of which, N , is a one-holed torus. In this case the elements of T (N ) are called interior classes of β .

Lemma 3.10 (1) If P is a 3-simpants and N P has two connected components, then λ(P) is a 3-simpants.

(2) If α ∈ T 0 (N) is non-separating and N α is non-orientable, then λ(α) is nonseparating.

(3) If β bounds a torus and α is an interior class of β , then λ(β) bounds a torus and λ(α) is an interior class of λ(β).

Proof We take a subsurface K of N homeomorphic to S 1,2 endowed with the configuration of circles drawn in Figure 3.3 (i). We assume that a 3 , a 4 are essential circles and [a 3 ], [a 4 ] ∈ T 0 (N). We set α i = [a i ] for all i ∈ {1, 2, 3, 4} and

β 2 = [b 2 ].
The following claims are easy to prove.

Claim 1. Suppose that P = {γ 1 , γ 2 , γ 3 } is a 3-simpants and that N P has two connected components. Then γ 1 , γ 2 , γ 3 are non-separating and at most one of the surfaces N γi is orientable. Moreover, up to renumbering γ 1 , γ 2 , γ 3 , we can choose K so that γ i = α i for all i ∈ {1, 2, 3}. In fact, the only constrain we have is that, if N γi is orientable, then γ i = γ 3 = α 3 .

Claim 2. If α is a non-separating class such that N α is non-orientable, then we can choose K so that α = α 1 .

Claim 3. If β bounds a torus and α is an interior class of β , then we can choose K so that β = β 2 and α = α 1 . We complete {α 1 , α 2 , α 3 , α 4 } in a simplex A = {α 1 , . . . , α r } of T 0 (N) of cardinality r = rk(N). Let B = {α 1 , β 2 , α 3 , α 4 , . . . , α r }. Then B is also a simplex of T 0 (N) of cardinality r = rk(N). We choose minimally intersecting representatives a i ∈ λ(α i ) for all i ∈ {1, . . . , r} and b 2 ∈ λ(β 2 ). By Lemma 3.9 the elements of λ(B) that are adjacent to λ(β 2 ) with respect to λ(B) are exactly λ(α 1 ), λ(α 3 ), λ(α 4 ), the class λ(α 1 ) is adjacent only to λ(β 2 ) with respect to λ(B), and the class λ(α 3 ) is adjacent to λ(α 4 ) with respect to λ(B). It follows that we have one of the following alternatives.

a 1 a 2 a 3 a 4 b 2 a 1 b 2 a 3 a 4 x 1 x 2 x 3 ' ' ' ' a 1 a 3 a 4 b 2 ' ' ' ' a 1 a 2 a 3 a 4 ' ' ' ' (i) (ii) (iii) (iv)
• There exists a subsurface K of N homeomorphic to S 0,5 endowed with the configuration of circles drawn in Figure 3.3 (ii), where each circle x i either is a boundary component of N , or bounds a Möbius band.

• There exists a subsurface K of N homeomorphic to S 1,2 endowed with the configuration of circles drawn in Figure 3.3 (iii).

Since i(λ(α 2 ), λ(α i )) = 0 for all i ∈ {1, 3, . . . , r} and i(λ(α 2 ), λ(β 2 )) = 0, the circle a 2 must lie in K . By Lemma 3.9, λ(α 1 ) is adjacent to λ(α 3 ) and λ(α 4 ) with respect to λ(A), hence a 1 , a 3 , a 4 are included in the same connected component of K \ a 2 . Such a circle a 2 does not exist in the configuration of The next lemma can be proved in the same way as Irmak-Paris [13, Lemma 2.7] (see also Ivanov [15,Lemma 1]), so we do not give any poof.

Lemma 3.11 Let α 1 , α 2 ∈ T 0 (N). If i(α 1 , α 2 ) = 1, then i(λ(α 1 ), λ(α 2 )) = 1.
Lemma 3.12 Suppose that ρ is even.

(1) Let P be a 2-simpants such that N P has two connected components and the connected component of N P non-homeomorphic to S 0,3 is non-orientable. Then λ(P) is a 2-simpants.

(2) Let P be a simpskirt such that N P has two connected components. Then λ(P) is a simpskirt.

Proof Let P = {α 1 , α 2 } be a 2-simpants such that N P has two connected components and the connected component of N P non-homeomorphic to S 0,3 is non-orientable. We take a subsurface K of N homeomorphic to S 1,3 endowed with the configuration of circles drawn in Figure 3.4 (i). In this configuration a 1 , a 2 , a 3 , a 4 , a 5 are essential circles that represent elements of T 0 (N) and x is a boundary component of N . We can and do choose the configuration so that α

c 1 c 3 c ρ-3 a d 1 d 2 d n-1 e 0 (i) (ii) a 1 a 2 a 3 a ρ-3 b 1 b 3 b ρ-3 c 1 c 3 c ρ-3 a' d 1 d 2 d n-1 x 1 x 2
x n e' Now we take a simpskirt P such that N P has two connected components and we show that λ(P) is a simpskirt. We assume that n ≥ 1. The case n = 0 can be proved in the same way. Consider the configuration of circles in N drawn in Figure 3.4 (ii). We can and do assume that

P = {[a], [b 1 ]}. We choose a ∈ λ([a]), a 1 ∈ λ([a 1 ]), . . . , a ρ-3 ∈ λ([a ρ-3 ]), b 1 ∈ λ([b 1 ]), . . . , b ρ-3 ∈ λ([b ρ-3 ]), c 1 ∈ λ([c 1 ]), . . . , c ρ-3 ∈ λ([c ρ-3 ]), d 1 ∈ λ([d 1 ]), . . . , d n-1 ∈ λ([d n-1 ]
) that minimally intersect. By Lemma 3.10, Lemma 3.11, and the above, there exists an orientable subsurface M of N of genus ρ-2 2 endowed with the configuration of circles drawn in Figure 3.4 (iii), where x 1 , . . . , x n are the boundary components of N . For topological reasons, the circle e bounds a one-holed Klein bottle K in N . This implies that either y 1 is isotopic to y 2 , or y 1 , y 2 both bound Möbius bands. By Lemma 2.10 ( 2 (1) Let P be a 2-simpants such that N P has two connected components. Then λ(P) is a 2-simpants.

(2) Let P be a simpskirt such that N P has two connected components. Then λ(P) is a simpskirt.

Proof Let P = {α 1 , α 2 } be a simplex of T 0 (N) of cardinality 2. We suppose that P is either a 2-simpants or a simpskirt, and that N P has two connected components. We take a subsurface K of N homeomorphic to S 1,3 endowed with the configuration of circles drawn in Figure 3.5 (i). In this configuration a 1 , a 2 , a 3 , a 4 , a 5 are essential circles that represent elements of T 0 (N), the circle x is a boundary component of N if P is a 2-simpants, and x bounds a Möbius band if P is a simpskirt. We can and do choose the configuration so that α 1 = [a 1 ] and α 2 = [a 2 ]. We set α i = [a i ] for i ∈ {3, 4, 5} and we complete {α 1 , . . . , α 5 } in a simplex A = {α 1 , . . . , α r } of T 0 (N) of cardinality r = rk(N). We set β i = λ(α i ) and we choose b i ∈ β i such that b 1 , . . . , b r are pairwise disjoint. We denote by N λ(A) the natural compactification of N \ (∪ r i=1 b i ) and by π λ(A) : N λ(A) → N the gluing map. We denote by b i and b i the two boundary components of N λ(A) that are sent to b i under π λ(A) . By Lemma 3.9, β 3 is adjacent to β 1 , β 2 , β 4 , β 5 with respect to λ(A) and β 1 is adjacent to β 2 but not to β 3 and β 4 with respect to λ(A). It follows that, up to replacing b i by b i , there exists a connected component P 1 (resp. P 2 ) of N λ(A) which is a pair of pants and whose boundary components are b 1 , b 2 , b 3 (resp. b 3 , b 4 , b 5 ). Let P 3 (resp. P 4 ) be the connected component of N λ(A) containing b 1 (resp. b 2 ) in its boundary. By Lemma 3.10, β 1 is non-separating and by Lemma 3.9 the only elements of λ(A) adjacent to β 1 with respect to λ(A) are β 2 , β 3 . This implies that P 3 = P 4 and either P 3 is a pair of pants and its third boundary component is sent to a boundary component of N under π λ(A) , or P 3 is homeomorphic to N 1,2 . We conclude that either λ(P) is a 2-simpants, or λ(P) is a simpskirt. Suppose that P is a simpskirt and that the connected component of N P non-homeomorphic to N 1,2 is non-orientable. Let K be a subsurface of N homeomorphic to N 2,2 endowed with the configuration of circles drawn in Figure 3.5 (ii). We suppose that the complement of K in N is connected. In particular, a 1 , a 2 , a 3 , b 1 , b 2 are nonseparating, hence they represent elements of T 0 (N). We can and do choose K such that

a 1 a 2 a 3 a 4 a 5 x a 1 a 3 a 2 b 1 b 2 a 4 a 5 a 1 a 2 a 3 b 1 b 2 (i) (ii) (iii)
α 1 = [a 1 ] and α 2 = [a 2 ]. We set α 3 = [a 3 ], β 1 = [b 1 ], and β 2 = [b 2 ]
. We choose minimally intersecting representatives a i ∈ λ(α i ) for i ∈ {1, 2, 3} and b j ∈ λ(β j ) for j ∈ {1, 2}. By Lemma 3.10, {λ(α 1 ), λ(β 1 ), λ(β 2 )} and {λ(α 3 ), λ(β 1 ), λ(β 2 )} are 3-simpants. It follows that there exists a subsurface K 1 of N homeomorphic to S 1,2 or N 2,2 , whose boundary components are a 1 and a 3 , and which contains b 1 and b 2 in its interior. On the other hand, by the above, each of {λ(α 1 ), λ(α 2 )} and {λ(α 2 ), λ(α 3 )} is either a 2-simpants or a simpskirt. It follows that there exists a subsurface K 2 of N homeomorphic either to S 0,4 , or to N 1,3 , or to N 2,2 , whose boundary components are a 1 , a 3 and possibly boundary components of N , and which contains a 2 in its interior.

Since i(λ(α 2 ), λ(β 2 )) = 0, we have a 2 ∩ b 2 = ∅, hence K 1 = K 2 . We conclude that K 1 = K 2 is homeomorphic to N 2,2 and λ(P) = {λ(α 1 ), λ(α 2 )} is a simpskirt.
Suppose that P = {α 1 , α 2 } is a simpskirt and the connected component of N P nonhomeomorphic to N 1,2 is orientable. Let K be a subsurface of N homeomorphic to N 3,2 endowed with the configuration of circles drawn in Figure 3.5 (iii). We assume that the complement of K in N is connected. In particular, a 1 , a 2 , a 4 , a 5 , b 1 , b 2 are non-separating, and therefore they represent elements of T 0 (N). We choose K so that N [a 3 ] has two connected components, a non-orientable one of genus 3 and an orientable one. In particular, we also have [a 3 ] ∈ T 0 (N). We can and do choose K

such that α 1 = [a 1 ] and α 2 = [a 2 ]. We set α i = [a i ] for i ∈ {3, 4, 5}, β j = [b j ]
for j ∈ {1, 2}, and we complete {α 1 , . . . , α 5 } in a simplex A = {α 1 , . . . , α r } of T 0 (N) of cardinality r = rk(N). We choose minimally intersecting representatives a i ∈ λ(α i ) for i ∈ {1, . . . , r} and b j ∈ λ(β j ) for j ∈ {1, 2}. By the above, {λ(α 4 ), λ(β 1 )}, {λ(β 1 ), λ(β 2 )}, and {λ(β 2 ), λ(α 5 )} are simpskirts. It follows that there exists a subsurface K 1 of N homeomorphic to N 3,2 whose boundary components are a 4 , a 5 and which contains b 1 , b 2 in its interior. By Lemma 3.9, λ(α 3 ) is adjacent to λ(α 1 ), λ(α 2 ), λ(α 4 ), λ(α 5 ) with respect to λ(A), hence there exists a subsurface R of N homeomorphic to S 0,4 whose boundary components are a 1 , a 2 , a 4 , a 5 and which contains a 3 in its interior. Recall that we know that {λ(α 1 ), λ(α 2 )} is either a 2-simpants or a simpskirt. If {λ(α 1 ), λ(α 2 )} was a 2-simpants, then there would exist a subsurface K 2 of N homeomorhic to S 1,3 or N 2,3 , containing R , and whose boundary components are a 4 , a 5 and a boundary component of N . Since i(α 3 , β 1 ) = 0, we would have

a 3 ∩ b 1 = ∅, hence K 1 = K 2 : contradiction. So, {λ(α 1 ), λ(α 2 )} is a simpskirt.
Suppose now that P = {α 1 , α 2 } is a 2-simpants. We know that λ(P) is a simpskirt or a 2-simpants. We complete P = {α 1 , α 2 } in a simplex A = {α 1 , . . . , α r } of T 0 (N) of cardinality r = rk(N). We can and do choose A so that P = {α 2 , α 3 } is a simpskirt and N P has two connected components. By Lemma 3.5 there is at most one pair {λ(α i ), λ(α j )} in λ(A) which is a simpskirt. By the above the pair {λ(α 2 ), λ(α 3 )} is a simpskirt, hence {λ(α 1 ), λ(α 2 )} is a 2-simpants.

Proof of Theorem 3.1

In this subsection G denotes a finite index subgroup of M(N) and ϕ : G → M(N) an injective homomorphism. Let µ : T 0 (N) → T 0 (N) be the super-injective simplicial map induced by ϕ (see Proposition 3.4).

The following lemma is well-known for orientable surfaces (see Castel [7, Proposition 2.1.3] for example). It can be proved for non-orientable surfaces in the same way as for orientable surfaces.

Lemma 3.14 Let {x 1 , . . . , x }, {y 1 , . . . , y } be two collections of essential circles such that:

(a) x 1 , . . . , x (resp. y 1 , . . . , y ) are pairwise non-isotopic and they minimally intersect;

(b) x i is isotopic to y i for all i ∈ {1, . . . , };

(c) there exist no three indices i, j, k ∈ {1, . . . , } such that i(

[x i ], [x j ]) = 0, i([x j ], [x k ]) = 0, and i([x k ], [x i ]) = 0.
Then there exists a homeomorphism H : N → N isotopic to the identity such that H(x i ) = y i for all i ∈ {1, . . . , }.

Lemma 3.15 Let γ ∈ T 1 (N). Suppose that one of the connected components of N γ is (non-orientable) of genus 1. Then there exist γ ∈ T 1 (N) and z, z ∈ Z \ {0} such that t z γ ∈ G and ϕ(t z γ ) = t z γ .

Proof Consider the configuration of circles in N drawn in Figure 3.6. We set

α i = [a i ] for i ∈ {1, . . . , ρ -2}, β = [b], γ j = [c j ] for j ∈ {1, . . . , n}, δ j = [d j ]
for j ∈ {1, . . . , n}, and j = [e j ] for j ∈ {2, . . . , n}. We can and do choose this configuration so that γ = γ j for some j ∈ {1, . . . , n}. By the results of Subsection 3.2, there exists a homeomorphism H :

N → N such that [H(u)] = µ([u]
) for all u ∈ {a 1 , . . . , a ρ-2 , b, d 1 , . . . , d n }. We show by induction on j that [H(e j )] = µ( j ) for all j ∈ {2, . . . , n}. The class µ( 2 ) is the unique element of T 0 (N) that satisfies i(µ( 2 ), η) = 0 for all η ∈ {µ(α 1 ), . . . , µ(α ρ-2 ), µ(β), µ(δ 1 ), µ(δ 3 ), . . . , µ(δ n )}.

Since [H(e 2 )] also satisfies this property, we have µ( 2 ) = [H(e 2 )]. We assume that 3 ≤ j ≤ n and µ( k ) = [H(e k )] for all k ∈ {2, . . . , j -1}. The class µ( j ) is the unique element of T 0 (N) that satisfies i(µ( j ), η) = 0 and µ( j ) = η for all η ∈ {µ(α 1 ), . . . , µ(α ρ-2 ), µ(β), µ(δ 1 ), µ(δ j+1 ), . . . , µ(δ n ), µ( 2 ), . . . , µ( j-1 )}. Since [H(e j )] also satisfies this property, we have [H(e j )] = µ( j ). We set γ j = [H(c j )] and we show by induction on j that there exist z j , z j ∈ Z \ {0} such that t z j γ j ∈ G and ϕ(t

z j γ j ) = t z j γ j
. Here and in the other proofs we use the wellknown fact (which is a consequence of Lemma 2.4 and Lemma 2.5) that, if an element f ∈ M(N) commutes with a non-trivial power of a Dehn twist t α , then f (α) = α.

Suppose that j = 1. Let u ∈ Z \ {0} such that t u γ 1 ∈ G . Set f = ϕ(t u γ 1 )
. The element t u γ 1 commutes with every power of t η for all η ∈ C = {α 1 , . . . , α ρ-2 , β, δ 2 , . . . , δ n }. By Proposition 3.4 it follows that f commutes with a non-trivial power of t η for all η ∈ µ(C), hence f (η ) = η for all η ∈ µ(C). In particular each element of µ(C) is a reduction class for f . None of them is an essential reduction class since i(µ(α i ), µ(β)) = 0 for all i ∈ {1, . . . , ρ -2} and i(µ(β), µ(δ j )) = 0 for all j ∈ {2, . . . , n}. By Lemma 3.14, f admits a representative F : N → N such that F(H(a 1 )) = H(a 1 ), F(H(b)) = H(b), and F(H(d 2 )) = H(d 2 ). Take a closed regular neighborhood U of H(a 1 ) ∪ H(b) ∪ H(d 2 ) such that H(c 1 ) is a boundary component of U . Observe that there is a unique connected component K of N \ U homeomorphic to N 1,2 . We have ∂K ∩ ∂U = H(c 1 ), hence F(H(c 1 )) is isotopic to H(c 1 ), that is, f (γ 1 ) = γ 1 . Let A = {µ(α 1 ), . . . , µ(α ρ-2 ), µ(δ 2 ), . . . , µ(δ n ), γ 1 }. Note that A is a simplex of T (N) and f (η ) = η for all η ∈ A. Let Λ A : M A (N) → M(N A ) be the reduction homomorphism along A. Each connected component of N A is homeomorphic to S 0,3 or N 1,2 hence, by Lemma 2.10 (1), M(N A ) is finite. It follows that there exists v ∈ Z \ {0} such that f v ∈ Z A = Ker Λ A . By the above the only element of A which can be an essential reduction class for f v is γ 1 , hence, by Lemma 2.5, f v is a power of t γ 1 . We set z 1 = uv. Then there exists z 1 ∈ Z \ {0} such that

ϕ(t z 1 γ 1 ) = t z 1 γ 1 .
We assume that j ≥ 2 plus the inductive hypothesis. Let u ∈ Z \ {0} such that t u γ j ∈ G . Set f = ϕ(t u γ j ). The element t u γ j commutes with every power of t η for all η ∈ {α 1 , . . . , α ρ-2 , β, δ j+1 , . . . , δ n , 2 , . . . , j , γ 1 , . . . , γ j-1 }. By Proposition 3.4 and the inductive hypothesis it follows that f commutes with a non-trivial power of t η for all η ∈ C = {µ(α 1 ), . . . , µ(α ρ-2 ), µ(β), µ(δ j+1 ), . . . , µ(δ n ), µ( 2 ), . . . , µ( j ), γ 1 , . . . , γ j-1 }, hence f (η ) = η for all η ∈ C . In particular, each element of C is a reduction class for f . None of them is an essential reduction class since i(µ(α i ), µ(β)) = 0 for all i ∈ {1, . . . , ρ-2}, i(µ(β), µ(δ k )) = 0 for all k ∈ {j+1, . . . , n}, and i(µ( k ), γ k-1 ) = 0 for all k ∈ {2, . . . , j}. By Lemma 3.14, f admits a representative F : andF(H(d j+1 )) = H(d j+1 ). Take a closed regular neighborhood U of H(a 1 )∪H(b)∪H(d j+1 ) such that H(c j ) is a boundary component of U . Observe that there is a unique connected component K of N \ U homeomorphic to N 1,j+1 . We have ∂K ∩ ∂U = H(c j ), hence F(H(c j )) is isotopic to H(c j ), that is, f (γ j ) = γ j . Let A = {µ(α 1 ), . . . , µ(α ρ-2 ), µ(δ j+1 ), . . . , µ(δ n ), γ 1 , . . . , γ j-1 , γ j }. Note that A is a simplex of T (N) and f (η ) = η for all η ∈ A. Let Λ A : M A (N) → M(N A ) be the reduction homomorphism along A. Every connected component of N A is homeomorphic to S 0,3 or N 1,2 hence, by Lemma 2.10 (1), M(N A ) is finite. It follows that there exists v ∈ Z \ {0} such that f v ∈ Z A = Ker Λ A . By the above the only element of A which can be an essential reduction class for f v is γ j , hence, by Lemma 2.5, f v is a power of t γ j . We set z j = uv. Then there exists z j ∈ Z \ {0} such that ϕ(t

N → N such that F(H(a 1 )) = H(a 1 ), F(H(b)) = H(b),
z j γ j ) = t z j γ j . Lemma 3.16 Let γ ∈ T 1 (N). Assume that both connected components of N γ are (non-orientable) of genus ≥ 3. Then there exist γ ∈ T 1 (N) and z, z ∈ Z \ {0} such that t z γ ∈ G and ϕ(t z γ ) = t z γ .
Proof Consider the configuration of circles in N drawn in Figure 3.7. We can and do choose this configuration so that γ

= [c]. We set α i = [a i ] for i ∈ {1, . . . , ρ -2}, β i = [b i ] for i ∈ {1, 2
, z ∈ Z \ {0} such that t z γ ∈ G and ϕ(t z γ ) = t z γ . Let u ∈ Z \ {0} such that t u γ ∈ G . Set f = ϕ(t u γ )
. The element t u γ commutes with every power of t η for all η ∈ C = {α 1 , . . . , α ρ-2 , β 1 , β 3 , δ 1 , . . . , δ n }. By Proposition 3.4 it follows that f commutes with a non-trivial power of t η for all η ∈ µ(C), hence f (η ) = η for all η ∈ µ(C). In particular each element of µ(C) is a reduction class for f . None of them is an essential reduction class since i(µ(α i ), µ(β 1 )) = 0 for all i ∈ {1, . . . , p}, i(µ(α i ), µ(β 3 )) = 0 for all i ∈ {p + 1, . . . , ρ -2}, i(µ(δ j ), µ(β 1 )) = 0 for all j ∈ {1, . . . , q}, and i(µ(δ j ), µ(β 3 )) = 0 for all j ∈ {q + 1, . . . , n}. By Lemma 3.14, f admits a representative F : N → N such that F(H(a i )) = H(a i ) for all i ∈ {1, . . . , p}, F(H(b 1 )) = H(b 1 ), and F(H(d j )) = H(d j ) for all j ∈ {1, . . . , q}. Take a closed regular neighborhood U of H(a 1 ) ∪

• • • ∪ H(a p ) ∪ H(b 1 ) ∪ H(d 1 ) ∪ • • • ∪ H(d q ) such that H(c) is a boundary component of U . Observe that there is a unique connected component K of N \ U non-orientable of genus ≥ 3. We have ∂K ∩ ∂U = H(c), hence F(H(c)) is isotopic to H(c), that is, f (γ ) = γ . Let A = {µ(α 1 ), . . . , µ(α ρ-2 ), µ(δ 1 ), . . . , µ(δ n ), γ }. Note that A is a simplex of T (N) and f (η ) = η for all η ∈ A. Let Λ A : M A (N) → M(N A ) be the reduction homomorphism along A. Every connected component of N A is homeomorphic to S 0,3 or N 1,2 hence, by Lemma 2.10 (1), M(N A ) is finite. It follows that there exists v ∈ Z \ {0} such that f v ∈ Z A = Ker Λ A .
By the above the only element of A which can be an essential reduction class for f v is γ , hence, by Lemma 2.5, f v is a power of t γ . We set z = uv. Then there exists z ∈ Z \ {0} such that ϕ(t z γ ) = t z γ .

Lemma 3.17 Let γ ∈ T 2 (N) and let γ 0 be the interior class of γ . Then there exist γ ∈ T 2 (N), z, z ∈ Z \ {0} and z 0 ∈ Z such that µ(γ 0 ) is the interior class of γ ,

t z γ ∈ G , and ϕ(t z γ ) = t z γ t z 0 µ(γ 0 ) . a 1 a 2 a p-1 a p a p+1 a ρ-3 a ρ-2 b 1 b 2 b 3 d 1 d q-1 d q d q+1 d n c Figure 3.7: Circles in N (Lemma 3.16)
Proof We suppose that the connected component of N γ non-homeomorphic to N 2,1 is non-orientable. The case where this connected component is orientable can be proved in a similar way. We consider the configuration of circles in N drawn in Figure 3.8. We can and do choose this configuration so that γ = [c]. We set

α i = [a i ] for i ∈ {1, . . . , ρ -1}, β = [b], γ 0 = [c 0 ]
, and δ j = [d j ] for j ∈ {1, . . . , n -1}. By the results of Subsection 3.2, there exists a homeomorphism

H : N → N such that [H(u)] = µ([u]) for all u ∈ {a 1 , . . . , a ρ-1 , b, d 1 , . . . , d n-1 }.
We set γ = [H(c)]. Note that γ ∈ T 2 (N) and γ 0 = [H(c 0 )] is the interior class of γ . It is easily seen that γ 0 is the unique element of T 0 (N) that satisfies i(γ 0 , η) = 0 for all η ∈ {µ(α 1 ), µ(α 3 ), . . . , µ(α ρ-1 ), µ(β), µ(δ 1 ), . . . , µ(δ n-1 )}. Since µ(γ 0 ) also satisfies this property, we have µ(γ 0 ) = γ 0 . Let u ∈ Z \ {0} such that t u γ ∈ G . We set f = ϕ(t u γ ). The element t u γ commutes with every power of t η for all η ∈ C = {α 1 , α 3 , . . . , α ρ-1 , β, δ 1 , . . . , δ n-1 }. By Proposition 3.4 it follows that f commutes with a non-trivial power of t η for all η ∈ µ(C), hence f (η ) = η for all η ∈ µ(C). In particular every element of µ(C) is a reduction class for f . None of them is an essential reduction class since i(µ(α i ), µ(β)) = 0 for all i ∈ {1, 3, . . . , ρ -1} and i(µ(δ j ), µ(β)) = 0 for all j ∈ {1, . . . , n -1}. We have f (γ 0 ) = γ 0 for the same reason (but we do not know if γ 0 is an essential reduction class for f ). By Lemma 3.14, f admits a representative F :

N → N such that F(H(a i )) = H(a i ) for all i ∈ {1, 3, . . . , ρ-1}, F(H(b)) = H(b), and F(H(d j )) = H(d j ) for all j ∈ {1, . . . , n-1}. Take a closed regular neighborhood U of H(a 1 ) ∪ H(a 3 ) ∪ • • • ∪ H(a ρ-1 ) ∪ H(b) ∪ H(d 1 ) ∪ • • • ∪ H(d n-1 ) such that H(c) is a boundary component of U . Observe that there is a unique connected component K of N \ U homeomorphic to N 2,1 , and the boundary component of K is H(c). So, F(H(c)) is isotopic to H(c), that is, f (γ ) = γ . Let A = {µ(α 1 )
, µ(α 3 ), . . . , µ(α ρ-1 ), µ(δ 1 ), . . . , µ(δ n-1 ), γ 0 , γ }. Note that A is a simplex of T (N) and f (η ) = η for all η ∈ A. Let Λ A : M A (N) → M(N A ) be the reduction homomorphism along A. All the connected components of N A are homeomorphic to S 0,3 or N 1,2 hence, by Lemma 2.10 (1), M(N A ) is finite. It follows that there exists v ∈ Z \ {0} such that f v ∈ Z A = Ker Λ A . By the above the only elements of A that can be essential reduction classes for f v are γ and γ 0 , hence, by Lemma 2.5, f v ∈ t γ , t γ 0 . We set z = uv. Then there exist z , z 0 ∈ Z such that Proof of Theorem 3.1 Let α ∈ T (N). If α ∈ T 0 (N), then we set λ(α) = µ(α). Suppose that α ∈ T 1 (N). By Lemma 3.15 and Lemma 3.16 there exist α ∈ T 1 (N) and x, x ∈ Z \ {0} such that t x α ∈ G and ϕ(t x α ) = t x α . We choose such a α and we set λ(α) = α . Suppose that α ∈ T 2 (N). Let α 0 be the interior class of α. By Lemma 3.17 there exist α ∈ T 2 (N), x, x ∈ Z \ {0}, and y ∈ Z, such that t x α ∈ G , µ(α 0 ) is the interior class of α , and ϕ(t x α ) = t x α t y µ(α 0 ) . We choose such a α and we set λ(α) = α .

ϕ(t z γ ) = t z γ t z 0 γ 0 . a 1 a 2 a 3 a 4 a ρ-2 a ρ-1 d 1 d 2 d n
We need to prove that λ : T (N) → T (N) is a super-injective simplicial map. To do that we take two distinct classes α, β ∈ T (N) and we show that i(α, β) = 0 if and only if i(λ(α), λ(β)) = 0. It is easily shown using the same arguments as in the proof of Proposition 3.4 that this is true if α, β ∈ T 0 (N) ∪ T 1 (N). To prove the other cases we will use the following claims.

Claim 1. Let γ ∈ T 2 (N), let γ 0 be the interior class of γ , and let δ ∈ T (N). If i(δ, γ) = 0, then i(δ, γ 0 ) = 0.

Proof of Claim 1. Suppose that i(δ, γ) = 0 and i(δ, γ 0 ) = 0. Let N 1 be the connected component of N γ homeomorphic to N 2,1 . Since i(δ, γ) = 0 and i(δ, γ 0 ) = 0, we should have δ ∈ T (N 1 ) = {γ 0 }, hence δ = γ 0 , and therefore i(δ, γ 0 ) = 0: contradiction. This proves Claim 1.

Claim 2. Let γ ∈ T 2 (N) and let γ 0 be the interior class of γ . There exist z, z ∈ Z \ {0} and q ∈ Z such that t z γ t q γ 0 ∈ G and ϕ(t z γ t q γ 0 ) = t z λ(γ) .

Proof of Claim 2. Let x, x ∈ Z \ {0} and p ∈ Z such that ϕ(t x γ ) = t x λ(γ) t p λ(γ 0 ) . Let y, y ∈ Z \ {0} such that t y γ 0 ∈ G and ϕ(t y γ 0 ) = t y λ(γ 0 ) . Then t xy γ t -p y γ 0 ∈ G and ϕ(t xy γ t -p y γ 0 ) = t x y λ(γ) . This proves Claim 2. Suppose that α ∈ T 0 (N) ∪ T 1 (N) and β ∈ T 2 (N). Let β 0 be the interior class of β . We choose x, x , y, y , z, z ∈ Z\{0} and p , q ∈ Z such that t x α , t y β , t z β t q β 0 ∈ G , ϕ(t x α ) = t x λ(α) , ϕ(t y β ) = t y λ(β) t p λ(β 0 ) , and ϕ(t z β t q β 0 ) = t z λ(β) . Suppose that i(λ(α), λ(β)) = 0. By Claim 1 we have i(λ(α), λ(β 0 )) = 0. Then ϕ(t x α ) = t x λ(α) and ϕ(t y β ) = t y λ(β) t p λ(β 0 ) commute, hence t x α and t y β commute, and therefore i(α, β) = 0. Suppose that i(α, β) = 0. By Claim 1 we have i(α, β 0 ) = 0. Then t x α and t z β t q β 0 commute, hence ϕ(t x α ) = t x λ(α) and ϕ(t z β t q β 0 ) = t z λ(β) commute, and therefore i(λ(α), λ(β)) = 0. Suppose that α, β ∈ T 2 (N). Let α 0 be the interior class of α and let β 0 be the interior class of β . Let x, x , y, y , z, z , u, u ∈ Z \ {0} and p , q, r , s ∈ Z such that t x α , t y α t q α 0 , t z β , t u β t s β 0 ∈ G , ϕ(t x α ) = t x λ(α) t p λ(α 0 ) , ϕ(t y α t q α 0 ) = t y λ(α) , ϕ(t z β ) = t z λ(β) t r λ(β 0 ) , and ϕ(t u β t s β 0 ) = t u λ(β) . Suppose that i(λ(α), λ(β)) = 0. By Claim 1 we have i(λ(α 0 ), λ(β)) = i(λ(α), λ(β 0 )) = 0, hence, again by Claim 1, i(λ(α 0 ), λ(β 0 )) = 0. Then ϕ(t x α ) = t x λ(α) t p λ(α 0 ) and ϕ(t z β ) = t z λ(β) t r λ(β 0 ) commute, hence t x α and t z β commute, and therefore i(α, β) = 0. Suppose that i(α, β) = 0. By Claim 1 we have i(α 0 , β) = i(α, β 0 ) = 0, hence, again by Claim 1, i(α 0 , β 0 ) = 0. Then t y α t q α 0 and t u β t s β 0 commute, hence t y λ(α) = ϕ(t y α t q α 0 ) and t u λ(β) = ϕ(t u β t s β 0 ) commute, and therefore i(λ(α), λ(β)) = 0.

4 Proofs of Theorem 1.1 and Corollary 1.3 Proof of Theorem 1.1 Let G be a finite index subgroup of M(N) and let ϕ : G → M(N) be an injective homomorphism. We denote by λ : T (N) → T (N) the superinjective simplicial map induced by ϕ (see Theorem 3.1). We know by Irmak-Paris [13, Theorem 1.1] that there exists f 0 ∈ M(N) such that λ(α) = f 0 (α) for all α ∈ T (N). We turn to show that ϕ(g) = f 0 gf -1 0 for all g ∈ G .

We say that a collection C ⊂ T (N) has trivial stabilizer if the only element f in M(N) that satisfies f (α) = α for all α ∈ C is f = id. It is shown in Irmak-Paris [13, Lemma 3.6, Lemma 3.7] that there exists a finite collection C ⊂ T 0 (N) which has trivial stabilizer. Take such a collection C . Let g ∈ G . Let α ∈ C . Since α, g(α) ∈ T 0 (N), by Theorem 3.1, there exist x, x , y, y ∈ Z \ {0} such that t x α , t y g(α) ∈ G , ϕ(t x α ) = t x λ(α) , and ϕ(t y g(α) ) = t y λ(g(α)) . We have t x y (ϕ(g)f 0 )(α) = ϕ(g)t x y f 0 (α) ϕ(g) -1 = ϕ(g)t x y λ(α) ϕ(g) -1 = ϕ(g)ϕ(t xy α )ϕ(g) -1 = ϕ(gt xy α g -1 ) = ϕ(t xy g(α) ) = t xy λ(g(α)) = t xy (f 0 g)(α) .

So, (ϕ(g)f 0 )(α) = (f 0 g)(α), that is, (g -1 f -1 0 ϕ(g)f 0 )(α) = α. Since C has trivial stabilizer, it follows that g -1 f -1 0 ϕ(g)f 0 = id, hence ϕ(g) = f 0 gf -1 0 .

Proof of Corollary 1.3 For f ∈ M(N) we denote by c f : M(N) → M(N), g → fgf -1 , the conjugation by f . We have an obvious homomorphism Ψ : M(N) → Com(M(N)) which sends f to the class of (M(N), M(N), c f ). Theorem 1.1 clearly implies that Ψ is surjective, and the following lemma implies that Ψ is injective. Proof As in the proof of Theorem 1.1, we may choose a finite collection C ⊂ T 0 (N) having trivial stabilizer. Let g be an element in the centralizer of G . Let α ∈ C . Take x ∈ Z \ {0} such that t x α ∈ G . Then t x α = gt x α g -1 = t x g(α) , hence g(α) = α. Since C has trivial stabilizer, we conclude that g = id.

Theorem 3 . 1

 31 Let G be a finite index subgroup of M(N) and let ϕ : G → M(N) be an injective homomorphism. There exists a super-injective simplicial map λ : T (N) → T (N) satisfying the following properties. (a) Let α ∈ T (N) which does not bound a Klein bottle. There exist k, ∈ Z \ {0} such that t k α ∈ G and ϕ(t k α ) = t λ(α) . (b) Let α ∈ T (N) which bounds a Klein bottle and let α 0 be its interior class. The class λ(α) bounds a Klein bottle, λ(α 0 ) is the interior class of λ(α), and there exist k, ∈ Z \ {0} and 0 ∈ Z such that t k α ∈ G and ϕ(t k α ) = t λ(α) t 0 λ(α 0 ) . Remark It will follow from Theorem 1.1 that 0 = 0 in Part (b) of the above theorem.

Figure 3 . 1 :

 31 Figure 3.1: Circles in N (Lemma 3.7)

Figure 3 . 1 (

 31 ii). In this figure each x i either is a boundary component of N A , or bounds a Möbius band, and there is at most one x i bounding a Möbius band. Consider the arcs b and b drawn in Figure 3.1 (ii) and set b = π A (b ∪ b ) and β = [b]. Then i(α 1 , β) = 2 and i(α i , β) = 0 for all i ∈ {2, . . . , r}. Moreover, we have β ∈ T 2 (N), since there is at one x i bounding a Möbius band, and we have β ∈ T 1 (N), since {β, α 2 , . . . , α r } is a simplex in T (N) of cardinality r = rk(N) (see Lemma 3.2), hence β ∈ T 0 (N).

Figure 3 .

 3 Figure 3.2 (i). Each boundary component of K either is isotopic to an element of {a 3 , . . . , a r }, or is a boundary component of N , or bounds a Möbius band. Moreover, we have K ∩ a i = ∅ for all i ∈ {3, . . . , r}. Let b be the circle drawn in Figure 3.2 (i) and let β = [b]. Then β ∈ T 0 (N), i(α 1 , β) = 1, i(α 2 , β) = 2, and i(α i , β) = 0 for all i ∈ {3, . . . , r}.

Figure 3 . 2 :

 32 Figure 3.2: Circles in N (Lemma 3.8)

Figure 3 . 3 :

 33 Figure 3.3: Circles in N (Lemma

Figure 3 . 3 (

 33 ii), hence K is homeomorphic to S 1,2 and a 1 , b 2 , a 3 , a 4 are as shown inFigure 3.3 (iii). This shows Part (2) and Part (3) of the lemma. This also shows that, up to homeomorphism, K is endowed with the configuration of circles drawn in Figure3.3 (iv), which shows Part (1).

1 =

 1 [a 1 ] and α 2 = [a 2 ]. We set α i = [a i ] for i ∈ {3, 4, 5} and we complete {α 1 , . . . , α 5 } in a simplex A = {α 1 , . . . , α r } of T 0 (N) of cardinality r = rk(N). We set β i = λ(α i ) and we choose b i ∈ β i such that b 1 , . . . , b r are pairwise disjoint. We denote by N λ(A) the natural compactification of N \ (∪ r i=1 b i ) and by π λ(A) : N λ(A) → N the gluing map. We denote by b i and b i the two components of π -1 λ(A) (b i ). Recall that, by Lemma 3.5, each connected component of N λ(A) is a pair of pants. By Lemma 3.9, β 3 is adjacent to β 1 , β 2 , β 4 , β 5 with respect to λ(A) and β 1 is adjacent to β 2 but not to β 3 and β 4 with respect to λ(A). It follows that, up to replacing b i by b i , there exists a connected component P 1 of N λ(A) whose boundary components are b 1 , b 2 , b 3 and there exists another connected component P 2 of N λ(A) whose boundary components are b 3 , b 4 , b 5 . Let P 3 (resp. P 4 ) be the connected component of N λ(A) containing b 1 (resp. b 2 ) in its boundary. By Lemma 3.10, β 1 is non-separating and by Lemma 3.9 the elements of λ(A) adjacent to β 1 with respect to λ(A) are precisely β 2 , β 3 . This implies that P 3 = P 4 and the third boundary component of P 3 is sent to a boundary component of N under π λ(A) . So, λ(P) is a 2-simpants.

Figure 3 . 4 :

 34 Figure 3.4: Circles in N (Lemma 3.12)

  ), T (K) is a singleton { 0 }. Since i(λ([e 0 ]), [z ]) = 0 for all z ∈ {a 1 , . . . , a ρ-3 , b 1 , . . . , b ρ-3 , c 1 , . . . , c ρ-3 , d 1 , . . . , d n-1 } and i(λ([e 0 ]), [a ]) = 0, we have λ([e 0 ]) ∈ T (K), that is, λ([e 0 ]) = 0 (we cannot have λ([e 0 ]) = [e ] since [e ] ∈ T 0 (N)). If y 1 was isotopic to y 2 , then we would have [y 1 ] = 0 , hence 0 = i([y 1 ], [a ]) = i(λ([e 0 ]), [a ]) = 0: contradiction. So, y 1 , y 2 both bound Möbius bands and therefore λ(P) = {[a ], [b 1 ]} is a simpskirt. Lemma 3.13 Assume that ρ is odd.

Figure 3 . 5 :

 35 Figure 3.5: Circles in N (Lemma 3.13)

Figure 3 . 6 :

 36 Figure 3.6: Circles in N (Lemma 3.15)

  , 3}, and δ j = [d j ] for j ∈ {1, . . . , n}. By the results of Subsection 3.2 there exists a homeomorphism H :N → N such that [H(u)] = µ([u]) for all u ∈ {a 1 , . . . , a ρ-2 , b 1 , b 2 , b 3 , d 1 , . . . , d n }. We set γ = [H(c)]and we turn to show that there exist z

Figure 3 . 8 :

 38 Figure 3.8: Circles in N (Lemma 3.17)

Lemma 4 . 1

 41 Let G be a finite index subgroup of M(N). Then the centralizer of G in M(N) is trivial.

  12, Lemma 2.2]. Proof We choose a pants decomposition A which contains A. Clearly, |A T | ≥ |A T | and |A O | ≥ |A O |. Let p be the number of connected components of N A . Since every connected component of N A is a pair of pants, we have 2|A T| + |A O | + n = 3p and 2 -ρn = χ(N) = -p. These equalities imply that |A T | + 1 2 |A O | = 3 2 ρ + n -3. It follows that |A T |+ 1 2 |A O | ≤3 2 ρ+n-3, and equality holds if and only if A = A . Proof We have A T = A and A O = ∅. By Lemma 2.8, it follows that |A| ≤

	Lemma 2.8 Let A be a simplex of C(N). Set A T = A ∩ T (N) and A O = A \ A T .
	Then	|A T | +	1 2	|A O | ≤	3 2	ρ + n -3 .
	Moreover, equality holds if and only if A is a pants decomposition.
	Corollary 2.9 If A is a simplex of T (N), then |A| ≤ rk(N).
	3 2 ρ + n -3. If ρ is even, then the right hand side of this inequality is rk(N), hence
	|A| ≤ rk(N). If ρ is odd, then the right hand side of this inequality is not an integer,
	but the greatest integer less than the right hand side is rk(N) = 1 2 (3ρ -1) + n -3. So,
	we have |A| ≤ rk(N) in this case, too.

  We denote by Γ(N S ) the set of connected components of N S , by S(Γ(N S )) the group of permutations of Γ(N S ), and by ϕ :M(N S ) → S(Γ(N S )) the natural homomorphism. Note that rk(Λ S (H) ∩ Ker ϕ) = rk(Λ S (H)) since Λ S (H) ∩ Ker ϕ is of finite index in Λ S (H). Let f ∈ Λ S (H)∩Ker ϕ and let N be a connected component of N S . Let f ∈ H such that f = Λ S (f ).The set S is an adequate reduction system for f , hence the restriction of f to N is either pseudo-Anosov or periodic.

	Lemma 2.11 Let M be a connected surface (orientable or non-orientable), and let
	f ∈ M(M) be a pseudo-Anosov element. Then the centralizer of f in M(M) is
	virtually cyclic.
	Proof of Proposition 2.7 Let H be an abelian subgroup of M(N). Set S = S(H) and
	S T = S ∩ T (N). Denote by Λ S : M S (N) → M(N S ) the reduction homomorphism
	along S . Note that H ⊂ M S (N) since S is a reduction system for H . By Proposition
	2.1 we have the following short exact sequence
	1 → H ∩ Z S -→ H -→ Λ S (H) → 1 ,
	hence rk(H) = rk(H ∩ Z S ) + rk(Λ S (H)). Moreover, again by Proposition 2.1, rk(H ∩
	Z S ) ≤ rk(Z S ) = |S T |.
	Moreover, by Lemma 2.10,
	M(N ) does not contain any pseudo-Anosov element if N is homeomorphic to S 0,3
	or N 1,2 , and, by Lemma 2.11, the centralizer of a pseudo-Anosov element of M(N )
	is virtually cyclic. Let N 1 , . . . , N be the connected components of N S that are not
	homeomorphic to S 0,3 or N 1,2 . Then, by the above, rk(Λ S (H)) = rk(Λ S (H)∩Ker ϕ) ≤
	.

Let i ∈ {1, . . . , }. By Lemma 2.10, we have T (N i ) = ∅, hence we can choose a class

  So, t kk α ∈ H ∩ H . Since H ∩ H = f , it follows that there exists ∈ Z \ {0} such that f = t kk α . If β was not an element of S(H), then N would be a connected component of N S(H) and this would contradict Proposition 2.7 (3). Hence β ∈ S(H). Moreover, again by Proposition 2.7, there exists p ≥ 1 such that t p β ∈ H . Similarly, there exists p ≥ 1 such that t p β ∈ H . So, t pp β ∈ H ∩ H . On the other hand, we know that there exist ≥ 1 and u ∈ Z \ {0} such that f = t u α ∈ H ∩ H . Then the elements t u α , t pp β generate a free abelian group of rank 2 lying in H ∩ H Z: contradiction. So, α ∈ T 0 (N).

hence f is of finite order: contradiction. Now we show that there exists α ∈ T (N) such that f ∈ R(α). By the above, we can choose a class α ∈ S(f ) ∩ T (N). By Proposition 2.7 there exists k ≥ 1 such that

Z S(H) [k] ⊂ H , hence t k α ∈ H . Similarly there exists k ≥ 1 such that t k α ∈ H .

Since α ∈ S(H), by Proposition 2.7 there exists a simplex A ⊂ T (N) of cardinality rk(N) containing α. By Lemma 3.2 it follows that α ∈ T 1 (N). Suppose that α ∈ T 2 (N). Then N α has two connected components, one of which, N , is a one-holed Klein bottle. By Lemma 2.10, T (N ) is a singleton {β}.

Now we consider a finite index subgroup G of M(N), a class α ∈ T 0 (N), and an integer k ∈ Z \ {0} such that t k α ∈ G . By Lemma 3.2 there exist two simplices


I Refregiers

Yousef

L Buriankova

Z Nadova

D Jancura

M Refregiers

I Yousef

J Mikes

Pavol Miskovsky
email: pavol.miskovsky@upjs.sk

Synchrotron based

published or not. The documents may come

Introduction

Synchrotron based Fourier-transform infrared micro-spectroscopy (SR-FTIRM) has become a valuable technique for examining the chemical make-up of biological molecules by probing their vibrational motions on a microscopic scale [1]. This technique allows in situ structure determination of the most important biomolecules (nucleic acids, proteins, lipids) in chosen sub-cellular compartments [1,2]. The higher spatial resolution and spectral quality, in comparison with "classic" Fourier-transform infrared microscopy, is due to the fact that a synchrotron IR source is 100-1000 times brighter than a conventional thermal source for IR spectroscopy [3]. The higher brightness (flux density) of synchrotron is due to the small effective source and its very narrow emittance. The high brightness of synchrotron source allows the analysis of smaller region with acceptable signal-to-noise ratio and the spatial resolution reaches the diffraction limit (λ/2) [1,4,5]. Thanks to the high spatial resolution of the SR-FTIR technique, the structure of biomolecules inside cells can be probed with a sub-cellular resolution.

The SR-FTIRM has already been successfully applied in several studies of biological systems. The comprehensive review on this subject was provided by Miller and Dumas [1]. The broad spectrum of biological and medical applications of SR-FTIRM includes determination of the protein structure aggregates in brain tissue of patients with Alzheimer´s disease, study of protein folding, investigation of the bone composition during osteoporosis, observation of the changes in the lipid and collagen content and in the structure of myocardium during heart disease and many others [see review 1]. This technique was also successfully used to study cell cycle, differentiation and proliferation of different cell lines and cell death mode [6][7][START_REF] Jamin | Proc. Natl. Acad. Sci[END_REF][START_REF] Holman | [END_REF][10][11][12].

With respect to the scope of this work, we would like to emphasize that for the study of programmed cell death (apoptosis) not only SR-FTIRM, but also "classic" IR spectroscopy has been relatively widely used [13][14][15][16][17][18][19][20]. Apoptosis is accompanied by changes in the structure of the three main types of biomolecules (nucleic acids, proteins, lipids). Techniques based on IR spectroscopy are able to analyze simultaneously these structural changes in all three types of biomolecules [21].

The main IR spectral features such as modifications of the protein structure determined by shift in the positions of Amide I and Amide II bands, fragmentation of DNA revealed by the alterations of vibrational modes of DNA, and changes of stretching vibrations in CH 2 and CH 3 groups of lipid molecules have been shown to be a consequence of apoptosis. Changes in the position and intensity of Amide I peak are in a good correlation with the change in the percentage of apoptotic population of Jurkat cells [13].

In leukemia cells under apoptosis, it was determined that the protein structure shifts from predominantly β-sheet to unordered coil structure, and that quantification of apoptotic cells correlates very well with the determination of apoptosis by two standard methods (flow-cytometry and DNA ladder formation) [14,15]. Further, it has been shown that significant changes in the IR spectra of apoptotic cells, mainly attributed to conformational changes of DNA, are associated with decreasing amounts of detectable DNA [16]. It was also observed that the apoptotic index is inversely proportional with the spectral area 900-1200 cm -1 assigned to the nucleic acids (NAs) [18]. There is evidence that apoptosis induces a conformational disorder of lipid acyl chains as indicated by a marker band in the IR spectrum of an apoptotic cell corresponding to the exposure of phosphatidylserine on the outer leaflet of the membrane [19]. All these findings strongly show that IR spectroscopy, especially SR-FTIRM can provide unique insights into properties of apoptotic processes. However, it should be pointed out that the above mentioned results are often contradictory. This can be attributed to the fact that in different studies, different types of cell lines are used, and the IR spectra are often obtained from cells at different stages of apoptosis. Furthermore, the mode of the apoptosis induction (drugs, radiation, photodynamic action ) can influence the IR spectra of apoptotic cells.

In present work, we induced apoptosis by photodynamic action caused by illumination of either Hyp or Hyp/LDL complex localized in U-87 MG cells. Hyp (Fig. 1) is a natural photosensitizing pigment occurring in plants of the genus Hypericum [for reviews see [22][23][24]. Previously, it has been clearly demonstrated that under suitable concentration and illumination conditions Hyp is a very effective and potent inducer of apoptosis in many cell lines, including glioma cells such as U-87 MG cells [24][25][26][27].

The reason for the use of Hyp/LDL complex in addition to Hyp alone in this study stems from the following. Upon administration into the blood stream Hyp associates with serum proteins, mainly with LDL. Because of the enhanced expression of specific LDL receptors in many types of transformed cells (cancerous cells) in contrast to nontransformed cells, LDL could play a key role in the targeted delivery of hydrophobic and amphiphilic photosensitizers to tumor cells in photodynamic therapy (PDT) [28]. Our previous studies have shown binding of both monomer and aggregate forms of Hyp to LDL [29][30][31]. It was also demonstrated that over-expression of LDL receptors results in an increased Hyp accumulation within U-87 MG cells, and that a different delivery strategy leads to a different type of cellular response to Hyp photoactivation [32].

The aim of this study is to show that SR-FTIRM spectroscopy, can be successfully used for the detection of the early apoptotic processes in the cells after photodynamic action.

In particular, we show that this technique is capable to distinguish between very subtle differences in cellular response induced by photoactivation of Hyp, delivered to cells alone or complexed with LDL. In the SR-FTIRM spectra, the Amide I band seems to be the most sensitive marker to these subtle differences. The correlation of SR-FTIRM results with the outcome of standard flow cytometry method for the detection of apoptosis is also discussed. and glucose 4500 mgl -1 supplemented with 10% Fetal bovine serum (FBS), 1% penicillin/streptomycin and 1% sodium-pyruvate (control cells). In order to induce an increased expression of LDL receptors in U-87MG cells, the 10% FBS was replaced by 2% Ultroser ® G-serum without lipids in cell media 24 hours before experiment (cells with elevated LDL receptors). The cells were kept at 5% CO 2 atmosphere. To prevent an accidental photoactivation of Hyp, all experiments were done in dark conditions.

Materials and Methods

Chemicals

Sample´s preparation for SR-FTIRM experiments: U-87MG cell line was cultured on

low-e slides at the concentration of 10 5 cells per ml in different cultivation media (10 % FBS, 2% Ultroser ® G). Thereafter cells were incubated with Hyp and/or with Hyp/LDL complex for 3 hours. Cells with non-elevated LDL receptors (10%FBS) were cultured in DMEM in presence of 500 nM Hyp. Cells with elevated LDL receptors (2% Ultroser ® G) were cultured in DMEM in presence of Hyp/LDL=30:1 complex ratio, which corresponds to 500 nM of total concentration of Hyp. Samples were then irradiated with 4J/cm 2 at 600 nm wavelength. At several times after irradiation (4 and 24 hours) cells were washed with PBS (pH=7.2) and then fixed with 3.8% formalin for 20 minutes. After fixation with formalin, cells were washed with water and air dried.

SR-FTIRM experiments:

The experiments were performed at Synchrotron SOLEIL (St.

Aubin, Gif Sur Yvette, France) at the SMIS infrared beamline. The experimental set-up was previously described in detail by P. Dumas, and M. J. Tobin [2]. Briefly, the synchrotron light is collected from a bending magnet in the electron storage ring. The IR beam is extracted by the combination of plane water cooled and ellipsoidal mirror. The IR beam is re-collimated and directed toward the IR microscope. The collimated beam of synchrotron IR light enters the FTIR spectrometer before the IR microscope. The sample area is spectroscopically sampled and the image is built up by raster-scanning specimen through the focused beam. The single element detector is used for this configuration. The microscope is equipped with reflecting Schwarshild-type objective to avoid absorption and chromatic aberration over the large mid-IR-spectral range.

The IR spectra were recorded in reflection mode by the IR Continuum XL microscope coupled to a ThermoNicolet 5700 micro-spectrometer. Each individual cell was selected visually and the IR spectra were taken at the nucleus location of the cell with an aperture of 8×8 µm 2 . Each spectrum was acquired after 100 accumulations with 4 cm -1 spectral resolution. Data acquisition and processing were performed using Omnic software (Version 7.3, Thermo Electron Corporation). Each final IR spectrum is the average of at least of 11 individual cell spectra.

Results and discussion

IR spectra of untreated U-87 MG cells: An average IR spectrum obtained from several U-87 MG cell is shown on Figure 2B. The broad band centered at ~3300 cm -1 comes from O-H stretching and N-H vibrations (proteins, carbohydrates and water). These vibrations are not usually used for the investigation of cell's processes like apoptosis, differentiation, or cell death. Asymmetric stretching of CH 3 groups of lipids is localized at 2962 cm -1 and symmetric one has maximum at 2877 cm -1 . Asymmetric and symmetric modes of CH 2 stretching vibrations appear at 2927 and 2853 cm -1 , respectively. Proteins are characterized by two dominant peaks: Amide I at 1650 cm -1 and Amide II at 1633 cm - 1 . The Amide I band is sensitive to the secondary structure of proteins and corresponds to the vibration of C=O group of the peptide bond with a small contribution of the C-N stretching vibration [33]. The Amide II band arises predominantly from the coupled N-H in-plane bending and C-N stretching vibrations in peptide bond and is not usually used in the conformational analysis because it overlaps with amino acid side chain vibrations.

The middle intense peak at 1240 cm -1 is assigned to the asymmetric phosphate stretching vibrations (υ as PO 2 -) and the band at around 1080 cm -1 results from symmetric phosphates stretching vibrations (υ s PO 2 -) of phosphodiester groups of nucleic acids [START_REF] Mantsch | Infrared Spectroscopy of Biomolecules[END_REF].

IR spectra of U-87 MG cells after photodynamic action induced by Hyp and Hyp/LDL:

Figures 3 and4 represent the proteins contribution region of IR spectra collected from U-87 MG cells treated with either Hyp alone or with Hyp/LDL complex (Hyp/LDL=30:1) measured 4 hours (Fig. 3) and 24 hours (Fig. 4) after illumination. The most significant changes appear in the region of Amide I vibrations. The position of the maximum of Amide I band in the cells treated with Hyp alone (at 4 hrs) is shifted from 1650 cm -1 to 1624 cm -1 (Fig. 3, Table 1) and to 1640 cm -1 (at 24 hrs) (Fig. 4, Table 1).

Similar shifts of the Amide I band to the final position at 1630 cm -1 (at 4 hrs) (Fig. 3, Table 1) and 1644 cm -1 (at 24 hrs) (Fig. 4 Cell survival (staining by PI) and occurrence of apoptotic signs were tested 4 and 24 hours after Hyp photoactivation in U-87 MG cells (4 J/cm -2 ) (Fig. 5 and Table 2).

Annexin V/PI double-negative cells are defined as live cells (Fig. 5, lower left quadrant), Annexin V-positive/PI-negative cells are defined as early apoptotic cells (Fig. 5 2). Fig. 5C shows the distribution of cells incubated with Hyp/LDL complex. In this case, almost half of cells population is still alive even after 24 hours after the photodynamic action. The rest of the cells undergo late apoptosis or necrosis (Fig. 5 upper right quadrant).

The results of the FACS analysis are summarized in Table 2. In cells incubated with Hyp alone, the number of early apoptotic cells increases in a time dependant manner from 14 % at 4 hours after illumination to 61 % at 24 hours after illumination. The cells treated by Hyp/LDL complex showed both apoptosis as well as necrosis signs. 24 hours after illumination the percentage of cells in the late apoptosis/necrosis was even higher than early apoptotic cells (36 % vs. 6 % ). The determination of the late apoptosis/necrosis 24 hour after illumination of U-87 MG in the presence of Hyp/LDL complex is an a good correlation with the SR-FTIRM experiments where the existence of unordered coil structure of proteins was observed. Thus, it can be suggested that the process of necrosis is accompanied by conformational transition from ordered to unordered structure of proteins. Although flow-cytometry measurements were not able to detect apoptosis in U-87 MG cells treated with Hyp/LDL complex 4 hours after illumination (only 3 % of cell population demonstrated signs of apoptosis), IR spectra are sufficiently sensitive to reveal conformational transition from α-helix to β-sheet in protein structure.

Conclusion

The

  : Hypericin (Hyp), Dulbecco´s modified Eagle´s medium (DMEM), Fetal Bovine Serum (FBS) and penicillin/streptomycin were purchased from Gibco invitrogen (France), low-density lipoproteins (LDL) from Merck (Darmstadt, Germany) and Ultroser® G from Pall Life Sciences (France) Cell cultivation: Human glioma cells (U-87MG) were cultured as monolayers and grown in Dulbecco´s modified Eagle´s medium (DMEM) containing L-glutamine (862mg L -1 )

Flow-cytometry

  experiments: The flow-cytometry experiments were performed as reference experiments. The samples were prepared under identical incubation conditions as in SR-FTIRM experiments. Evaluation of apoptotic cells was performed using the Alexa Fluor ® 488 Annexin V (Molecular Probes)/ propidium iodine (PI) staining. Cells with photoactivated Hyp were harvested by low-speed centrifugation and washed with PBS. Cells were resuspended in 100 µl of Annexin-V binding buffer and incubated with 5 µl of Alexa Fluor ® 488 Annexin-V for 15 min at room temperature. After the incubation, the cells were diluted with 400 µl of Annexin-V binding buffer and stained with PI (final concentration 1 µg/ml). Control cells stained with Alexa Fluor ® 488 Annexin V or PI alone were used to compensate for the flow cytometric analysis. The measurements were performed on a BD FACSCalibur flow-cytometer (Becton Dickinson, San Jose, CA, USA) equipped with a 15 mW/488 nm argon laser. The externalization of phosphatidylserine (PS) into outer leaflet of the plasma membrane as a sign of apoptosis was detected by a green fluorescence of bounded Annexin V to PS in the FL-1 channel. The dead cells were identified by a red fluorescence of PI marker (bound to DNA in nucleus) in the FL-3 channel.

  , Table1) are observed for cells treated with the Hyp/LDL complex. For a better resolution of Amide I band and to determine the exact position of the maxima of each vibration contributing to the Amide I band, the secondderivative IR spectra are shown in Figures3 and 4. It is evident that the Amide I band in untreated U-87 MG (control) cells has only one peak centered at 1652 cm -1 . This finding indicates the dominancy of α-helix secondary structure of the proteins present in untreated cells. In contrast, the second-derivative IR spectra in the Amide I region of the Hyp and Hyp/LDL treated cells (at 4 hrs after illumination) are characterized by the appearance of a new band localized at 1622 cm -1 . This is a consequence of the conformational transition of the secondary structure of proteins from α-helix to β-sheet structure. The final position of Amide I band in SR-FTIRM spectra at 1640 cm -1 and at 1644 cm -1 is observed 24 hour after illumination for cells treated with Hyp alone and for Hyp/LDL treated cells, respectively. The second-derivative spectrum has only one peak centered at 1643 cm -1 and at 1647 cm -1 , respectively. This position corresponds to the unfolding of proteins to unstructured coil. Thus, the overall secondary structure of proteins in U-87 MG cells after photodynamic action induced by the irradiation of either Hyp alone or Hyp/LDL complex demonstrates a shift from α-helix to β-sheet (4 hrs after illumination) and from β-sheet to disordered conformation (24 hrs). Similar structural changes in the proteins during apoptosis were also observed by others. Liu et al. demonstrated structural shifts from β-sheet in the normal cells to unordered coil in the apoptotic cells[14,15]. Zhou et al. observed an increase of the β-sheet structure occurrence in membrane proteins of HL60 apoptotic cells[17]. Pozzi et. al. suggested a correlation between the value of the shift of β-sheet wavenumber and the percentage of apoptotic cells[13]. Taking all together, one can conclude that the analysis of the Amide I position provides useful tool for the detection of early apoptotic signs. Flow-cytometry (Annexin V+PI staining) study of U-87 MG cells after treatment with Hyp or Hyp/LDL: To make a correlation between the detection of apoptosis by SR-FTIRM and flow-cytometry experiments, we performed experiments where apoptotic cells quantification was assessed by detecting of Annexin V staining (externalization of the PS molecules in plasma membrane during apoptosis).

  lower right quadrant), and Annexin V/PI double-positive cells are defined as late apoptotic/ necrotic cells (Fig. 5 upper right quadrant). PI only positive cells are considered dead (Fig. 5 upper left quadrant). Fig. 5A shows the typical distribution of control cells (with Hyp alone and without irradiation). Fig. 5B shows the distribution of cells incubated with Hyp alone measured 4 and 24 hours after irradiation. There is a slight shift of the cell population toward apoptotic region (lower right quadrant) 4 hours after irradiation. The amount of apoptotic cells is dramatically increased 24 hours after Hyp photoactivation (Fig. 5 and Table

  modification of the protein secondary structure in U-87 MG cells loaded with either Hyp or Hyp/LDL complex after photodynamic treatment could be explained by on-going apoptosis confirmed by flow-cytometry experiments. In the IR spectrum of the cells 4 hours after irradiation, the overall secondary structure of proteins has been significantly shifted from α-helix to β-sheet (1640 -1622 cm -1 shift). The shift of the Amide I band in the IR spectrum strongly indicates events of the early apoptosis. The unordered coil structure of proteins has been observed in the cells 24 h after irradiation. This finding likely corresponds to the final stages of apoptosis, in which the cells loses their structure and proteins are degraded. The flow-cytometry results have further supported this possibility as the majority of cells was at late apoptosis/necrosis stage at 24 hours after irradiation. Taking all together, our results demonstrate that SR-FTIRM is a powerful technique for the detection of the early stage of apoptosis (the apoptotic signs are detected even earlier than by flow-cytometry), and in combination with other apoptotic assays can provide unambiguously a determination of the apoptosis induced by various factors (e.g. oxidative stress induced by photodynamic action, drugs, radiation).

Fig. abstract .Fig. 1 Fig. 2 (Fig. 3 (Fig. 4 (

 abstract1234 Fig. abstract . Visible image of human glioma cell U-87 MG cell and corresponding Infrared spectrum collected from the point marked in the VIS image of cell

Fig. 5 Fig

 5 Fig.5AnnexinV (FL1-H channel) and propidium iodide (FL3-H channel) staining of U-87 MG cells (A.) control cells, (B.) cells treated with Hyp alone (4 hours after irradiation), (C.) cells treated with Hyp alone (24 hours after irradiation), (D) cells treated with Hyp/LDL complex, (ratio 30:1 ) (4 hours after irradiation), (E) cells treated with Hyp/LDL complex, (ratio 30:1 ) (24 hours after irradiation)

Table 2

 2 Response of U-87MG cells to the PDT treatment in the different period after irradiation (4h, 24h)

		ANNEXIN V -	ANNEXIN V + /
		/ PI -(LIVE	PI -(EARLY	ANNEXIN V/ PI +
		CELLS)	APOPTOTIS)	(LATE
				APOTOSIS/NECROSIS)
	Control	82 %	3 %	13 %
	Hyp, 4 hrs after irrad	68 %	14 %	16 %
	Hyp, 24 hrs after	3 %	61 %	34 %
	irrad
	Hyp/LDL, 4 hrs after	43 %	3 %	54 %
	irrad
	Hyp/LDL, 24 hrs	58 %	6 %	36 %
	after irrad

Acknowledgement: This work was supported by the Slovak Research and Development

Agency under the contracts APVV-0449-07 and lpp-0337-06. We would like to thanks Synchrotron SOLEIL for the access to the SMIS beamline and the SMIS staff for technical help and advices.

Tables:

Table 1