Six dimensional counterexample to the Milnor Conjecture

Abstract

We extend our previous work by building a smooth complete manifold $(M^6,g,p)$ with $\mathrm{Ric}\geq 0$ and whose fundamental group $\pi_1(M^6)=\mathbb{Q}/\mathbb{Z}$ is infinitely generated. The example is built with a variety of interesting geometric properties. To begin the universal cover $\tilde M^6$ is diffeomorphic to $S^3\times \mathbb{R}^3$, which turns out to be rather subtle as this diffeomorphism is increasingly twisting at infinity. The curvature of $M^6$ is uniformly bounded, and in fact decaying polynomially. The example is {\it locally} noncollapsed, in that $\mathrm{Vol}(B_1(x))>v>0$ for all $x\in M$. Finally, the space is built so that it is {\it almost } globally noncollapsed. Precisely, for every $\eta>0$ there exists radii $r_j\to \infty$ such that $\mathrm{Vol}(B_{r_j}(p))\geq r_j^{6-\eta}$. The broad outline for the construction of the example will closely follow the scheme introduced in our previous work. The six-dimensional case requires a couple of new points, in particular the corresponding Ricci curvature control on the equivariant mapping class group is harder and cannot be done in the same manner.