Re: loop braid isotopies


Blake Winter wrote:
Any ideas of the reference? I have done some checking around and can't
seem to find any of his work on the matter.

Thanks

I'm unaware of any work on your specific question in the literature,
although I think existing work may essentially cover what you're
looking for with a little effort.

Your question is a "standard" kind of question for which there is a
formalism -- pseudoisotopy theory. This allows you to restate your
question in terms of other questions that one might be able to solve.

You mentioned the case of braid groups. Let me show you how the main
constructions work in that case.

Let B_n be the braid group on n strands, this we will think of as the
fundamental group of the space of n-element subsets of R^2. Call this
space C_n(R^2). ie: B_n = \pi_1 C_n(R^2).

Let P_n(IxR^2) be the space of embeddings of n disjoint, unknotted
intervals in IxR^2 such that there are n fixed endpoints on the {0}xR^2
face, and n free endpoints on the {1}xR^2 face where we demand that
each interval goes from one face to the other. Here I=[0,1] is the
unit interval.

So there is a fibration: P_n(IxR^2) --> C_n(R^2)

given by restriction to the {1}xR^2 face, where the fiber is the space
of embeddings of n disjoint unknotted intervals in IxR^2 which connect
n fixed points in R^2 (ie: unknotted ropes connecting n hooks on the
top with n hooks on the bottom face), call this space R_n(IxR^2).

So in the homotopy long exact sequence of this fibration you get a map
which is exactly what you'd expect (one simply "graphs" the braid):

\pi_1 C_n(R^2) --> \pi_0 R_n(IxR^2)

and our goal is to show this map is an isomorphism -- ie: all such
"ropes" can be realized as genuine braids where the strands descend at
constant velocity.

To do this, one needs to show that the two maps below are trivial:

1) \pi_0 R_n(IxR^2) --> \pi_0 P_n(IxR^2)

2) \pi_1 P_n(IxR^2) \to \pi_1 C_n(R^2)

For (1) we show P_n(IxR^2) is path-connected. This uses standard Dehn's
lemma type argument that is typical in 3-manifold theory (these go back
to Schubert, Waldhausen, etc). One could avoid this step by simply
restricting to the path-component of the trivial braid in P_n(IxR^2) as
well.

To show the map (2) is trivial one could reduce this to an application
of Artin's embedding of the braid group in Aut(F_n) (automorphisms of
the free group on n elements), because one can compute the
corresponding automorphism "upstairs" via \pi_1 P_n(IxR^2) and check
that there it is trivial.

So you see now roughly what you're up against for "loop braid groups"
(which I usually call "the Dahm group of unlinks" or "the fundamental
group of the space of unlinks"). It seems to me like Dahm's theorem,
that the fundamental group of the space of unlinks embeds in Aut(F_n)
-- this theorem, via the above type of argument, would imply a positive
answer to your initial question.

-ryan

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