Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g: B \rightarrow X$ such that $pg = f$.

Q1: What spectral sequences or other technology exists for computing the (homotopy groups of the) space of such E-infinity lifts?

Any $E_\infty$ lift $g: B \rightarrow X$ provides a lift of the map of commutative monoids $\pi_0 B \rightarrow \pi_0 Y$ across the map $\pi_0 X \rightarrow \pi_0 Y$.

Q2: Do all the techniques for computing E-infinity lifts in effect require that you first solve this $\pi_0$ lifting problem in commutative monoids, and begin an obstruction calculation from there, or are there techniques that solve both the $\pi_0$ problem and the E-infinity problem 'simultaneously' and perhaps in a way that eases both computations?

I am particularly interested in merely knowing if there exists an $E_\infty$ lift, thus might have asked the seemingly more basic question:

Q1': When is there an E-infinity lift of an E-infinity map?

I think the obstruction groups answering Q1' are liable to come packaged in the answers to Q1/Q2, but if there are separate techniques for the existence question, that would also be helpful.

Remark: I could imagine that one answer to Q1 involving relative Andre-Quillen cohomology might be extracted from Goerss-Hopkins, Moduli Problems for Structured Ring Spectra, but perhaps there are more elementary means. I'd be very interested for answers to Q1 along those or especially other lines of thought, and any ideas about Q2. Thanks!

anybodyhas ever computed something like this, where X isn't grouplike, and B isn't something simple (like a free E-infinity space). I can't think of any such case, though that may just reflect my ignorance. $\endgroup$ – Charles Rezk Feb 17 '10 at 20:46