论文标题
配置映射空间和同源稳定性
Configuration-mapping spaces and homology stability
论文作者
论文摘要
对于给定的捆绑包$ξ\ colon e \ to m $，在歧管上，$ξ$ parametrise parametrise有限子集上的配置截面空间$ z \ subseteq m $，配备了$ m \ smallsetminus z $定义的$ξ$的部分，并在点$ z $ z $ z $ z $ z $ n offere。这些空间可以物理地解释为在有限的许多点上允许奇异的田地空间，并且在奇异之处附近的行为受到限制。作为一种特殊情况，它们包括Hurwitz空间，该空间与指定的甲板转换组参数分支覆盖了$ 2 $ disc的空间。 我们证明，每当基础歧管$ m $连接并且具有非空边界，并且在某些意义上，收费是“小”的，配置截面的空间在同源物上是属物稳定的（具有积分系数）。这与艾伦伯格，维卡茨和韦斯特兰的Hurwitz空间的工作有部分交集。
For a given bundle $ξ\colon E \to M$ over a manifold, configuration-section spaces on $ξ$ parametrise finite subsets $z \subseteq M$ equipped with a section of $ξ$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the $2$-disc with specified deck transformation group. We prove that configuration-section spaces are homologically stable (with integral coefficients) whenever the underlying manifold $M$ is connected and has non-empty boundary and the charge is "small" in a certain sense. This has a partial intersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland.
