Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$

Question

What is the rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$? Here $\Omega$ denotes based loop space.

Answer 1

Let us first consider the homotopy fiber of the map $f\colon\mathbb CP^n\vee S^d \to \mathbb CP^\infty$ which is the inclusion on $\mathbb CP^n$ and is trivial on $S^d$. The homotopy fiber of the map $\mathbb CP^n\to \mathbb CP^\infty$ is $S^{2n+1}$. The homotopy fiber of the map $*\to \mathbb CP^\infty$ is $S^1$, and the homotopy fiber of the trivial map $S^d\to \mathbb CP^\infty$ is $S^1\times S^d$.

It is well-known that taking homotopy pushout commutes with homotopy fibers, therefore the homotopy fiber of $f$ is equivalent to the homotopy pushout of the following diagram. $$ S^{2n+1}\leftarrow S^1 \to S^1\times S^d. $$ I will assume that $n>0$. In this case the map $S^1\to S^{2n+1}$ can only be null homotopic. It follows that the homotopy fiber is equivalent to $$S^{2n+1}\vee S^1_+ \wedge S^d\simeq S^{2n+1}\vee S^d \vee S^{d+1}$$ (for the second equivalence, assume $d\ge 1$).

Applying $\Omega^2$ we get a homotopy fibration sequence $$\Omega^2(S^{2n+1}\vee S^d\vee S^{d+1})\to \Omega^2 (\mathbb CP^n \vee S^d)\to \Omega^2 \mathbb CP^\infty\simeq \mathbb Z.$$ This is a fibration sequence over a homotopically discrete space $\mathbb Z$. Moreover, it is a fibration of (double) loop spaces, so all the homotopy fibers are homotopy equivalent to each other. It follows that there is a homotopy equivalence of spaces $$ \Omega^2 (\mathbb CP^n \vee S^d)\simeq \mathbb Z \times \Omega^2(S^{2n+1}\vee S^d\vee S^{d+1}). $$ So in order to calculate the homology of $\Omega^2 (\mathbb CP^n \vee S^d)$ (with any coefficients), it is enough to calculate the homology of $\Omega^2(S^{2n+1}\vee S^d\vee S^{d+1})$ (with same coefficients).

Let me assume for simplicity that $n\ge 1$ and $d\ge 2$. In this case we can write $$\Omega^2(S^{2n+1}\vee S^d\vee S^{d+1})\simeq \Omega^2\Sigma^2(S^{2n-1}\vee S^{d-2}\vee S^{d-1}).$$

The rational homology of spaces of the form $\Omega^2\Sigma^2 X$ is well-understood. If $X$ is path connected, then it is isomorphic to a certain type of free algebra on $X$. I think some people will call it the free Gerstenhaber algebra and some will say it is the free Poisson algebra. I am not sure what is the most accepted terminology. To get a description of $H_*(\Omega^2 \Sigma^2 X;\mathbb Q)$ as a vector space, first take the free Lie algebra on $H_*(X;\mathbb Q)$, where the Lie bracket has degree $+1$, and then take the free graded commutative algebra on that.

I believe the result about $H_*(\Omega^2\Sigma^2 X)$ is due to Fred Cohen ("The homology of $C_{n+1}$-Spaces", LNM 533). You can find a perhaps somewhat more modern explanation and proof in this paper of Alexander Berglund (Corollary 8 in particular).