I will consider a functor L_H(-) from the category of nice topological spaces to itself. This functor satisfies several important qualities, but one of them is that it turns maps that induce an isomorphism with respect to integral homology into homotopy equivalences. In fact, I will call the above functor localization with respect to integral homology. It’s intended to be analogous to localization in a ring, instead of inverting certain elements we invert certain maps. It turns out one can do such a thing for any generalized homology theory, but I’ll be considering mainly L_K(-), which is localization with respect to complex K-theory. To investigate such a thing, it’s natural to ask which spaces are sent by this functor to something contractible. I’ll discuss a result by Emmanuel Dror-Farjoun along these lines and work I’ve been doing in this area.