We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant map $f \colon A \to Y$ under the stability assumption $\dim X^H \leq 2 \operatorname{conn} Y^H$ and $\operatorname{conn} Y^H \geq 1$, for all subgroups $H\leq G$. For fixed $n = \operatorname{dim} X$, the algorithm runs in polynomial time. When the stability condition is dropped, the problem is undecidable already in the non-equivariant setting. The algorithm is obtained as a special case of a more general result: For finite diagrams of simplicial sets $X$, $A$, $Y$, i.e. functors $\mathcal{I}^\mathrm{op} \to \mathsf{sSet}$, in the stable range $\operatorname{dim} X \leq 2 \operatorname{conn} Y$ and $\operatorname{conn} Y > 1$, we give an algorithm that computes the set $[X, Y]^A$ of homotopy classes of maps of diagrams $\ell \colon X \to Y$ extending a given $f \colon A \to Y$. Again, for fixed $n = \dim X$, the running time of the algorithm is polynomial. The algorithm can be utilized to compute homotopy invariants in the equivariant setting -- for example, one can algorithmically compute equivariant stable homotopy groups. Further, one can apply the result to solve problems from computational topology, which we showcase on the following Tverberg-type problem: Given a $k$-dimensional simplicial complex $K$, is there a map $K \to \mathbb{R}^{d}$ without $r$-tuple intersection points? In the metastable range of dimensions, $rd \geq (r+1)k +3$, the result of Mabillard and Wagner shows this problem equivalent to the existence of a particular equivariant map. In this range, our algorithm is applicable and, thus, the $r$-Tverberg problem is algorithmically decidable (in polynomial time when $k$, $d$ and $r$ are fixed).

中文翻译：

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计算图的同伦类

我们给出了一种算法，给定有限简单组$ X $，$ A $，$ Y $和有限组$ G $的作用，该算法计算等变映射的同伦类的集合$ [X，Y] ^ A_G $在稳定假设$ \ dim X ^ H \ leq 2 \ operatorname {conn} Y ^ H $和$ \ operatorname下，$ \ ell \ colon X \ to Y $扩展给定的等价映射$ f \ colon A \ to Y $ {conn} Y ^ H \ geq 1 $，用于所有子组$ H \ leq G $。对于固定的$ n = \ operatorname {dim} X $，该算法以多项式时间运行。当稳定性条件下降时，在非等变设置中问题已经无法确定。该算法是作为更一般结果的特例获得的：对于简单集合$ X $，$ A $，$ Y $的有限图，即函子$ \ mathcal {I} ^ \ mathrm {op} \ to \ mathsf {sSet} $，在稳定范围$ \ operatorname {dim} X \ leq 2 \ operatorname {conn} Y $和$ \ operatorname {conn} Y> 1 $的范围内，我们给出了一种计算集合$ [X，Y] ^ A $的算法图$ \ ell \冒号X \到Y $的同伦类图扩展给定的$ f \冒号A \到Y $。同样，对于固定的$ n = \ dim X $，算法的运行时间为多项式。该算法可用于在等变设置中计算同构不变量-例如，可以通过算法计算等变稳定同构群。此外，可以将结果应用于解决计算拓扑的问题，我们将在以下Tverberg型问题上进行展示：给定一个维数为$ k $的单纯形复数$ K $，是否有映射$ K \到\ mathbb {R } ^ {d} $没有$ r $元组的交点？在尺寸的亚稳范围内，$ rd \ geq（r + 1）k + 3 $，Mabillard和Wagner的结果表明此问题等同于特定等变图的存在。在此范围内，我们的算法适用，因此，$ r $ -Tverberg问题在算法上是可确定的（在多项式时间内，$ k $，$ d $和$ r $是固定的）。