The homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more accessible Luminy notes of Bisson-Joyal). Has anyone used this to construct a spectral sequence converging under some assumptions to $[X,Y]$, the homotopy classes of infinite-loop-maps between $X$ and $Y$, which starts with some kind of derived (Ext/Tor) maps between their homology in the category of algebras over the Dyer-Lashof algebra? Have any calculations been done with such a spectral sequence?
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$\begingroup$ Note: because of "duality" between the Dyer-Lashof algebra action on the homology of \Omega^\infty \Sigma^\infty X and the Steenrod action on the cohomology of X, one might recover something close to the Adams Spectral sequence in these cases. $\endgroup$– Dev SinhaCommented Apr 12, 2010 at 23:03
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$\begingroup$ Does Peter May say something about this in his paper "A general algebraic approach to Steenrod operations" (ams.org/mathscinet-getitem?mr=281196) ? I can't find my copy, so I don't remember what spectral sequences he writes down. Probably there are no calculations in that paper, in any event. $\endgroup$– Bill KronholmCommented Apr 13, 2010 at 15:14
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1$\begingroup$ No, not there, but a spectral sequence of the sort requested does appear in ``The geometry of iterated loop spaces'', pages 155-156. That is probably the first reference to such a spectral sequence, but not the best. I believe Kraines and Lada made some calculational use of such a spectral sequence. $\endgroup$– Peter MayCommented Jan 2, 2012 at 0:07
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2 Answers
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This might not quite be what you're looking for, Dev, but you should check out Paul Goerss and Mike Hopkins' "Multiplicative ring spectra project," on Paul's webpage. They construct such a spectral sequence using Andre-Quillen cohomology in "Moduli spaces of commutative ring spectra," and "Andre-Quillen (co-)homology for simplicial algebras over simplicial operads." A relevant theorem would be 4.3 in the first reference, which gives the spectral sequence.
Though this doesn't use Dyer-Lashof operations, they appear in section 6 (especially Prop 6.4) where Goerss and Hopkins give a second spectral sequence which computes the $E_2$ term of the original spectral sequence. The new $E_2$ term is given in terms of an $Ext$ functor in the category of unstable modules over the Dyer-Lashof algebra.
They use this machinery to show in section 7 that the space of $E_\infty$ maps between Lubin-Tate spectra is homotopically discrete. If you're looking for computations using these spectral sequences, that's a great place to start.
answered Apr 14, 2010 at 4:43
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I'm not sure if this is what you want, but Haynes Miller constructs a spectral sequence computing the homology of a connective spectrum $E$ from the homology of $E_0$ as a Hopf algebra over the Dyer-Lashof algebra in the 1978 Pacific Journal of Mathematics paper "A spectral sequence for the homology of an infinite delooping."
