Computing homotopy classes for diagrams

Abstract

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).