We prove compactness theorems for noncompact 4-dimensional shrinking and steady
gradient Ricci solitons, respectively, satisfying: (1) every bounded open subset can be
embedded in a closed 4-manifold with vanishing second homology group, and (2) are strongly
-noncollapsed on all scales
with respect to a uniform
.
These solitons are of interest because they are the only ones that can arise as
finite-time singularity models for a Ricci flow on a closed 4-manifold with vanishing
second homology group.