Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension

For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed $k\ge 2$, there is a polynomial-time algorithm that, for a $1$-connected topological space $X$ given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the $k$th homotopy group $\pi_k(X)$, as well as the first $k$ stages of a Postnikov system of $X$. Combined with results of an earlier paper, this yields a polynomial-time computation of $[X,Y]$, i.e., all homotopy classes of continuous mappings $X\to Y$, under the assumption that $Y$ is $(k-1)$-connected and $\dim X\le 2k-2$. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes $X$, $Y$, where $Y$ is $(k-1)$-connected and $\dim X\le 2k-1$, plus a subspace $A\subseteq X$ and a (simplicial) map $f:A\to Y$, and the question is the extendability of $f$ to all of $X$. The algorit...