Errata to “A presentation for Hilden’s subgroup of the braid group”

After publication Allen Hatcher found a gap in the proof that the complex $X_n$ is simply connected. This complex is defined in terms of isotopy classes of discs, but the argument uses representatives of the isotopy classes. There was an implicit assumption that for an edge path in the complex there exists sufficiently nice representatives of each isotopy class. In this paper the properties of these representatives will be made explicit. It is clear that such representatives exist for a path, the problem is that for a loop it is not obvious that the representative at the beginning and end can be chosen to coincide. This paper addresses this problem and contains the complete proof that $X_n$ is simply connected, incorporating all of the necessary changes. There were also small errors in Figure 7 and Figure 8 and the correct versions of these are included.

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Published 28 February 2011