Minimal Collapsing Extensions of Models of ZFC

We present some results concerning extensions of models of ZFC in which cofinalities of cardinals are changed and/or cardinals are collapsed, in particular on minimal such extensions. Our main tools are perfect tree forcing PF(S) and Namba forcing Nm(S). We prove that if N 2 M is an extension such that (i) M k K = I+ > H,, (ii) 1 fl N E M and (iii) N = Mlf] for some cofinal f : oo+ K, then N 2 M is cf(K) = o,-minimal. On the other hand Namba forcing Nm(S) where S is a normal ultrafilter on a measurable cardinal K produces an extension satisfying (iii) and (ii) for every 1< K, which is not cf(rc) = w,-minimal. We show that if S is an K,-complete splitting criterion on K then Pf(S) collapses K+ to X, (assuming GCH). Moreover, we prove, under some reasonable assumptions, that every extension changing the cofinality of a successor cardinal K must collapse K+. Using these results and results on trees from Sections 2, 3 we construct, assuming e.g. GCH, for every regular uncountable K a IKJ = &-minimal extension, a cf(K) = o,-minimal extension and a IK+[ = X,-minimal extension.