10
$\begingroup$

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?

$\endgroup$
14
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Not at all, its always nice to get a chance to advertise one's thesis advisor's papers! $\endgroup$ – Mark Grant Jan 27 '12 at 7:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.