Continuation: Consider a relative two-complex (L,K) where K is an aspherical CW complex (of arbitrary dimension). The pair (L,K) is said to be “aspherical" if the second relative homotopy group of the pair (L,K) is trivial. In this talk, I will consider the alternatives to asphericity, examining in detail the case where the quotient complex L/K is modeled on the presentation (x:xxx^-1) [and possibly (x:xxx) depending on how far I get over spring break]. Asphericity was classified in these settings by M. Edjvet 1993 [and B/Pride in 1992]. I will survey the non-aspherical cases and conclude that except for a small handful of unresolved cases, if (L,K) is not aspherical, then the fundamental group of L either contains a "forbidden" finite subgroup or else splits as an amalgamated free product over a virtual three-manifold group. Some interesting and well-known groups and three-manifolds occur in this survey. Also some three-manifolds that I do not yet know how to identify.