Infinitely Generated Braid Groups
There are many ways to generalize this notion to an infinite number of strands. The simplest way is take the direct limit of braid groups, where the attaching maps send the generators of to the first generators of (i.e., by attaching a trivial strand). Fabel has shown that there are two topologies that can be imposed on the resulting group each of whose completion yields a different group. One is a very tame group and is isomorphic to the mapping class group of the infinitely punctured disk — a discrete set of punctures limiting to the boundary of the disk.
The second group can be thought of the same as with finite braid groups. Place a strand at each of the points and the set of all braids — where a braid is defined to be a collection of paths from the points to the points so that the function yields a permutation on endpoints — is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups and to the fundamental group of the Hilbert cube minus the set .
Read more about this topic: Braid Group
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