Roots of crosscap slides and crosscap transpositions

Let $N_{g}$ denote a closed nonorientable surface of genus $g$. For $g \geq 2$ the mapping class group $\mathcal{M}(N_{g})$ is generated by Dehn twists and one crosscap slide ($Y$-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists have nontrivial roots. We give necessary and sufficient conditions for the existence of a root of a crosscap slide and a crosscap transposition.


Introduction
Let N n g,s be a connected, nonorientable surface of genus g with s boundary components and n punctures, that is a surface obtained from a connected sum of g projective planes N g by removing s open disks and specifying the set Σ = {p 1 , . . . , p n } of n distinguished points in the interior of N g . If s or/and n equals zero, we omit it from notation. The mapping class group M(N n g,s ) consists of isotopy classes of self-homeomorphisms h : N n g,s → N n g,s fixing boundary components pointwise and such that h(Σ) = Σ. The mapping class group M(S n g,s ) of an orientable surface is defined analogously, but we consider only orientation-preserving maps. If we allow orientation-reversing maps, we obtain the extended mapping class group M ± (S n g,s ). By abuse of notation, we identify a homeomorphism with its isotopy class.
In the orientable case, the mapping class group M(S g ) is generated by Dehn twists [3]. As for nonorientable surfaces, Lickorish proved that Dehn twists alone do not generate M(N g ), g ≥ 2. This group is generated by Dehn twists and one crosscap slide (Y -homeomorphism) [4]. A presentation for M(N g ) using these generators was obtained by Stukow [11]. This presentation was derived from the presentation given by Paris and Szepietowski [7], which used as generators Dehn twists and yet another homeomorphisms of nonorientable surfaces, so-called crosscap transpositions.
Margalit and Schleimer discovered a surprising property of Dehn twists: in the mapping class group of a closed, connected, orientable surface S g of genus g ≥ 2, every Dehn twist has a nontrivial root [5]. It is natural to ask if crosscap slides and crosscap transpositions also have a similar property. The main goal of this paper is to prove the following: Main Theorem. In M(N g ) a nontrivial root of a crosscap transposition [or a crosscap slide] exists if and only if g ≥ 5 or g = 4 and the complement of the support of this crosscap transposition [or crosscap slide] is orientable.

Preliminaries
Crosscap transpositions and crosscap slides. Let N = N g be a nonorientable surface of genus g ≥ 2. Let α and µ be two simple closed curves in N intersecting in one point, such that α is two-sided and µ is one-sided. A regular neighborhood of µ ∪ α is homeomorphic to the Klein bottle with a hole denoted by K. A convenient model of K is a disk with 2 crosscaps, see Figure 1. In this figure shaded disks represent crosscaps, thus the boundary points of these disks are identified by the antipodal map. A crosscap transposition U µ,α specified by µ and α is a homeomorphism of K which interchanges two crosscaps keeping the boundary of K fixed [7]. It may be extended by the identity to a homeomorphism of N . If t α is the Dehn twist about α (with the direction of the twist indicated by small arrows in Figure 1), then Y µ,α = t α U µ,α is a crosscap slide of µ along α, that is the effect of pushing µ once along α keeping the boundary of K fixed. Note that U 2 µ,α = Y 2 µ,α = t ∂K . If g is even, then the complement of K in N g can be either a nonorientable surface N g−2,1 or an orientable surface S g−2 2 ,1 , therefore on surfaces of even genus two conjugacy classes of crosscap slides and crosscap transpositions exist.
Notation. Represent N g as a connected sum of g projective planes and let µ 1 , . . . , µ g be one-sided circles that correspond to crosscaps as in indicated in Figure 2. By abuse of notation, we identify µ i with the corresponding crosscap.
If α 1 , . . . , α g−1 are two-sided circles indicated in the same figure, then for each i = 1, . . . , g − 1 by t α i , u i , y i we denote the Dehn twist about α i , the crosscap transposition U µ i+1 ,α i , and the crosscap slide Y µ i+1 ,α i , respectively. Relations in the mapping class group of a nonorientable surface. A full presentation for M(N g ) is given in [7,11]. Among others, the following relations hold in M(N g ): . . u g−1 ) g−1 = 1 Geometrically u 1 . . . u g−1 is a cyclic rotation of µ 1 , . . . , µ g and u 2 1 u 2 . . . u g−1 is a cyclic rotation of µ 2 , . . . , µ g around µ 1 . In particular, (R6) (u 1 . . . u g−1 ) g = (u 2 1 u 2 . . . u g−1 ) g−1 = t ∂N g,1 in M(N g,1 ). We also have the following chain relation between Dehn twists (Proposition 4.12 of [1]): if k ≥ 2 is even and c 1 , . . . , c k is a chain of simple closed curves in a surface S, such that the boundary of a closed regular neighborhood of their union is isotopic to d, then

Proof of the Main Theorem
Remark 1. Automorphisms of H 1 (N g ; R) induced by crosscap transpositions and crosscap slides have determinants equal to −1, so if a root of a crosscap slide or a crosscap transposition exists, it must be of odd degree.
Let K be a subsurface of N g that is a Klein bottle with one boundary component δ and which contains µ 1 and µ 2 (Figure 2). In particular u 2 1 = y 2 1 = t δ .
The case of g ≥ 4 even and N g \K orientable. Suppose now that crosscap transposition u and crosscap slide y are supported in a Klein bottle with a hole K such that N g \ K is orientable. If c 1 , . . . , c g−2 is a chain of two-sided circles in N g \ K, then by relation (R7), Analogously, The case of g = 2. Crosscap slides and a crosscap transpositions are primitive in M(N 2 ) because [4] M(N 2 ) ∼ = t α 1 , y 1 | t 2 The case of g = 3.
Remark 2. It is known that the mapping class group M(N 3 ) is hyperelliptic [12] and has the central element such that M(N 3 )/ is the extended mapping class group M ± (S 3,1 0 ) of a sphere with 4 punctures. Two upper subscripts mean that we have four punctures on the sphere, but one of them must be fixed. This implies [2] that the maximal finite order of an element in M ± (S 3,1 0 ) is 3, and hence the maximal finite order of an element in M(N 3 ) is 6. Moreover, each two rotations of order 3 in M ± (S 3,1 0 ) are conjugate, which easily implies that each two elements of order 6 in M(N 3 ) are conjugate. The details of the proof of the last statement are completely analogous to that used in [9], hence we skip them.
The same conclusion can be obtained also purely algebraically: it is known [8] that M(N 3 ) ∼ = GL(2, Z) and the maximal finite order of an element in GL(2, Z) is 6. Moreover, there is only one conjugacy class of such elements in GL(2, Z) -for details see for example Theorem 2 of [6].
The case of g = 4 and N 4 \K nonorientable. If N 4 \K is nonorientable, then δ cuts N 4 into two Klein bottles with one boundary component: K and K 1 . Moreover, as was shown in Appendix A of [10], If x ∈ M(N 4 ) exists such that x 2k+1 = u 1 and k ≥ 1 (see Remark 1), then In particular, x commutes with t δ and By Proposition 4.6 of [10], up to isotopy of N 4 , x(δ) = δ. Because u 1 does not interchange two sides of δ and does not reverse the orientation of δ, x has exactly the same properties. Therefore, we can assume that x is composed of maps of K and K 1 . Moreover u 1 = x 2k+1 interchanges µ 1 and µ 2 and does not interchange µ 3 and µ 4 , hence which is a contradiction, because Dehn twists about disjoint circles generate a free abelian group (Proposition 4.4 of [10]).
In the case of a crosscap slide the argument is completely analogous, hence we skip the details. 4. Roots of elementary braids in the mapping class group of n-punctured sphere.
Margalit and Schleimer observed in [5] that if g ≥ 5, then roots of elementary braids in M(S g 0 ) exist. The Main Theorem implies slightly stronger version of that result.