Group completion of string links - or a cute idea of Salvatore's

One afternoon I was sitting around with Paolo Salvatore, trying to prove that an operad of little cubes had no hope of acting on spaces of string links. I was certain the operad couldn't act and kept on insisting on it. Eventually Paolo came up with a proof. This post is about Paolo's idea.

To make things precise, let S be an k-element subset of n-1-dimensional Euclidean space. A k-component string link in Rn is a smooth embedding of IxS into IxRn-1 so that {0}xS is sent to {0}xS, and {1}xS is similarly sent to {1}xS. So a string link is a generalized braid. The difference being that projection onto the I factor has no critical points on a braid, while there can be such critical points for a string link.

The isotopy classes of braids form a group under concatenation, and provided k>2, the group is non-abelian. When k is 3, the braid group is known to be the trefoil group, which has presentation <a,b | aba=bab>, among other things this group is a split semi-direct product of Z with a free group on two generators. So it's "very" not abelian.

Quite a bit of my recent work has been about operads acting on spaces. Frequently one deals with groups, and sometimes they're abelian. And quite often your groups are the homotopy groups of a space, and that space has additional structure -- an almost tautologically simple example would be to say that  π1(X) =π0(\Omega X), i.e. the fundamental group of a space can be viewed as the space of path-components of the loop-space on the space. Loop spaces have additional structure, in the form of concatenation maps \Omega X x \Omega X --> \Omega X. These concatenation maps are part of a larger structure -- an action of the operad of little 1-cubes on \Omega X. In this regard, cubes actions on spaces are a type of measurement of "how commutative" a group structure or an operation on a space is. If the operad of 2-cubes were to act on \Omega X, that would say the fundamental group is commutative, among other things. The point being, this is an enhanced formalism to talk about issues such as commutativity.

So we were interested in the case of string links. Could concatenation of string links have a compatible action of the operad of 2-cubes? Well, at least for string links in 3-dimensional Euclidean space the answer is clearly no since string links contain braids, and those are non-abelian. But the answer gets more subtle. Given a topological monoid M there is something called the "group completion" of it. Spaces of string links can be made into topological monoids via a "Moore loop space" construction, so you could ask whether *those* have actions of the operad of 2-cubes. The corresponding argument fails even in R3, since it's not clear what happens to the braid group when you group-complete string links. Group completion is not always an injective process.

Paolo's idea is rather simple. It starts with this paper:

A Lagrangian Representation of Tangles by D. Cimasoni and V. Turaev.

What Cimasoni and Turaev do is find the natural generalization of the Burau representation of the braid groups to string links. String links don't make a group since there's no inverses. But they do make a monoid. So you can talk about their representations. In particular, Cimasoni and Turaev construct a representation of this monoid into a group, and the image is not abelian (because it's not for the braid groups!). So the group-completion of string links is a non-abelian group.

For spaces of string links in R^n (n>3) I believe there's also no action of the operad of 2-cubes although the above argument certainly does not generalize.