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Is there an explicit description of the Pontryagin product on the homology of $CP^{\infty}$? Also, what is the homology of the classifying spaces $BU(n)$?

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Look it up in "Topology of Lie groups I, II" by Mimura and Toda. –  Fernando Muro Apr 14 '13 at 10:01
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The Hopf algebra structure on homology is dual to cohomology. One often uses that the diagonal of a generator in cohomology is easy to work out (for degree reasons) and that it is an algebra map. The Hopf algebra structure on $H^{*}(CP^{\infty};R)$ is given by $R[[x]]$ where x is primitive and so the dual Hopf algebra structure on homology will be a divided power algebra where the element dual to $x$ is primitive (and this determines the rest of the structure). One can think of this as the algebra with generators $\frac{1}{k!}\frac{d^{k}}{dx^{k}}$ and the coproduct as given by the Leibniz rule –  Callan McGill Apr 14 '13 at 10:30
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