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In this paper, we prove finite topological type theorems for open manifoldswith non-negative Ricci curvature under a new additional regularity assumptionon metrics, almost maximal local rewinding volume. In contrast to past relatedworks, our theorems get rid of restrictions of sectional curvature or conjugateradius. Specially our settings do not necessarily satisfy a triangle comparisonof Toponogov type, and our new approach also applies to past works as far as weknow.

A finite Topological Type Theorem for open manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume