论文标题
稳定的嵌入塔和配置空间上的经营结构
The stable embedding tower and operadic structures on configuration spaces
论文作者
论文摘要
给定平滑的歧管$ m $和$ n $，歧管微积分研究通过“嵌入塔”嵌入嵌入的空间$ \ operatatorName {emb}（m，n）$，该空间是使用$ m $ presheaves构建的。相同的理论使我们能够通过“稳定的嵌入塔”来研究$ \ permatatorname {emb}（m，n）$的稳定同型类型。通过分析框架配置空间的立方体，我们证明稳定嵌入塔的层是$ n $的切向同型不变性。 如果$ m $被构架，则磁盘的模量空间$ e_m $与通过$ e_n $ operad的稳定和不稳定的嵌入塔密切相关。 $ e_n $在$ e_m $上的操作引起了泊松operad $ \ mathrm {pois} _n $在配置空间的同源性上的动作。 为了研究此操作，我们介绍了庞加莱 - 科萨尔（Poincare-Koszul）的概念和模块，并表明$ e_n $和$ e_m $就是示例。作为一个应用程序，我们计算了$ h _*（f（m， - ））$的Lie Operad的诱导动作，并表明它是$ M^+$的同型不变性。
Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $\operatorname{Emb}(M,N)$ via the "embedding tower", which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to study the stable homotopy type of $\operatorname{Emb}(M,N)$ via the "stable embedding tower". By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of $N$. If $M$ is framed, the moduli space of disks $E_M$ is intimately connected to both the stable and unstable embedding towers through the $E_n$ operad. The action of $E_n$ on $E_M$ induces an action of the Poisson operad $\mathrm{pois}_n$ on the homology of configuration spaces $H_*(F(M,-))$. In order to study this action, we introduce the notion of Poincare-Koszul operads and modules and show that $E_n$ and $E_M$ are examples. As an application, we compute the induced action of the Lie operad on $H_*(F(M,-))$ and show it is a homotopy invariant of $M^+$.
