Topology of tame contractible $n$-manifolds, $n=6,7$, with Positive BiRicci Curvature and Ricci lower bounds

Abstract

We show that if an open smooth $6$-manifold $M^6$ is the interior of a compact, contractible manifold with boundary $X^6$, such that ${\pi }_3(X,\partial X)=0$, and $M$ supports a complete metric with uniform positive BiRicci curvature and Ricci curvature bounded below, then $M$ is diffeomorphic to ${\mathbb{R}}^6$. Additionally, we show that if an open smooth $n$-manifold $M^n$, $n\in \{6,7\}$, is the interior of a compact, contractible manifold with boundary $X^n$, such that ${\pi }_3(X,\partial X)=0$, and $M$ supports a complete metric with positive BiRicci curvature with at most $C$-quadratic decay at infinity for some $C>C_n$ and nonnegative Ricci curvature, then $M$ is diffeomorphic to ${\mathbb{R}}^n$. Lastly, for each $n\ge 6$, we construct a compact, contractible Newman $n$-manifold, $N^n$, based on a presentation of the Higman group such that ${\pi }_i(N,\partial N)=0$, $3\le i\le n-3$, and $\mathrm{int}(N)$ admits a complete metric with uniform positive scalar curvature and Ricci curvature bounded below.

arXiv comment18 pages, 1 figure. Statement of Theorem A has been corrected. Introduction has been expanded and improved. References were added