Six-dimensional counterexample to the Milnor conjecture

Abstract

We extend the previous work [Ann. of Math. (2) 201, 225–289 (2025)] by building a smooth complete manifold $(M^6,g,p)$ with $\mathrm{Ric}\ge 0$ 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 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 almost globally noncollapsed. Precisely, for every $\eta >0$ there exist radii $r_j\to \infty$ such that $\mathrm{Vol}(B_{r_j}(p))\ge r_j^{6-\eta }$. The broad outline for the construction of the example will closely follow the scheme introduced in [Ann. of Math. (2) 201, 225–289 (2025)]. 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.