Question

Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the Dyer-Lashof algebra.

Now there is a monomorphism $H(X) \rightarrow$ $H(QX)$ induced from $X \rightarrow QX$.

My question is if $H(QX)$ is generated over $H(X)$ in some sense. Or is there some other relation between these two homologies?

Answer 1

Let $X$ be a connected space, and let $\lbrace x_\lambda\rbrace$ be a homogeneous basis for $H_\ast(X;\mathbb{F}_2)$. Then $$H_\ast(QX;\mathbb{F}_2) = \mathbb{F}_2 [Q^I x_{\lambda} \mid I\mbox{ admissible of excess }e(I)>\mathrm{dim}\,\, x_\lambda ].$$

That is, the homology of $QX$ is a polynomial algebra with generators certain iterated Kudo-Araki operations on the basis of the homology of $X$. There is a similar result with coefficients mod $p$, $p$ an odd prime, involving Dyer-Lashof operations and the Bockstein operator. (The situation is reminiscent of Serre's Theorem on the cohomology of Eilenberg-Mac Lane spaces.)

The reference is (Section 5 of)

Dyer, Eldon; Lashof, R. K. Homology of iterated loop spaces. Amer. J. Math. 84 1962 35–88.

You will also find a nice discussion in

Eccles, P. J. Characteristic numbers of immersions and self-intersection manifolds. Topology with applications (Szekszárd, 1993), 197–216, Bolyai Soc. Math. Stud., 4, János Bolyai Math. Soc., Budapest, 1995.