Let {X} be a connected pointed space. The spaces {\Omega^n \Sigma^n X} 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 {n = 1}. In this case, the James construction gives a homotopy equivalence

\displaystyle JX \rightarrow \Omega \Sigma X,

where {JX} is the free topological monoid on {X}, subject to the relation that the basepoint {\ast \in X} be the identity. In other words, one can describe {JX} by taking the disjoint union {\bigsqcup_{n \geq 0} X^n} (the free topological monoid on {X}) and making identifications to make {\ast} be the identity. The space {JX} comes with a canonical filtration

\displaystyle \ast \subset X \subset J_2 X \subset J_3 X \subset \dots \subset JX,

where {J_n X} denotes elements of {JX} which can be expressed as products of {\leq n} elements of {X}. The “associated graded” quotients of {JX} are given by the smash powers {X^{\wedge n}, n > 0}.

More generally, let {\mathcal{O}^{(n)}} be the little {n}-cubes operad; then {\mathcal{O}^{(n)}} comes with a canonical action on any {n}-fold loop space. For a connected space {X}, it is a theorem of May (related to delooping machinery) that {\Omega^n \Sigma^n X} is homotopy equivalent to the free {\mathcal{O}^{(n)}}-algebra on {X}, again subject to the condition that the base-point be the unit element. In other words, we can model {\Omega^n \Sigma^n X} via the space

\displaystyle C_n X \stackrel{\mathrm{def}}{=} \left( \bigsqcup_{j \geq 0} \mathcal{O}^{(n)}(j) \times_{\Sigma_j} X^j \right)/\sim,

where the relation {\sim} is precisely the one that makes {\ast} the identity element. This space, too, has a filtration via the subspaces

\displaystyle \ast \subset C_n^{1}(X) \subset C_n^{2}(X) \subset \dots,

where {C_n^{k}(X)} consists of the image of the first {k+1} pieces of the disjoint union above. The associated graded pieces are given by the quotients

\displaystyle \mathrm{gr}_j C_n(X) = (\mathcal{O}^{(n)}(j) \times_{\Sigma_j} X^{\wedge j})/( \mathcal{O}^{(n)}(j) \times_{\Sigma_j} \ast), \quad j > 0:

in other words, we take the coinvariants of {\mathcal{O}^{(n)}(j) \times X^{\wedge j}} 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

\displaystyle \Sigma^\infty \Omega^n \Sigma^n X = \bigvee_{j \geq 1} \Sigma^\infty \mathrm{gr}_j C_n(X).

 1. The general operadic statement

When we let {n \rightarrow \infty}, there is a model for {\Omega^\infty \Sigma^\infty X} for a pointed space {X}: it is the free {E_\infty}-space on {X} with {\ast} as identity. In other words, choose an {E_\infty}-operad {\mathcal{O}} (that is, one with contractible spaces with free {\Sigma}-action) and repeat the same construction with {\mathcal{O}}, and get a filtered space which models the homotopy type of {\Omega^\infty \Sigma^\infty X}. The associated graded pieces are the homotopy quotients

\displaystyle (X^{\wedge j})_{h \Sigma_j}

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

\displaystyle \Sigma^\infty \Omega^\infty \Sigma^\infty X = \bigvee_{j \geq 1} \Sigma^\infty (X^{\wedge j})_{h \Sigma_j}.

In particular, this means that we can compute the homology of {\Omega^n \Sigma^n X} 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 {\mathcal{O}} be any operad in spaces, so that we can talk about {\mathcal{O}}-algebras in spaces; to make things homotopy invariant, we should assume the {\Sigma}-action on {\mathcal{O}} is free, and let’s also assume {\mathcal{O}(0) = \ast}. Then we can define a free functor

\displaystyle \mathbf{Free}: \mathbf{Spaces} \rightarrow \mathrm{Alg}(\mathcal{O}),

which sends a space {X} (unpointed) to the free {\mathcal{O}}-algebra on {X}: this is

\displaystyle \mathbf{Free}(X) = \bigsqcup_{j \geq 0} \mathcal{O}(j) \times_{\Sigma_j} X^{j}.

Given a pointed space {(X, \ast)}, we can define an {\mathcal{O}}-algebra {\mathbf{Free}_*(X)} via the pushout of {\mathcal{O}}-algebras

note that this is a push-out in the category of {\mathcal{O}}-algebras rather than in the category of spaces. When {\mathcal{O}} is the associative operad (or any {A_\infty}-operad), then the construction {\mathbf{Free}_*} is precisely the James construction. When {\mathcal{O}} is the little {n}-cubes operad, one gets the above model for {\Omega^n \Sigma^n X}.

In other words, one has that

\displaystyle \mathbf{Free}_*(X) = \left(\bigsqcup_{j \geq 0} \mathcal{O}(j) \times_{\Sigma_j} X^{j}\right)/\simeq,

where the relation {\simeq} declares that {\ast} be a unit. That is, {\mathbf{Free}_*(X)} is the free {\mathcal{O}}-algebra in the category of pointed spaces. This is not given by something of the form {\bigvee \mathcal{O}(j) \times_{\Sigma_j} X^{j}}; 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 {\mathbf{Free}_*(X)}, whose associated graded quotients are the homotopy quotients {(\mathcal{O}(j) \times X^{\wedge j})_{h \Sigma_j}} taken in the category of pointed spaces.

Theorem 2 (Cohen) For a pointed space {X}, one has a stable splitting

\displaystyle \Sigma^\infty \mathbf{Free}_*(X) = \bigvee_{j \geq 1} ( \mathcal{O}(j) \otimes \Sigma^\infty X^{\wedge j})_{\Sigma_j},

where {\otimes} refers to the enrichment of spectra over spaces.

In other words, {A \otimes X = \Sigma^\infty_+ A \wedge X}.

2. Proof of the splitting

In order to prove this, consider first the functor

\displaystyle \Sigma^\infty_+ : \mathbf{Spaces} \rightarrow \mathbf{Sp};

this is a symmetric monoidal, colimit-preserving functor, and it sends {\mathcal{O}}-algebras in spaces to {\mathcal{O}}-algebras in spectra. Moreover, it commutes with the natural “free” functors for forming free {\mathcal{O}}-algebras in spaces or in spectra. This means that if we take the pushout diagram in {\mathcal{O}}-algebras in spaces,

and apply {\Sigma^\infty_+}, we get a push-out diagram in {\mathcal{O}}-algebras in spectra

Now, the key observation in the argument is that, for a pointed space {X}, the natural cofiber sequence

\displaystyle S^0 \rightarrow X_+ \rightarrow X

splits after applying {\Sigma^\infty}, so that we have an equivalence

\displaystyle \Sigma^\infty_+ X \simeq S^0 \vee \Sigma^\infty X,

in such a way that the inclusion {S^0 \rightarrow \Sigma^\infty_+ X} corresponds to the inclusion {\Sigma^\infty_+ \ast \rightarrow \Sigma^\infty_+ X}. We find that

\displaystyle \mathbf{Free}(\Sigma^\infty_+ X) \simeq \mathbf{Free}(\Sigma^\infty X) \sqcup \mathbf{Free}( S^0)

where the coproduct is taken in {\mathcal{O}}-algebras in spectra: that is, it is a type of tensor product. But we have, again in {\mathcal{O}}-algebras,

\displaystyle \Sigma^\infty_+ \mathbf{Free}_*(X) \simeq \left(\mathbf{Free}(\Sigma^\infty X ) \sqcup \mathbf{Free}( S^0) \right) \sqcup_{\mathbf{Free}(S^0)} S^0,

where {S^0} is the initial object in {\mathcal{O}}-algebras. This gives formally an equivalence

\displaystyle \Sigma^\infty_+ \mathbf{Free}_*(X) \simeq \mathbf{Free}(\Sigma^\infty X) \simeq \bigvee_{j \geq 0} \left( \mathcal{O}(j) \otimes \Sigma^\infty X^{\wedge j}\right)_{h \Sigma_j},

which is the Snaith splitting, after quotienting by {S^0}.

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