Hope all of your loved ones are alright. There are 9 new papers on Hopf in the last two weeks. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on the homology of the mapping space.
Similar comments hold when a cohomology theory is applied.
In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to the suspension spectrum of its 0th space. The main technical tool involves working with simplicial resolutions of spaces rather than with spaces themselves. It seems clear that the deformation theory can be applied with little change to study other moduli questions in topology.