The codimension one homology of a complete manifold with nonnegative Ricci curvature (Q2731393)

In 1976, \textit{S. T. Yau} proved that a complete noncompact manifold with positive Ricci curvature has a trivial codimension one real homology [Indiana Univ. Math. J. 25, 659-670 (1976; Zbl 0335.53041)]. It was then conjectured that the integer homology might be trivial as well. In 1993, the first author proved that if the manifold is also proper, then the codimension one integer homology is trivial [Trans. Am. Math. Soc. 338, 289-310 (1993; Zbl 0783.53026)]. In 1999, \textit{Y. Itokawa} and \textit{R. Kobayashi} proved that if a manifold with nonnegative Ricci curvature has bounded diameter growth then the integer codimension one homology is either \(\mathbb{Z}, \mathbb{Z}_2\) or \(0\) [Am. J. Math. 121, 1253-1278 (1999; Zbl 0980.53044)].NEWLINENEWLINENEWLINEIn this paper the authors re-prove the results of Y. Itokawa and R. Kobayashi [op. cit.] and complete the classification of the codimension one integer homology using Poincaré duality, the Cheeger Gromoll splitting theorem [\textit{J. Cheeger} and \textit{D. Gromoll}, J. Differ. Geom. 6, 119-128 (1971; Zbl 0223.53033)] and properties of noncontractibility loops developed by the second author [Indiana Univ. Math. 50, 1867-1883 (2001; Zbl 1040.53043)].NEWLINENEWLINENEWLINEIn fact the following theorems and corollaries are proved: Theorem 1.1. Let \(M^n\) be an orientable complete noncompact with nonnegative Ricci curvature and \(G\) is an Abelian group. Then one of the following holds: NEWLINE\[NEWLINE\text{(i) }H_{n-1}(M,G)=0,\quad\text{(ii) }H_{n-1}(M,G)=G,\quad\text{(iii) }H_{n-1}(M,G)= \text{ker} (G @>\times 2>> G).NEWLINE\]NEWLINE Case (ii) can only occur when \(M^n\) is an isometrically split manifold over a compact totally geodesic orientable submanifold. Case (iii) can only occur when \(M^n\) is a one-ended flat normal bundle over a compact totally geodesic orientable submanifold.NEWLINENEWLINENEWLINETheorem 1.2. Let \(M^n\) be a complete noncompact unorientable manifold with nonnegative Ricci curvature and \(G=\mathbb{Z}_2\) or \(\mathbb{Z} \). Then one of the following holds: NEWLINE\[NEWLINE\text{(i) }H_{n-1}(M,G)=0,\quad\text{(ii) }H_{n-1}(M,G)=G,\quad\text{(iii) }H_{n-1}(M,G)= \text{ker}(G @>\times 2>> G).NEWLINE\]NEWLINE Case (ii) can only occur when \(M^n\) is a one-ended flat normal bundle over a compact totally geodesic unorientable submanifold. Case (iii) can only occur when \(M^n\) is an isometrically split manifold over a compact unorientable submanifold.NEWLINENEWLINENEWLINECorollary 1.1. Let \(M^n\) be a complete noncompact manifold with nonnegative Ricci curvature. If there is a point \(p\in M^n\) such that \(\text{Ric}_p (v,v)>0\), then \(H_{n-1} (M,\mathbb{Z})\) is trivial.NEWLINENEWLINENEWLINECorollary 2. Let \(M^n\) be an orientable complete noncompact manifold with nonnegative Ricci curvature. Then \(M\) is either an isometrically split manifold or a flat normal bundle over a compact totally geodesic submanifold or, for any Abelian group \(G\), \(H_{n-2} (M,G)*G=0\), in which case \(H_{n-2}(M,G)\) has no elements of finite order.