Homology of infinite loop spaces $QX$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:15:00Z http://mathoverflow.net/feeds/question/86718 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86718/homology-of-infinite-loop-spaces-qx Homology of infinite loop spaces $QX$ berl13 2012-01-26T14:02:42Z 2012-01-27T07:54:11Z <p>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. </p> <p>Now there is a monomorphism $H(X) \rightarrow$ $H(QX)$ induced from $X \rightarrow QX$.</p> <p>My question is if $H(QX)$ is generated over $H(X)$ in some sense. Or is there some other relation between these two homologies?</p> http://mathoverflow.net/questions/86718/homology-of-infinite-loop-spaces-qx/86721#86721 Answer by Mark Grant for Homology of infinite loop spaces $QX$ Mark Grant 2012-01-26T14:25:02Z 2012-01-27T07:54:11Z <p>Let $X$ be a connected space, and let $\lbrace x_\lambda\rbrace$ be a homogeneous basis for $H_\ast(X;\mathbb{F}_2)$. Then <code>$$H_\ast(QX;\mathbb{F}_2) = \mathbb{F}_2 [Q^I x_{\lambda} \mid I\mbox{ admissible of excess }e(I)&gt;\mathrm{dim}\,\, x_\lambda ].$$</code> </p> <p>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.)</p> <p>The reference is (Section 5 of)</p> <p>Dyer, Eldon; Lashof, R. K. <em>Homology of iterated loop spaces</em>. Amer. J. Math. 84 1962 35–88.</p> <p>You will also find a nice discussion in </p> <p>Eccles, P. J. <em>Characteristic numbers of immersions and self-intersection manifolds</em>. Topology with applications (Szekszárd, 1993), 197–216, Bolyai Soc. Math. Stud., 4, János Bolyai Math. Soc., Budapest, 1995.</p>