On the nonrealizability of braid groups by homeomorphisms

Chen, Lei¹

¹ Department of Mathematics, California Institute of Technology, Pasadena, CA, United States

Geometry & topology, Tome 23 (2019) no. 7, pp. 3735-3749.

Résumé

We show that the projection ${Homeo}_{+}\left(D_n^2\right)\to B_n$ does not have a section for $n\ge 6$; ie the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary pointwise and $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic (2007) that the mapping class group of a surface of genus $g$ cannot be geometrically realized as a group of homeomorphisms when $g\ge 2$.

Détail

DOI : 10.2140/gt.2019.23.3735

Classification : 37E30, 57M60
Keywords: dynamics of surfaces, braid groups, Nielsen realization

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