Math.GT] 29 Jan 2021 SO,∂Free SO Ωn+1 (G) −→ Ωn (G)

Extending free group action on surfaces

Jes´usEmilio Dom´ınguez∗and Carlos Segovia†

February 1, 2021

Abstract

The present work introduces new perspectives in order to extend finite group actions from surfaces to 3-manifolds. We consider the Schur multiplier associated to a finite group G in terms of principal G-bordisms in dimension two, called G- cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3- manifold. We show the answer to this question is affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show that every non-necessarily free action of abelian groups (under certain conditions) and dihedral groups on a closed oriented surface extends to 3-dimensional handlebody.

Introduction

SO Let Ωn (G) be the free G-bordism group in dimension n from Conner-Floyd [CF64] and SO,∂free denote by Ωn+1 (G), the G-bordism group of (n + 1)-dimensional manifolds with a non necessarily free G-action which restricts to a free action over the boundary. We are interested in knowing what the image of the following map is arXiv:2012.02464v2 [math.GT] 29 Jan 2021 SO,∂free SO Ωn+1 (G) −→ Ωn (G) . (1)

SO For n = 2, the group Ω2 (G) has been studied extensively with the name of the Schur mul- tiplier [Kar87] (denoted by M(G)). When this group vanishes, the map (1) is surjective, such is the case for cyclic groups, groups of deficiency zero, see [Kar87]. For free actions of abelian groups and dihedral groups, the extension was given by Reni-Zimmermann [RZ96] and Hidalgo [Hid94]. Obstructions for the surjectivity of the map (1) are constructed by Samperton [Sam20], considering the quotient by the homology classes represented by tori.

∗Universidad Autonoma de Sinaloa, M´exico. e-mail: [email protected] †Instituto de Matem´aticasUNAM-Oaxaca, M´exico. e-mail: [email protected]

1 SO Our approach considers the elements of the group Ω2 (G) represented by what we call G-cobordisms in dimension two. These are diffeomorphism classes of principal G- bundles over (closed) surfaces [GS16]. We say that a G-cobordism is extendable if it has a representative given by a principal G-bundle over a surface S, which is the boundary of a 3-dimensional manifold M with an action of G. For G a finite abelian group, we show that every G-cobordism over a closed surface is extendable, see Theorem 9. For this, we decompose any G-cobordism into small pieces given by G-cobordisms over a closed surface of genus one, which are extendable, as we will see in Proposition 7.

For the dihedral group D2n, we focus in the case n = 2k since the Schur multiplier M(D2n) vanishes for n = 2k + 1. Similar to the abelian case, we decompose every D2n- cobordism into a finite product of the generator with base space of genus one, which is induced by a reflection and the rotation by 180 degrees, see Corollary 16.

For the symmetric group Sn, the Schur multiplier is non-trivial for n > 3 and in that case it is equal to Z2. In Proposition 18, we prove that there is a generator with base space of genus one, which is induced by any two disjoint transpositions. A similar argument works for the alternating group An, where for n = 6, 7, we use the Sylow theory of the Schur multiplier that is shown in Proposition 14. In summary, we have the following result.

Theorem 1. For G a finite abelian group or G ∈ {D2n,Sn,An}, every G-cobordism over a closed oriented surface is extendable.

We have the following applications for extending non-necessarily free actions over surfaces to 3-dimensional handlebodies:

i) In Theorem 23, the actions of abelian groups have two types of fixed points, which are the ones induced by hyperelliptic involutions and pairs of ramification points with complementary monodromies (signature > 2). An unfolding process is per- formed by first considering the quotient by the hyperelliptic involutions and after some modifications, we reduce the problem to the extension of free actions. ii) In Theorem 24, the actions of dihedral groups reduce to a finite product of an specific generator. We extend the action for this generator and for the surfaces realized by the products.

These results were proven before by different methods, by Reni-Zimmermann [RZ96] and Hidalgo [Hid94]. This article is organized as follows. In Section 1, we review the concept of G-cobordism and the Schur multiplier, as well as the relations between them. In Section 2, we give explicit generators for the Schur multiplier of the dihedral, the symmetric and the alter- nating groups. Finally, in Section 3 we construct the extensions of the free actions on

2 closed oriented surfaces for the dihedral, symmetric and alternating groups. Additionally, for non necessarily free actions on closed oriented surfaces of abelian groups (under cer- tain conditions) and dihedral groups, we construct the extensions given by 3-dimensional handlebodies. Acknowledgements: the first author thanks the Academia Mexicana de Ciencias for the opportunity of participating in the Scientific Research Summer of 2020. The second author is supported by c´atedrasCONACYT and Proyecto CONACYT ciencias b´asicas 2016, No. 284621. We would like to thank Bernardo Uribe and Eric Samperton for their helpful conversations.

1 Preliminaries

In this section we review in detail the definitions and properties of the theory of G- cobordisms introduced in [GS16, Seg12]. In addition, we discuss some important facts about the Schur multiplier.

1.1 G-cobordisms

Throughout the article, G denotes a finite group and 1 ∈ G the neutral element. Also, we consider right actions of the group G, and all the surfaces are oriented. Definition 2. Let Σ and Σ0 be d-dimensional closed, oriented smooth manifolds. A cobordism between Σ and Σ0 is a (d + 1)-dimensional oriented smooth manifold M, with boundary diffeomorphic to Σ t −Σ0, where −Σ0 is Σ0 with the reverse orientation. Two cobordisms M and M 0 are equivalent if there exists a diffeomorphism φ : M −→ M 0 such that we have the commutative diagram

M (2) > a

Σ φ Σ0

 } M 0 . Definition 3. A principal G-bundle over a topological space X, consists of a fiber bundle π : E → X where the group G acts freely and transitively over each fiber. Example 1.1. In dimension one, for every g ∈ G, we construct the principal G-bundle 1 Pg → S obtained by attaching the ends of [0, 1] × G via multiplication by g, i.e., (0, h) is identified with (1, gh) for every h ∈ G. This construction Pg = [0, 1] × G/ ∼g projects to the circle by restriction to the first coordinate, and the action Pg × G → Pg is defined

3 by right multiplication on the second coordinate. Any principal G-bundle over the circle is isomorphic to some Pg, and Pg is isomorphic to Ph if and only if h is conjugate to g.

Throughout the paper, we refer to the element g ∈ G as the monodromy associated to the corresponding principal G-bundle Pg. In the case of the neutral element of the group G, we say that the monodromy is trivial. Definition 4. Let ξ : P → Σ and ξ0 : P 0 → Σ0 be principal G-bundles. A G-cobordism between ξ and ξ0 is a principal G-bundle  : Q → M, with diffeomorphisms for the boundaries ∂M ∼= S t −S0 and ∂Q ∼= P t −P 0, which match with the projections and the restriction of the action. Two G-cobordisms  : Q → M and 0 : Q0 → M 0 define the same class if M and M 0 are equivalent as cobordisms by a diffeomorphism φ : M → M 0, Q and Q0 are equivalent as cobordisms by a G-equivariant diffeomorphism ψ : Q −→ Q0, and in addition, we have the commutative diagram

ψ Q / Q0 (3)

 0   M / M 0 . φ

Example 1.2. 1.A G-cobordism from Pg to Ph (g, h ∈ G) with base space the cylinder, is given by an element k ∈ G such that h = kgk−1.

2.A G-cobordism with entry the disjoint union Pg t Ph and exit Pgh, with base space the pair of pants, is a G-deformation retract1 of a principal G-bundle over the wedge S1 ∨ S1. 3. There is only one G-cobordism over the disk and every representative is a trivial bundle. 4. Take as base space a two dimensional handlebody of genus n with one boundary cir- cle. A G-cobordism depends on elements gi, ki ∈ G, for 1 ≤ i ≤ n, with monodromy Qn for the boundary circle given by the product i=1[ki, gi]. In Figure 1, we have pictures for the G-cobordisms over the cylinder, the pair of pants and the disc. For these pictures, we draw from left to right the direction for our cobordisms. Also, every circle is labelled with the correspondent monodromy and for every cylinder we write inside the element of the group with which we do the conjugation.

In the left side of Figure 2, we have a G-cobordism over a genus one handlebody. Additionally, in the right side, we represent an equivalent manner to see this G-cobordism.

1By a G-deformation retract we mean that the homotopy is by means of principal G-bundles.

4 g

g k kgk-1 gh 1

h

Figure 1: G-cobordism over the cylinder, the pair of pants and the disc.

g kgk -1 k

k

=

[k,g] [k,g] g

g-1 k-1

Figure 2: Two equivalent G-cobordisms over a handle of genus one.

If a G-cobordism over a closed connected surface is cut along a simple closed sepa- rating curve2, the monodromy of the resulting curve lies inside the commutator group, as shown in the following proposition. Proposition 5. For a G-cobordism over a closed connected surface S, the monodromy of every embedded simple closed separating curve in S lies in the commutator group [G, G]. Definition 6. A G-cobordism of dimension two, over a closed surface, is extendable if for some representative principal G-bundle P → S, with the action α : P × G → P , there exits a 3-dimensional manifold M with boundary ∂M = P , with an action of G of the form α : M × G → M, which extends α, i.e., we have the commutative diagram

α P × G / P (4)  _  _

  M × G / M. α Proposition 7. Any G-cobordism over a closed surface of genus one is extendable.

Proof. Consider a principal G-bundle representing the given G-cobordism over the closed surface of genus one. It is enough to prove the case in which the total space of the bundle

2A simple closed curve in a surface is separating if the cut surface is not connected.

5 is a connected space. Moreover, the action of G over the total space can be modified by an isotopy resulting in an action which depends completely on a pair of monodromies (g, k), which are associated to two curves in the torus that intersect once. Denote by Pg and Pk the two principal G-bundles associated to these two curves. It follows that the action of G over the total space is given by the product of the total spaces Pg and Pk. Because of the assumptions, at least one of Pg or Pk is a connected space, let us assume that it is Pg. The extension of the action of G is through the 3-dimensional handlebody constructed as follows. First, consider the disc D as the union (S1 × (0, 1]) ∪ {0}, where 0 is the center. For each circle S1 × {r}, with r ∈ (0, 1], we take as monodromy the element g so that the principal G-bundle is Pg, and we take the center 0 as fixed point. Thus, over the disc we have the rotation by 2π/|g|, with |g| the order of g ∈ G. Taking the product of this disc together with the induced principal G-bundle Pk, we obtain the extension which makes the G-cobordism extendable.

Remark: 8. We want to emphasize why the construction given in the previous proposition does not work for closed surfaces with genus > 1. The reason is that the set of fixed points should be a smooth submanifold, which we can not assure for genus > 1, since we have points where three lines meet. Now, we apply the previous results to abelian groups.

Theorem 9. For G a finite abelian group, any G-cobordism is extendable.

Proof. Consider a G-cobordism over a connected closed surface. By Proposition 5, we can write this G-cobordism as a connected sum of G-cobordisms with base space of genus one. This connected sum is done for the total space along trivial bundles over a circle. Since the connected sum is in the same bordism class as the disjoint union, we are decomposing this G-cobordism as a disjoint union of G-cobordisms with base space a closed surface of genus one. Because of Proposition 7, any G-cobordism over a closed surface of genus one is extendable, so the theorem follows.

1.2 The Schur multiplier

The study of this theory began in 1904 by Isaai Schur in order to study the projective representations of groups. Nowadays, the Schur multiplier represents three different iso- SO morphic groups given by the second free bordism group Ω2 (G), the second homology 2 ∗ group H2(G, Z) and the second cohomology group H (G, C ).

Definition 10. Let hG, Gi be the free group on all pairs hx, yi, with x, y ∈ G. There is a natural homomorphism of hG, Gi onto the commutator group [G, G], which sends hx, yi into [x, y]. Consider the kernel Z(G) of this homomorphism and the normal subgroup

6 B(G) of hG, Gi generated by the relations

hx, xi ∼ 1 , (5) hx, yi ∼ hy, xi−1 , (6) hxy, zi ∼ hy, zix hx, zi , (7) hy, zix ∼ hx, [y, z]i hy, zi , (8) where x, y, z ∈ G and hy, zix = hyx, zxi = hxyx−1, xzx−1i. The Schur multiplier is defined as the quotient group Z(G) M(G) := . (9) B(G)

Miller [Mil52] shows that the quotient Z(G)/B(G) is canonically isomorphic to the Hopf’s integral formula R ∩ [F,F ] M(G) ∼= , (10) [F,R] where G = h F | R i. Moreover, in [Mil52] there are some consequent relations, which we enumerate in the following theorem.

Theorem 11 ([Mil52]). The following relations can be deduced from (5)-(8):

hx, yzi ∼ hx, yi hx, ziy , (11) hx, yiha,bi ∼ hx, yi[a,b] , (12) [hx, yi , ha, bi] ∼ h[x, y], [a, b]i , (13) 0 0 0 0 hb, b i ha0, b0i ∼ h[b, b ], a0i ha0, [b, b ]b0i hb, b i , (14) 0 0 0 0 hb, b i ha0, b0i ∼ h[b, b ]b0, a0i ha0, [b, b ]i hb, b i , (15) hb, b0i ha, a0i ∼ h[b, b0], [a, a0]i ha, a0i hb, b0i , (16) hxn, xsi ∼ 1 n = 0, ±1, ··· ; s = 0, ±1, ··· , (17)

0 0 for x, y, z, a, b, a , b , a0, b0 ∈ G.

The connection with bordism relates the elements of Z(G) by means of the assignment

hx1, y1ihx2, y2i · · · hxn, yni 7−→ (yn, xn)(yn−1, xn−1) ··· (y1, x1) , (18) where the sequence in the right defines the generating monodromies for a G-cobordism over a closed surface of genus n as in Figure 3. Indeed, the previous four relations (5), (6), (7) and (8), are interpreted in bordism as follows:

7 -1 -1 -1 yn x n y n xn yn-1 x n-1 yn-1 xn-1 y1 x 1 y 1 x1

xn xn-1 x1

[x n , y n ] [xn-1 ,y n-1 ][x n , y n ]

-1 y -1 [ x , y ] y -1 yn n-1 n n 1 [x2 ,y 2 ] ... [x n , yn ]

Figure 3: The G-cobordism associated to the sequence (yn, xn)(yn−1, xn−1) ··· (y1, x1), Qn with yi, xi ∈ G and i=1[xi, yi] = 1.

x 1 x

=

x

x-1

y y xyx-1

-1 x y

1 = [x,y] x

y-1 xyx-1 xy -1 x-1 1

Figure 4: The G-cobordisms associated to the pairs (x, x) and (x, y)(y, x).

(i) For (5) and (6), we consider the G-cobordism defined by the pairs (x, x) and (x, y)(y, x), respectively. We represent these G-cobordisms in the left side of Figure 4, respectively. In the top of Figure 4, we apply the Dehn twist diffeomorphism and obtain that the conjugation becomes the neutral element 1 ∈ G. In the bottom of Figure 4, we cut along a trivial monodromy to reduce the genus by one. Notice that the G-cobordisms in the left side of Figure 4 are null bordant since we can cut along a trivial monodromy eliminating the hole of the handle. (ii) For (7) and (8), we obtain a G-cobordism, over a handlebody of genus two, where we can find a curve with trivial monodromy which reduces the genus by one. In Figure 5, we represent these identifications, respectively.

Now, we focus on the Sylow theory of the Schur multiplier. We use Definition 12 and Theorem 13, in order to show Proposition 14. Definition 12. For a subgroup H ⊂ G, there are the following induces maps:

8 z xzx-1 xzx-1 z

x xyx-1 xy

1 = [x,z] [xy,z] [xy,z]

-1 -1 z xz-1 x-1[x,z] z

z yzy-1 [y,z] z

y x xy

x x = x [y,z] [y,z] [y,z]

-1 -1 z 1 -1 z

Figure 5: Reduction of genus through the cutting along a trivial monodromy.

i) the restriction map, denoted by res : M(G) → M(H), which associates to a G- cobordism over a closed surface, the restriction of the action to the subgroup H. ii) the corestriction map, denoted by cor : M(H) → M(G), which starts with a principal H-bundle P → S and associates the Borel construction P ×H G produced by the quotient of the product P × G with the action of H of the form (x, g)h = (xh, h−1g). The group G has a free action over the the Borel construction by [x, g]ˆg = [x, ggˆ].

In general, these maps extend to non-necessarily free actions, in particular, for the SO,∂free G-bordism groups Ω3 (G) of 3-dimensional manifolds with a non necessarily free G-action which restricts to a free action over the boundary. Theorem 13 ([Kar87]). Let P be a Sylow p-subgroup of G and let M(G) the p-component of the Schur multiplier M(G). Then the restriction map res : M(G) → M(P ) induces an injective homomorphism M(G)p → M(P ), and the corestriction map cor : M(P ) → M(G) induces a surjective homomorphism M(P ) → M(G)p. Proposition 14. For a finite group G, and Syl(G) the set of isomorphism classes of Sylow subgroups of G, if any element Q in Syl(G) satisfies that any Q-cobordism is extendable, then any G-cobordism is extendable.

k Proof. For n = |G| and n = p m with p 6 |m, consider an element f ∈ M(G)p, hence the composition F := cor ◦ res : M(G)p −→ M(G)p , (19) m is given by f 7−→ f , which is an automorphism of M(G)p. By the assumptions, the restriction res(f) is extendable by a 3-manifold M with an action of Q, therefore, applying

9 the corestriction we obtain that f m is extendable by the 3-manifold cor(M). Similarly, we can start with F −1(f) and we get that f is extendable and the proposition follows.

2 Generators for the Schur multiplier

In this section we give explicit generators for the Schur multiplier of the dihedral, the symmetric and the alternating groups.

2.1 Dihedral group

For n ≥ 3, the dihedral group is the group of symmetries of the n-regular polygon (with D2 = Z2, D4 = Z2 × Z2), and presentation

2 2 n D2n = ha, b : a = 1, b = 1, (ab) = 1i , (20) where c := ab is the rotation of 2π/n. The Schur multiplier has the form

 0 n = 2k + 1 , M(D2n) = (21) Z2 n = 2k .

In order to find a generator we show the following.

Proposition 15. We obtain the following identifications:

(i) hci, cji ∼ 1,

(ii) hci, acji ∼ hc, aii,

(iii) haci, cji ∼ hc, ai−j, and

(iv) haci, acji ∼ hc, aij−i.

Proof. The relation (i) follows by (17). The use of (7), (17) and (11) implies

ci, acj ∼ ci, a ci, cj a ∼ ci, a , (22) ci, acj = cci−1, acj ∼ ci−1, acj c c, acj ∼ ci−1, a hc, ai . (23)

Therefore, we obtain the relation

ci, a ∼ hc, ai hc, ai · · · hc, ai , (24) | {z } i

10 which implies (ii). By (6) we obtain (iii) as follows

aci, cj ∼ cj, aci −1 ∼ hc, ai−j . (25)

Finally, we use (11) and (5),

aci, acj ∼ ci, acj a a, acj ∼ c−1, a i a, cj a , (26) and by (6) we obtain (iv).

Corollary 16. For n = 2k, the generator of the group M(D2n) is represented by the element hck, ai.

2.2 Symmetric group

The symmetric group Sn is composed of permutation of the set [n] = {1, ··· , n}. This group is generated by the transpositions (ij) with i, j ∈ [n]. The Schur multiplier is given as follows  0 n ≤ 3 , M(Sn) = (27) Z2 n ≥ 4 .

Lemma 17. Let k ∈ [n], and hσ1, τ1i · · · hσr, τri be a sequence with σi, τi, ∈ Sn, for i ∈ {1, ··· , r}. There exist a positive number 0 ≤ s ≤ r and the following elements:

(i) ai, bi ∈ Sn, with 0 ≤ i ≤ s, such that all ai, bi fix k, and

(ii) cj, dj ∈ Sn, with 0 ≤ j ≤ r − s, such that for each j, at least one of cj, dj does not fix k, with the relation

hσ1, τ1i · · · hσr, τri ∼ ha1, b1i · · · has, bsihc1, d1i · · · hcr−s, cr−si . (28)

Moreover, s is the amount of pairs hσi, τii such that both σi and τi fix k.

Proof. It suffices to note that for pairs ha, bi and hx, yi, with a, b, x, y ∈ Sn, such that a and b fix k there is the relation

hx, yi ha, bi ∼ ha, bi hb, ai hx, yi ha, bi ∼ ha, bi x[b,a], y[b,a] , where we have used (12). An iterative application of this process, allows us to put all terms fixing k to the left in the sequence.

11 0 0 Proposition 18. Assume n ≥ 4, and take elements σi, τi, σj, τj ∈ Sn, for 1 ≤ i ≤ r and 1 ≤ j ≤ s, with the same commutator, i.e.,

0 0 0 0 [σ1, τ1] ··· [σr, τr] = [σ1, τ1] ··· [σs, τs] . (29)

Therefore, for the element u := h(1, 2), (3, 4)i, there is the relation

k 0 0 0 0 hσ1, τ1i · · · hσr, τri ∼ u hσ1, τ1i · · · hσs, τsi , (30) with k ∈ {0, 1}.

Proof. First, we observe that from (7) and (11), we can assume that all the elements 0 0 σi, τi, σj, τj ∈ Sn are transpositions. By (8), every pair hσ, τi with σ and τ disjoint transpositions is in the same class as the pair u := h(1, 2), (3, 4)i. Therefore, we can assume that the pairs are of the form h(i, j), (j, k)i, with i, j and k different numbers.

By exhaustion, the proposition follows for the symmetric group Sn, with 4 ≤ n ≤ 6. We proceed by induction for n ≥ 7 and we suppose that for k < n, the generator of the Schur multiplier M(Sn) is given by the element u := h(1, 2), (3, 4)i. Set by m the 0 0 0 0 maximum of r and s, for the sequences hσ1, τ1i · · · hσr, τri and hσ1, τ1i · · · hσs, τsi. For m = 1, the proposition follows from the triviality of M(S3). Suppose that our proposition follows for sequences with length l < m. We consider the sequence

0 0 0 0 −1 0 0 0 0 hσ1, τ1i · · · hσr, τri (hσ1, τ1i · · · hσs, τsi) ∼ hσ1, τ1i · · · hσr, τrihτs, σsi · · · hτ1, σ1i , (31) which has trivial commutator and length given by M := r+s ≤ 2m. Let x ∈ {1, ··· , n} be the number that is fixed by the most terms of the sequence (31). Given that the sequences have non trivial terms, each term permutes 3 different numbers in {1, 2, ··· , n}. Therefore, 3(r+s) 3(r+s) 3(2m) the number x is not fixed by at most n terms. Given that n ≤ 7 < m, hence x is not fixed by at most m − 1 terms. By Lemma 17, we can find an equivalent sequence for (31), with the following form

hα1, β1i · · · hαt, βti hαt+1, βt+1i · · · hαM , βM i , (32) | {z } | {z } fix number x do not fix number x where:

i) M − t < m;

ii) the αi, βi ∈ Sn, with 0 ≤ i ≤ t, fix x; and

iii) the αj, βj ∈ Sn, with t + 1 ≤ j ≤ M, at least one does not fix x.

12 0 0 Moreover, by the proof of Lemma 17, the elements αi, βi, αj, βj ∈ Sn are again transposi- tions. Now we consider the sequences

A := hα1, β1i · · · hαt, βti (33) and −1 B := (hαt+1, βt+1i · · · hαM , βM i) = hβM , αM i · · · hβt+1, αt+1i , (34) where both sequences have the same commutator. Furthermore, the sequence A has pairs composed by elements in Sn−1 because they fix x. By our induction hypothesis, for n, we conclude that the Schur multiplier M(Sn−1) is generated by u = h(1, 2), (3, 4)i. Therefore, i A ∼ u C for i ∈ {0, 1} and C is a sequence of pairs with elements in Sn−1. We can take C to be of length < m, as it is has the same commutator as the chain B of length < m. By the other induction hypothesis, for m, since B and C have length less than m, then there is j ∈ {0, 1} such that B ∼ ujC. This shows that the product of our initial sequences in (31) is equivalent to ui−j and the proof of the proposition follows.

Corollary 19. For n ≥ 4, the generator of the group M(Sn) is represented by the element u := h(1, 2), (3, 4)i.

2.3 Alternating group

The alternating group An is the normal subgroup of Sn with index 2. The Schur multiplier has the form  0 n ≤ 3 ,   Z2 n = 4, 5, M(An) = (35) Z6 n = 6, 7,   Z2 n ≥ 8 .

Proposition 20. For n ≥ 4, the element h(1, 2)(3, 4), (1, 3)(2, 4)i is nontrivial in M(An).

Proof. Because of the relations (7) and (11) in M(Sn), we have the following

h(1, 2)(3, 4), (1, 3)(2, 4)i ∼ h(3, 4), (2, 3)(1, 4)ih(1, 2), (1, 3)(2, 4)i ∼ h(3, 4), (2, 3)ih(2, 4), (1, 4)ih(1, 2), (1, 3)ih(2, 3), (2, 4)i

We also have from (8), (11) and h(2, 4), (1, 4)i = h(2, 3), (1, 3)i(3,4) that

h(2, 4), (1, 4)i ∼ h(3, 4), [(2, 3), (1, 3)]ih(2, 3), (1, 3)i = h(3, 4), (1, 2)(1, 3)ih(2, 3), (1, 3)i ∼ h(3, 4), (1, 2)ih(3, 4), (2, 3)ih(2, 3), (1, 3)i = uh(3, 4), (2, 3)ih(2, 3), (1, 3)i

13 As [(3, 4), (2, 3)] = [(2, 3), (2, 4)], [(2, 3), (1, 3)] = [(1, 3), (1, 2)] and M(S3) = 0, we have that h(3, 4), (2, 3)i ∼ h(2, 3), (2, 4)i and h(2, 3), (1, 3)i ∼ h(1, 3), (1, 2)i. Therefore,

h(1, 2)(3, 4), (1, 3)(2, 4)i ∼ h(2, 3), (2, 4)ih(2, 4), (1, 4)ih(1, 2), (1, 3)ih(2, 3), (2, 4)i ∼ uh(2, 3), (2, 4)i2h(1, 3), (1, 2)ih(1, 2), (1, 3)ih(2, 3), (2, 4)i ∼ uh(2, 3), (2, 4)i3 ∼ u = h(1, 2), (3, 4)i ,

3 3 where h(2, 3), (2, 4)i vanishes since [(2, 3), (2, 4)] = 1 and M(S3) = 0. As a consequence, the element h(1, 2)(3, 4), (1, 3)(2, 4)i is nontrivial in M(Sn), and hence it is also nontrivial in M(An) for n ≥ 4.

Corollary 21. For n ≥ 4 and n 6∈ {6, 7}, the generator of the group M(An) is represented by the element h(1, 2)(3, 4), (1, 3)(2, 4)i.

3 Extending group actions on surfaces

This section contains the main applications of this work. We start with free actions of abelian, dihedral, symmetric and alternating groups and then, we show that these actions extend to actions on 3-manifolds. Finally, we see the case of non-necessarily free actions for abelian and dihedral groups.

3.1 Free actions on surfaces

Definition 22. Consider a compact oriented surface S with a (free) group action

α : S × G −→ S. (36)

We say that the action is extendable if there exists a 3-manifold M with boundary ∂M = S, with an action of G of the form α : M × G −→ M, which extends α, i.e., we have the commutative diagram α S × G / S (37)  _  _

  M × G / M. α

Proof of Theorem 1. By Theorem 9 we know that any free action of a finite abelian group is extendable. For dihedral groups D2n, we have two cases to consider. One is for n = 2k + 1, but since M(D4k+2) = 0, then any free action is extendable. The other is for n = 2k, where by Corollary 16, the generator of the Schur multiplier is represented by a G-cobordism over a closed surface of genus one, therefore, by Proposition 7 these free actions are extendable. Now consider free actions of the symmetric groups Sn, since

14 Figure 6: Hyperelliptic involutions

M(Sn) = 0 for n ≤ 3, it remains to prove the extension for n ≥ 4. Similar as for dihedral groups, by Corollary 19 these free actions are extendable. For the alternating groups An the free actions are extendable for n ≤ 3. Again, for n ≥ 4 and n 6= 6, 7, by Corollary 21 these actions are extendable. In the case of free actions of An for n = 6, 7, we notice that the Sylow subgroups of A6 and A7 have the following isomorphic types {D8, Z3 ×Z3 ×Z3, Z5, Z7} and because of Proposition 14, we obtain that these free actions are extendable.

3.2 Non-necessarily free action on surfaces

Now we consider non-necessarily free actions of finite abelian groups and dihedral groups. The extension of these actions was already given by Reni-Zimmermann [RZ96] with 3- dimensional methods and by Hidalgo [Hid94] with 2-dimensional methods.

Theorem 23. Let G be a finite abelian group with an action on a closed oriented surface where the fixed points are of two types:

(i) fixed points produced by hyperelliptic involutions, see Figure 6, and

(ii) ramification points with complementary monodromies (signature > 2).

Then the action is extendable by a 3-dimensional handlebody.

Proof. The extension is performed in some steps. First, we consider the quotient of the surface by the hyperelliptic involutions in some order and smooth the corners with the aim to obtain a smooth closed oriented surface. The hyperelliptic involutions act in the set of ramification points with signature > 2 and in the quotient we still have ramifications points grouped into pairs. Then we connect the complementary monodromies by cylinders in order to have a free action over a closed oriented surface. From Theorem 9, we have that

15 T γ γ γ’ γ’ a α β α β α’ β’ α’ β’ b S S

δ δ δ’ δ’ T

Figure 7: Representative for hb, ai with n = 2k (k = 1). this free action is extendable and by the proof of Proposition 15, this extension is by means of a 3-dimensional handlebody. Now we disconnect the complementary monodromies by cutting in each of the cylinders that we glued. Each cylinder has only one fixed point by the proof of Proposition 7. Finally, we extend the action to the original surface by an unfolding process using the hyperelliptic involutions in the reverse order in which we constructed the initial quotient surface.

Theorem 24. Every action of a dihedral group D2n over a closed orientable surface is extendable by a 3-dimensional handlebody.

Proof. By Proposition 15, the extension problem reduces to a finite product of the same a D2n-cobordism induced by the pair hc, ai ∼ hab, ai ∼ hb, ai . Thus, it is enough to solve the extension problem for the pair hb, ai and for the D2n-cobordism over the pair of pants 2 with entries in [D2n,D2n] = hc i. The last reduces to the extension of G-cobordisms over pair of pants where G = [D2n,D2n], which follows because the group is cyclic. For the pair hb, ai we construct a representative D2n-cobordism and there are two cases to consider: (i) For n = 2k, we consider the disjoint union of two spheres, where each one is the gluing of two n-gons by the boundary. Denote by T the operation of switching from one sphere to the other and by S the operation of switching from one n-gon to the other in the same sphere. The action of a lifts to the composition Sa = aS and the action of b lifts to the composition T b = bT . We obtain n = 2k fixed points over each sphere plus the north a south poles. Then for each fixed point (except the north and south pole), we remove a small disc around it and we connect the holes for opposite fixed points with a cylinder. In a similar way as for abelian groups, this action extends with the north and south pole as unique ramification points. For k = 2, in Figure 7, we draw an illustrator of this construction. (ii) For n = 2k +1, we consider one sphere as the gluing of two n-gons by the boundary. Denote by S the operation of switching from one n-gon to the other. Thus the action

16 of a lifts to the composition Sa = aS and the action of b lifts to the composition Sb = bS. We obtain 2n fixed points plus the north a south poles. Then we perform the same procedure to construct the extension of the action, as in the even case.

References

[CF64] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Math- ematik und ihrer Grenzgebiete, N. F., Band 33 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1964.

[GS16] Ana Gonz´alezand Carlos Segovia, G-topological quantum field theory, Bol. Soc. Mat. Mex. (3) 23 (2016), no. 1, 439–456.

[Hid94] Rub´enA. Hidalgo, On schottky groups with automorphisms, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 2, 259–289.

[Kar87] Gregory Karpilovsky, The Schur Multipler, London Mathematical Society Mono- graphs. New Series, 2. The Clarendon Press, Oxford University Press, New York, 1987.

[Mil52] Clair Miller, The second homology group of a group; relations among commuta- tors, Proc. Amer. Math. Soc. 3 (1952), 588–595.

[RZ96] Marco Reni and Bruno Zimmermann, Extending finite group actions from sur- faces to handlebodies, Ann. Acad. Sci. Fenn. Ser. A I Math. 124 (1996), no. 9, 2877–2887.

[Sam20] Eric Samperton, Schur-type invariants of branched g-covers of surfaces, Topolog- ical phases of matter and quantum computation, Contemp. Math. Amer. Math. Soc., Providence 747 (2020), no. 2, 173–197.

[Seg12] Carlos Segovia, The classifying space of the 1+1 dimensional G-cobordism cate- gory, 2012, arXiv:1211.2144.

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