## Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop spaces (spectra).

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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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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. – Dev Sinha Apr 12 2010 at 23:03
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. – Bill Kronholm Apr 13 2010 at 15:14

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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. – Peter May Jan 2 2012 at 0:07

## 2 Answers

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.

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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."

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