计算同伦群元素的单纯表示
Computing simplicial representatives of homotopy group elements
摘要：
代数拓扑学的一个中心问题是理解拓扑空间$X$的同伦群$π_d(X)$。对于该问题的计算版本，众所周知，没有算法来判定给定的有限单纯复形$X$的基本群$π_1(X)$是否平凡。另一方面，有几种算法，给定一个简单连接的有限单纯复形$X$（即，对于任意给定的$π_1(X)$，计算高阶同伦群$π_d(X)$。第一个这样的算法是由Brown给出的，最近，Čadek等人。然而，这些算法带有一个警告：它们计算$π_d(X)$的同构类型，$d\geq 2$作为由生成元和关系给出的\emph{abstract}生成的阿贝尔群，但是它们使用$π_d(X)$的元素的非常隐式的表示。将这个抽象群的元素转换成从$d$维球面$S^d$到$X$的显式几何映射一直是计算同伦理论新兴领域中未解决的主要问题之一。在这里，我们提出了一个算法，给定一个~单连通空间$X$，计算$π_d(X)$，并将其元素表示为从$d$球$S^d$到$X$的适当三角剖分的单纯映射。对于固定的$d$，算法在$size(X)$中以时间指数运行，即$X$的单形数。此外，我们证明，这是最佳的：对于每一个固定的$d\geq 2$，我们构造了一个家庭的单连通空间$X$，使任何单纯映射表示生成器的$π_d(X)$，三角形的$S^d$上的地图定义的大小，是指数在$size(X)$。
英文摘要：
A central problem of algebraic topology is to understand the homotopy groups $π_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group $π_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is simply connected (i.e., with $π_1(X)$ trivial), compute the higher homotopy group $π_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, Čadek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $π_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $π_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $π_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $π_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$.
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