Is there a rings operad?

One of the things I find pretty neat about the splicing operad is how much goes into it. It's a relatively complicated structure and I'm a little surprised there aren't many similar, related structures in the literature.

I suspect one of the main reasons the splicing operad wasn't discovered earlier is that it isn't some variant on a common construction in one field. The motivation for the splicing operad construction was twofold: the action of the cubes operads on long embedding spaces. Moreover, this action does not see everything about the homotopy-type of embedding spaces. In dimension 3 splicing or the JSJ decomposition was the key other ingredient to the homotopy-type. That gave the desire to see splicing fit into a higher-algebraic framework.

This post is about how the idea of the splicing operad fails if one perturbs it just a little bit. An alternative title for this post could be Is there an operad of trivial linksI know I'm not the only one interested in such an operad.

As in my previous post, there is a splicing operad for every embedding space EC(j,M) where j is an integer and M is a compact manifold. When M is a point or a closed manifold this splicing operad isn't a big innovation. The splicing operad is mostly interesting when M is a manifold with non-empty boundary. It's particularly interesting to me when M a disc. At level k of the splicing operad for EC(j,M), an element is represented by a symbol L. L is a (k+2)-tuple L=(L0,L1,...,Lk,s), where L0 is an element of EC(j,M), and the other Li's are "hockey pucks", and s is a permutation of the set {1,...,k}. So given L in the splicing operad you can "forget" about L0. That gives you an object (L1,...,Lk,s) which, for lack of a better word, you could say is an object that belongs to a "space of trivial links". For several reasons you could hope for an operad structure on this "space of trivial links" making this map a map of operads. Unfortunately, that can't be done.

There a few good reasons for this. For one, the structure maps in the splicing operad make essential use of the L0 part, even if all you care about is the pucks. But you could step further back, and ask what would happen if you simply got rid of all the L0 parts and defined the structure maps like in the operad of overlapping cubes/discs. i.e. simple composition of functions -- no elaborate conjugations like in the splicing operad. So for example, given (L1,...,Lk,s) and (J1,...,Jl,t), the o1-operation applied to these two would be defined to be (L1 o J1, ..., L1 o Jl, L2, ..., Lk, the corresponding permutation). This definition leads to problems, too. The big one being that the disjointness one requires for the "annular part" of the hockey pucks is not preserved under composition. It happens in very simple cases, for example when k=2, L1=L2=the identity "puck", taking only two pucks as input, and simply choosing them to have non-disjoint annular parts. So I'm talking about the operad structure map O(2)xO(1)xO(1) --> O(2).

My suspicion is that there is some natural higher algebraic structure for spaces of trivial links but it seems like the splicing operad isn't quite the right direction to go. In particular I think if one wants any kind of efficient description of the homotopy and homology of spaces of trivial links, you're going to want to give these spaces a suitable higher algebraic structure. Exactly what that is, I'm not so sure. On any type of space of trivial links in Rn there is an action of the operad of framed n-discs, but this gives a relatively shallow perspective on the homotopy-type since pulling one component of a link through another is not represented by this action. Presumably there is some better way to study these spaces? That's one hope.