Examples of open manifolds with almost quadratic volume growth and infinite Betti numbers

TL;DR

This paper constructs examples of high-dimensional open manifolds with positive Ricci curvature, bounded sectional curvature, infinite Betti numbers, and volume growth close to quadratic, challenging assumptions about topological finiteness under such conditions.

Contribution

It provides explicit examples of open manifolds with positive Ricci curvature, infinite Betti numbers, and near-quadratic volume growth, addressing open questions in geometric topology.

Findings

Existence of open manifolds with positive Ricci curvature and infinite Betti numbers.

Volume growth can be arbitrarily close to quadratic.

Challenges assumptions about finite topological type under curvature bounds.

Abstract

We construct a family of examples of complete $(2+n)-$dimensional ($n\ge 2$) open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers $b_2,b_n$, moreover its volume growth can be arbitrarily close to quadratic volume growth. Compared with some known result of finite topology for manifolds with nonnegative Ricci curvature and lower sectional curvature bound, it makes sense to ask whether complete manifolds with such curvature bounds must be of finite topological type or not provided with at most quadratic volume growth.