A finite topological type theorem for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume (Q6624324)

A manifold is said to have finite topological type if it is homeomorphic to the interior of a compact manifold with boundary. Such manifolds arise naturally in the study of complete non-compact Riemannian manifolds of non-negative curvature \(M\) since a classical structure result of \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] provides a compact totally geodesic submanifold \(S\) (with non-negative curvature; somewhat poetically named a soul) so that the original non-compact manifold is diffeomorphic to the total space of its normal bundle \(\nu (S) \subset \mathrm{T}M\) over it (or equivalently its open unit disk-subbundle). Despite there being a similar structure result in the presence of non-negative Ricci curvature that shows \(M\) is isometric to a product \(S \times \mathbb{R}^k\) [\textit{J. Cheeger} and \textit{D. Gromoll}, J. Differ. Geom. 6, 119--128 (1971; Zbl 0223.53033)] one need not have that \(S\) is compact, see [\textit{J.-P. Sha} and \textit{D.-G. Yang}, J. Differ. Geom. 29, No. 1, 95--103 (1989; Zbl 0633.53064)], so the conclusions about finite topological type need not follow.\N\NIn this article, the author shows that an open \(n\)-manifold \(M\) with non-negative Ricci curvature has finite topological type provided \(M\) has almost maximal local rewinding number (appropriately understood to correspond to the satisfaction of conditions (1.1) and (1.2) in the paper). In this he follows the sentiment, expressed in [\textit{X. Rong}, Sci. Sin., Math. 48, No. 6, 791--806 (2018; Zbl 1499.53186)], that placing certain assumptions on the rewinding number on manifolds with lower Ricci curvature bound leads to topological and geometric properties similar to those displayed by manifolds with bounded sectional curvature.\N\NCrucial in his proof is a technique involving the construction of a smooth function by gluing Cheeger-Colding almost splitting functions to approximate the distance function [\textit{J. Cheeger} and \textit{T. H. Colding}, Ann. Math. (2) 144, No. 1, 189--237 (1996; Zbl 0865.53037)] and the transformation theorem for almost splitting functions (see [\textit{J. Cheeger} et al., Ann. Math. (2) 193, No. 2, 407--538 (2021; Zbl 1469.53083)]) to establish its non-degeneracy. The only additional assumption required for this is a generalized Reifenberg property see [\textit{H. Huang} and \textit{X.-T. Huang}, Adv. Math. 457, Article ID 109914, 54 p. (2024; Zbl 07939015)].