Six dimensional counterexample to the Milnor Conjecture
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912600288133120 |
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| author | Bruè, Elia Naber, Aaron Semola, Daniele |
| author_facet | Bruè, Elia Naber, Aaron Semola, Daniele |
| contents | We extend our previous work by building a smooth complete manifold $(M^6,g,p)$ with $\mathrm{Ric}\geq 0$ and whose fundamental group $π_1(M^6)=\mathbb{Q}/\mathbb{Z}$ 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 $η>0$ there exists radii $r_j\to \infty$ such that $\mathrm{Vol}(B_{r_j}(p))\geq r_j^{6-η}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12155 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Six dimensional counterexample to the Milnor Conjecture Bruè, Elia Naber, Aaron Semola, Daniele Differential Geometry We extend our previous work by building a smooth complete manifold $(M^6,g,p)$ with $\mathrm{Ric}\geq 0$ and whose fundamental group $π_1(M^6)=\mathbb{Q}/\mathbb{Z}$ 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 $η>0$ there exists radii $r_j\to \infty$ such that $\mathrm{Vol}(B_{r_j}(p))\geq r_j^{6-η}$. 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. |
| title | Six dimensional counterexample to the Milnor Conjecture |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2311.12155 |