Let be a connected pointed space. The spaces are very large and complicated, but a fairly concrete model for their homotopy type can be given using the theory of operads. As a toy example, let’s take . In this case, the James construction gives a homotopy equivalence
where is the free topological monoid on , subject to the relation that the basepoint be the identity. In other words, one can describe by taking the disjoint union (the free topological monoid on ) and making identifications to make be the identity. The space comes with a canonical filtration
where denotes elements of which can be expressed as products of elements of . The “associated graded” quotients of are given by the smash powers .
More generally, let be the little -cubes operad; then comes with a canonical action on any -fold loop space. For a connected space , it is a theorem of May (related to delooping machinery) that is homotopy equivalent to the free -algebra on , again subject to the condition that the base-point be the unit element. In other words, we can model via the space
where the relation is precisely the one that makes the identity element. This space, too, has a filtration via the subspaces
where consists of the image of the first pieces of the disjoint union above. The associated graded pieces are given by the quotients
in other words, we take the coinvariants of in the category of pointed spaces (rather than in spaces). It turns out that this filtration actually becomes much nicer stably.
Theorem 1 (Snaith) The above filtration splits after taking suspension spectra. That is, we have an equivalence
1. The general operadic statement
When we let , there is a model for for a pointed space : it is the free -space on with as identity. In other words, choose an -operad (that is, one with contractible spaces with free -action) and repeat the same construction with , and get a filtered space which models the homotopy type of . The associated graded pieces are the homotopy quotients
in the category of pointed spaces. (These are called extended powers and turn out to be quite useful.) When we pass to suspension spectra, the Snaith splitting states that
In particular, this means that we can compute the homology of entirely in terms of the (more tractable) homology of the associated graded.
The purpose of this post is to describe a simple argument, due to Cohen, for this splitting.
Let be any operad in spaces, so that we can talk about -algebras in spaces; to make things homotopy invariant, we should assume the -action on is free, and let’s also assume . Then we can define a free functor
which sends a space (unpointed) to the free -algebra on : this is
Given a pointed space , we can define an -algebra via the pushout of -algebras
note that this is a push-out in the category of -algebras rather than in the category of spaces. When is the associative operad (or any -operad), then the construction is precisely the James construction. When is the little -cubes operad, one gets the above model for .
In other words, one has that
where the relation declares that be a unit. That is, is the free -algebra in the category of pointed spaces. This is not given by something of the form ; in fact, one cannot form free algebras in pointed spaces via quite the same construction. One also has to crush the basepoint (this is related to the fact that taking products with a pointed space does not preserve wedges). However, one does have a filtration of , whose associated graded quotients are the homotopy quotients taken in the category of pointed spaces.
Theorem 2 (Cohen) For a pointed space , one has a stable splitting
where refers to the enrichment of spectra over spaces.
In other words, .
2. Proof of the splitting
In order to prove this, consider first the functor
this is a symmetric monoidal, colimit-preserving functor, and it sends -algebras in spaces to -algebras in spectra. Moreover, it commutes with the natural “free” functors for forming free -algebras in spaces or in spectra. This means that if we take the pushout diagram in -algebras in spaces,
and apply , we get a push-out diagram in -algebras in spectra
Now, the key observation in the argument is that, for a pointed space , the natural cofiber sequence
splits after applying , so that we have an equivalence
in such a way that the inclusion corresponds to the inclusion . We find that
where the coproduct is taken in -algebras in spectra: that is, it is a type of tensor product. But we have, again in -algebras,
where is the initial object in -algebras. This gives formally an equivalence
which is the Snaith splitting, after quotienting by .