The Splitting Theorem and Topology of Noncompact Spaces with Nonnegative N-Bakry \'Emery Ricci Curvature

In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry 'Emery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry 'Emery Ricci Curvature. In addition, we show that if $M^n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry 'Emery Ricci curvature where $N>n$, then $H_{n-1}(M,\mathbb{Z})$ is $0$.