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 ${S^0}$.