  In this paper, we prove finite topological type theorems for open manifolds
with non-negative Ricci curvature under a new additional regularity assumption
on metrics, almost maximal local rewinding volume. In contrast to past related
works, our theorems get rid of restrictions of sectional curvature or conjugate
radius. Specially our settings do not necessarily satisfy a triangle comparison
of Toponogov type, and our new approach also applies to past works as far as we
know.
