Noncollapsing examples with positive Ricci curvature and infinite topological type (Q1584726)

The first result of this paper is: There exists a complete 4-dimensional smooth manifold \(M\), a point \(p\) in \(M\), and a positive number \(c\) such that \(M\) has positive Ricci curvature, infinite topological type, \(\text{Vol}(B_r(p)) \geq cr^4\), where \(r\) denotes the distance function to \(p\). Using this theorem and taking products with Euclidean spaces one obtains examples of noncollapsing manifolds of positive Ricci curvature and infinite topological type in dimensions greater than or equal to four. The second result is the following: There exists a pointed sequence of complete 4-dimensional manifolds \((M_i,p_i)\) and positive number \(c\) such that each \(M_i\) has positive Ricci curvature, \(\text{Vol} (B_1(p_i))\geq c\), the sequence \((M_i,p_i)\) converges in the pointed Gromov-Hausdorff topology to a length space \((M_\infty, p_\infty)\), and \(B_1(p_\infty)\) has infinite topological type.