Mapping class groups of covers with boundary and braid group embeddings

Ghaswala, Tyrone¹ ; McLeay, Alan²

¹Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada
²Mathematics Research Unit, University of Luxembourg, Esch-sur-Alzette, Luxembourg

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 239-278

Résumé

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of nongeometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman–Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

Détail

DOI : 10.2140/agt.2020.20.239

Classification : 57M12, 20F36, 20L05
Keywords: mapping class groups, braid groups, covering spaces

Bibliographie

[1] M Alp, C D Wensley, Automorphisms and homotopies of groupoids and crossed modules, Appl. Categ. Structures 18 (2010) 473 | DOI

[2] J Aramayona, C J Leininger, J Souto, Injections of mapping class groups, Geom. Topol. 13 (2009) 2523 | DOI

[3] S J Bigelow, R D Budney, The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001) 699 | DOI

[4] J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. 97 (1973) 424 | DOI

[5] C F Bödigheimer, U Tillmann, Embeddings of braid groups into mapping class groups and their homology, from: "Configuration spaces" (editors A Bjorner, F Cohen, C De Concini, C Procesi, M Salvetti), CRM Series 14, Ed. Norm. (2012) 173 | DOI

[6] T E Brendle, D Margalit, The level four braid group, J. Reine Angew. Math. 735 (2018) 249 | DOI

[7] T Brendle, D Margalit, A Putman, Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = −1, Invent. Math. 200 (2015) 263 | DOI

[8] R Brown, Topology and groupoids: a geometric account of general topology, homotopy types and the fundamental groupoid, BookSurge (2006)

[9] P Dehornoy, A fast method for comparing braids, Adv. Math. 125 (1997) 200 | DOI

[10] H Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000) 237 | DOI

[11] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[12] T Ghaswala, R R Winarski, Lifting homeomorphisms and cyclic branched covers of spheres, Michigan Math. J. 66 (2017) 885 | DOI

[13] P J Higgins, Notes on categories and groupoids, 32, Van Nostrand Reinhold (1971)

[14] N Kawazumi, Y Kuno, Groupoid-theoretical methods in the mapping class groups of surfaces, preprint (2011)

[15] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235 | DOI

[16] B C Kim, Y Song, Configuration spaces, moduli spaces and 3–fold covering spaces, Manuscripta Math. 160 (2019) 391 | DOI

[17] S Lefschetz, On the fixed point formula, Ann. of Math. 38 (1937) 819 | DOI

[18] C Maclachlan, W J Harvey, On mapping-class groups and Teichmüller spaces, Proc. London Math. Soc. 30 (1975) 496 | DOI

[19] D Margalit, R R Winarski, The Birman–Hilden theory, preprint (2017)

[20] A Mcleay, Subgroups of mapping class groups and braid groups, PhD thesis, University of Glasgow (2018)

[21] C T Mcmullen, Braid groups and Hodge theory, Math. Ann. 355 (2013) 893 | DOI

[22] T Morifuji, On Meyer’s function of hyperelliptic mapping class groups, J. Math. Soc. Japan 55 (2003) 117 | DOI

[23] Y Song, The braidings in the mapping class groups of surfaces, J. Korean Math. Soc. 50 (2013) 865 | DOI

[24] Y Song, U Tillmann, Braids, mapping class groups, and categorical delooping, Math. Ann. 339 (2007) 377 | DOI

[25] M Stukow, Conjugacy classes of finite subgroups of certain mapping class groups, Turkish J. Math. 28 (2004) 101

[26] M Stukow, Small torsion generating sets for hyperelliptic mapping class groups, Topology Appl. 145 (2004) 83 | DOI

[27] B Szepietowski, Embedding the braid group in mapping class groups, Publ. Mat. 54 (2010) 359 | DOI

[28] B Wajnryb, Relations in the mapping class group, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 115 | DOI

[29] R R Winarski, Symmetry, isotopy, and irregular covers, Geom. Dedicata 177 (2015) 213 | DOI