论文标题
关于高维地理问题
On the high-dimensional geography problem
论文作者
论文摘要
1962年，沃尔（Wall）表明，光滑，封闭，定向，$（n-1）$ - 连接的$ 2N $ - 尺寸的manifolds至少$ 6 $被归类为通过代数的交叉点改进，将其称为$ n $ space的代数改进。在本文中，我们完成了哪些$ n $ - 空间可以通过平滑，封闭，定向，$（n-1）$连接的$ 2N $ -Manifolds实现的确定，用于所有$ n \ neq 63 $。在尺寸中，$ 126 $ kervaire不变一个问题仍然开放。在此过程中，我们完全解决了Galatius-Randal-Williams和Bowden-Crowley-Stipsicz的猜想，这表明它们在非凡的尺寸$ 23 $之外是真实的，我们提供反例。该反例与Witten属有关，并将其改进到Ando-Hopkins-Rezk的$ \ Mathbb {e} _ \ infty $ -ring Spectra的地图上。通过许多作者的先前作品，包括沃尔，舒尔茨，斯托尔兹和希尔·霍普金斯·雷纳尔，以及哈恩与作者的最新联合作品，这些问题已经解决了除了有限的许多方面，但本文的贡献是填补这些空白。
In 1962, Wall showed that smooth, closed, oriented, $(n-1)$-connected $2n$-manifolds of dimension at least $6$ are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form which he called an $n$-space. In this paper, we complete the determination of which $n$-spaces are realizable by smooth, closed, oriented, $(n-1)$-connected $2n$-manifolds for all $n \neq 63$. In dimension $126$ the Kervaire invariant one problem remains open. Along the way, we completely resolve conjectures of Galatius-Randal-Williams and Bowden-Crowley-Stipsicz, showing that they are true outside of the exceptional dimension $23$, where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of $\mathbb{E}_\infty$-ring spectra by Ando-Hopkins-Rezk. By previous work of many authors, including Wall, Schultz, Stolz and Hill-Hopkins-Ravenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.
