Roots of crosscap slides and crosscap transpositions

Let Ng\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{g}$$\end{document} denote a closed nonorientable surface of genus g. For g≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \ge 2$$\end{document} the mapping class group M(Ng)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}(N_{g})$$\end{document} 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 on orientable surfaces have nontrivial roots. We give necessary and sufficient conditions for the existence of roots of crosscap slides and crosscap transpositions.


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 = {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( ) = . 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 [14]. This presentation was derived from the presentation given by Paris and Szepietowski [9], 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 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 (resp. 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.

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 on 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 Fig. 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 [9]. 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 Fig. 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 Remark 2.1 If g is odd, then the complement of K in N g is a nonorientable surface N g−2,1 , thus in this case every two crosscap slides [or crosscap transpositions] are conjugate. If g is even, then the complement of K in N g can be either a nonorientable surface N g−2,1 or an Fig. 1 A crosscap transposition and a crosscap slide orientable surface S g−2 2 ,1 , therefore in the mapping class group of a surface 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 indicated in Fig. 2. By abuse of notation, we identify μ i with the corresponding crosscap.

Relations in the mapping class group of a nonorientable surface
A full presentation for M(N g ) is given in [9,14]. Among others, the following relations hold in M(N g ): It is straightforward to check that relations (R1)-(R3) imply Geometrically u 1 u 2 · · · u g−1 is a cyclic rotation of μ 1 , μ 2 , . . . , μ g and u 2 1 u 2 · · · u g−1 is a cyclic rotation of μ 2 , μ 3 , . . . , μ g around μ 1 . In particular, 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 on a surface S, such that the boundary of a closed regular neighborhood of their union is isotopic to d, then  Let u and y be respectively a crosscap transposition and a crosscap slide supported in the same Klein bottle K ⊂ N g and let δ = ∂ K . In particular, u 2 = y 2 = t δ . If N \ K is nonorientable, then we assume that u = u 1 , y = y 1 and K is a Klein bottle which contains μ 1 and μ 2 (Fig. 2).

The case of g ≥ 4 even and N g \K orientable
Suppose now 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, y = t c 1 t c 2 · · · t c g−2 g y −1 g−1 .

The case of g = 2
Crosscap slides and crosscap transpositions are primitive in M(N 2 ) because [4]

Remark 3.2 It is known that the mapping class group M(N 3 ) is hyperelliptic [15] 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 of two rotations of order 3 in M ± (S 3,1 0 ) are conjugate, which easily implies that each of 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 [12], hence we skip them.
The same conclusion can also be obtained purely algebraically: it is known [11] 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 [7]. Remark 3.1). Then

We will show that crosscap transpositions do not have nontrivial roots in
Note that the order of x can not be odd (because otherwise u 1 = x 2k+1 = 1), it can not equal 2 (because otherwise x = u 1 ), and it is not divisible by 4. Hence, by Remark 3.2, x has order 6. Moreover, by relation (R7), where d is a boundary of a closed regular neighborhood of α 1 ∪ α 2 . Hence x is conjugate to t α 1 t α 2 and this contradicts Remark 3.1, because Dehn twists induce automorphisms of H 1 (N 3 ; R) with determinant equal to 1 and x 2k+1 = u 1 .
In the case of a crosscap slide the argument is completely analogous, hence we skip the details.

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 [13], By abuse of language, we say that t k α 1 u m 1 ∈ M(K ) interchanges μ 1 and μ 2 if and only if m is odd. Equivalently, elements of M(K ) which does not interchange μ 1 and μ 2 are precisely elements of the twist subgroup of M(K ) (that is the subgroup of index 2 generated by Dehn twists). In the same way we define the notion of interchanging μ 3 and μ 4 for elements of M(K 1 ).
If x ∈ M(N 4 ) exists such that x 2k+1 = u 1 and k ≥ 1 (see Remark 3.1), then In particular, x commutes with t δ and By Proposition 4.6 of [13], 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 But then and Proposition 4.4 of [13] implies that k 2 = 0 and (2m 1 + 2m 2 + 1)(2k + 1) = 1 which is a contradiction.
In the case of a crosscap slide the argument is completely analogous, hence we skip the details. [6] and independently by Monden [8], roots of Dehn twists constructed by Margalit and Schleimer [5] are of maximal possible degree. Since the geometric nature of our constructions of roots of crosscap slides and crosscap transpositions is quite similar to that of Margalit and Schleimer, it is natural to ask if the roots constructed in the proof of Main Theorem are also of maximal degree. We plan to answer this question in subsequent work, but it turns out that even in the case of Dehn twists on nonorientable surfaces, the study of possible degrees of their roots is significantly more complicated than in the orientable case-see for example Sect. 4 [10].