Generating twist subgroup of mapping class group of non-orienatable surface by involutions
Let denote the closed non-orientable surface of genus and let denote the mapping class group of . Let denote the twist subgroup of which is the subgroup of is generated by all Dehn twists. In this thesis, we proved that is generated by six involutions for .
For , let denote the closed connected non-orientable surface of genus . The mapping class group is the group of isotopy classes of diffeomorphisms of . A simple closed curve in is two-sided (resp. one-sided) if a regular neighborhood of , denoted by , is an annulus (resp. a Möbius band). Let be a two-sided simple closed curve on . By the definition, the regular neighborhood of is an annulus, and it has two possible orientation. Now, we fix one of its two possible orientations. For two sided simple closed curve , we can define the Dehn twist . We indicate the direction of a Dehn twist by an arrow beside the curve as shown in Fig. 1.
For , Lickorish ([Li1], [Li2]) first proved that is not generated by Dehn twists, and is generated by Dehn twists and Y-homeomorphisms. Chillingworth [C] found a finite set of generators of this group. The number of Chillingworth’s generators is improved to by Szepietowski [S3]. Hirose [Hi] proved that his generating set is the minimal generating set by Dehn twists and Y-homemorphisms. Szepietowski show that is generated by four involutions for [S2]. Denote by the subgroup of is generated by all Dehn twists. We call the twist subgroup of . The group is an index subgroup of (see [Li1]). In paricular, is finitely generated. Chillingworth [C] proved that is generated by a Dehn twist if , two Dehn twists if , Dehn twists if is odd, Dehn twists if is even. Stukow [St2] proved that admits a finite presentation. Omori [O] showed that is generated by Dehn twists for . Recently, Du [Du] showed that is generated by three elements of finite order if is odd. One of his generator is of order and the other two are of order .
On the other hand, it is not known the generating set for consisting only of involutions. In this paper, we showed that is generated by involutions.
For or , is generated by six involutions.
The paper is organized as follows. In Section we recall the properties of Dehn twists and Y-homeomorphisms. In Section we construct involutions of and prove the theorem 1.1.
We represent the surface as a connected sum of projective planes as in Fig. 2. In this Figure, each encircled cross mark represents a crosscap: the interior of the encircled disk is to be removed and each pair of antipodal points on the boundary are to be identified.
Let be one-sided simple closed curves on as in Fig. 4.
A Y-homeomorphism is defined as follow. For a one-sided simple closed curve and a two-sided oriented simple closed curve which intersects transversely in one point, the regular neighborhood of is homomeomorphic to the Klein bottle with one hole. Let be the regular neighborhood of . Then the Y-homeomorphism is the isotopy class of the diffeomorphism obtained by pushing once along keeping the boundary of fixed (see Fig. 5).
Dehn twists and Y-homeomorphisms have the following properties.
For any element in and a two-sided simple closed curve , we have
where if is an orientation preserving diffeomorphism (resp. orientation reversing diffeomorphism), then (resp. ).
For a one-sided simple closed curve , a two-sided simple closed curve , and diffeomorphism on , we have the following.
Omori [O] reduced the number of Dehn twist generators for and showed:
For , is generated by Dehn twists and .
The minimum number of generators for by Dehn twists is at least for (see [O]). Omori asked the following problem: Which of and is the minimum number of generators for by Dehn twists when ?
We set a basis of as in Fig. 6.
It is known that a homology group has a basis in which every linear map , induced by a diffeomorphim , has a matrix with integral coefficients. Therefore we can define the homomorphism as follows: . We recall the properies of this homomorphism (see [St1]).
Let be an element of . Then (resp. ) if is (resp. is not) in .
Let be any Y-homeomorphism. Then .
3. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. For , We assume that .
so that in Fig .7 is symmetrical with respect to the plane, dnoted by mirror, illastrated in the middle of this surface. Let be a reflection in the mirror and let be a composition . Then is involution of . By an easy calculation, we found if is odd and if is even. When is odd, is not element in by lemma 2.5. So, we consider a product . This element is in . Since and since is orientation reversing, we found that is an involution by lemma 2.2.We rewirte as . Therefore, for , is involution in .
In the same way, we define as reflection in the mirror in Fig. 8.
We found if is odd and if is even. When is even, is not element in by lemma 2.5. So, we consider a product . This element is in . By lemma 2.2, is an involution. We rewirte as . Therefore, for , is involution in .
We construct third involution. We define as reflection in the mirror in Fig. 9 and we found . Hence, the element is involution in .
Let , , and be , , and , respectively. By lemma 2.1, is involution in for . Let be a subgroup of generated by , , , , , and . Since is a product , is a product of two involutions. In the same way, and are products of two involutions. We show that are products of involutions.
Let and be two-sided simple closed curves on . For , the symbol means that or . Since is mapped to by , , and (see Fig. 10), is a product of involutions by lemma 2.1 for . Since all Omori’s generators for is in , we prove that is equal to .
If (resp. ) is not in , we rewrite (resp. ) as (resp. ). Since the element is not in , we define as a product . This element is in and this element is an involution by lemma 2.2. Involution , , and are the same as the case of . In the same way as the case of , all Omori’s generators for is in .
So we proved Theorem 3.1:
For or , is generated by involutions , , , , , and .
4. Concluding Remarks
We found the construction of needs or . By constructing of three other involutions instead of , can be generated by eigth involutions in case of . In this case, is generated by in Section and additionally , , and . Let and are involutions of which are reflections in the mirrors in Fig. 14 and Fig. 15, respectively. We define and . A element is a product of . Then, , , and are involutions in by lemma 2.2 and 2.5. In the same way as Section 3, We can proved the following:
For , is generated by eight involutions.
In the case of , Szepietowski show that is generated by involutions for .
We can consider following problem:
For , can the twist subgroup be generated by involutions?
The author would like to thank Susumu Hirose and Naoyuki Monden for many advices. He would also like to thank Genki Omori for many helpful suggestions and comments of the twist subgroup.
- [C] D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Math. Proc. Cambridge. Philos. Soc., 65 (1969), 409–430.
- [Du] Xiaoming Du, The torsion generating set of the Dehn twist subgroups of non-orientable surfaces, arXiv:math.GT/04883 v3 19 Nov 2018.
- [FM] B. Farb and D.Margalit, A Primer on Mapping Class Groups, Princeton University Press, 2011.
- [Hi] S. Hirose, Generators for the mapping class group of the nonorientable surface, Kodai Math. J. 41 (2018), 154–159
- [Ko1] M. Korkmaz, Mapping class groups of nonorientable surfaces, Geom. Dedicata, 89 (2002), 109–133.
- [Li1] W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Cambridge. Philos. Soc., 59 (1963):307–317.
- [Li2] W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface, Math. Proc. Cambridge. Philos. Soc., 61 (1965):61–64.
- [O] G. Omori, A small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists, Hiroshima Math. J. 48 (2018), no.1, 81-â88.
- [St1] M. Stukow, The twist subgroup of the mapping class group of a nonorientable surface, Osaka J. Math. 46 (2009), no. 3, 717–738
- [St2] M. Stukow, A finite presentation for the twist subgroup of the mapping class group of a nonorientable surface, Bull. Korean Math. Soc. 53 (2016), no. 2, 601–614.
- [S1] B. Szepietowski, Involutions in mapping class groups of nonorientable surfaces, Collect. Math., 55 (2004), 253–260.
- [S2] B. Szepietowski, The Mapping Class Group of a Nonorientable Surface is Generated by Three Elements and by Four Involutions, Geom. Dedicata, 117 (2006), 1–9.
- [S3] B. Szepietowski, A finite generating set for the level 2 mapping class group of a nonorientable surface, Kodai Math. J. 36 (2013), 1–14.