标题： 一个扭结曲面补集的基本交叉模 显示英文标题

标题： The Fundamental Crossed Module of the Complement of a Knotted Surface

摘要： 我们证明如果$M$是一个CW复形，$M^1$是它的1骨架，则交叉模$\Pi_2(M,M^1)$仅依赖于空间$M$的同伦类型，除自由积外，在交叉模范畴中，与$\Pi_2(D^2,S^1)$相关。 由此可知，如果$G$是一个有限的交叉模，并且$M$是有限的，那么交叉模同态$\Pi_2(M,M^1) \to G$的数量可以重新缩放为一个同伦不变量$I_G(M)$，仅依赖于$M$的同伦 2-类型。 我们描述一个算法，用于计算当$M$是扭结曲面$\Sigma$在$S^4$中的补集，且$M^{(1)}$是从$M$的手柄分解中的 0- 和 1- 手柄构造的手柄体时，$\pi_2(M,M^{(1)})$作为$\pi_1(M^{(1)})$上的交叉模。 这里$\Sigma$由一个带结的图表示。 这尤其为我们提供了一种几何方法，用于从其双曲分解计算扭面补集的代数 2 型。 我们还证明不变量$I_G$在关于显式计算方面对 $S^4$中的扭面具有良好的性质，并给出非平凡的不变量。 显示英文摘要

摘要： We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.