Miscellaneous problems 3 — This one is due to Mike Hopkins.

This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-infinity self equivalences of E. I think this has a classifying space B, because of composition of self equivalences. Given a map X --> B you should be able to construct a Thom spectrum, and it should be some kind of half-smash product of E and X. Mike had a more explicit description, which I seem to have forgotten.