Lifting homotopy algebra maps to algebra maps

I will discuss an obstruction theory for lifting homotopy algebra maps
to algebra maps developed with Niles Johnson. Although our primary
interest is in a topological setting, I will mostly restrict myself to
the algebraic setting during this talk. I will examine the analogue of
this question in the setting of rational chain complexes and show how
examples arising from the study of manifolds rationally give
interesting examples in this situation.  Finally, I will explain why
these examples can be moved into stable homotopy theory to answer
questions about lifting H_\infty ring maps.