Given a simplicial pair \((X, A)\) and a map \(f \colon A \to Y\), the author gives an algorithm for computing the set \([X, Y]^f\) of homotopy classes of extensions of \(f\) to \(X\) under the assumption that \(Y\) is simply connected and a rational H-space through dimension \(d\) for all pairs \((X, A)\) of cohomological dimension at most \(d+1\). The author proves a converse result, as well, showing that, if \(Y\) is not a rational H-space, the extension problem is equivalent to a version of Hilbert’s tenth problem.
The main result builds upon and extends the results of [M. Čadek et al., Discrete Comput. Geom. 57, No. 4, 915–965 (2017; Zbl 1373.55018)]. The author proves that elements of finite order in the rational cohomology of \(Y\) vanish under the finitely many iterations of the multiplication map which allows for the removal of obstruction classes. The main construction is a computable action of the group \([X, H_d]^f\) on \([X, Y_d]^f\) with \(Y_d\) the \(d\)th Postnikov section of \(Y\) and \(H_d = \prod_{n=2}^{d}K(\pi_n(Y), n)\) the H-space approximation to \(Y_d\). The algorithm applies the tools of effective cohomology to compute the set \([X, Y]^f\) via this action [J. Rubio and F. Sergeraert, Bull. Sci. Math. 126, No. 5, 389–412 (2002; Zbl 1007.55019)].
The converse result follows the line of proof for the case \(Y = S^2\) given in [M. Čadek et al., Discrete Comput. Geom. 51, No. 1, 24–66 (2014; Zbl 1358.68297)]. The existence of a higher order Whitehead product in \(\pi_d(Y)\) gives rise to a simplicial pair \((A, X)\) of cohomological degree \(d,\) with \(A\) built from the fat wedge of spheres representing the higher order Whitehead product and \(f \colon A \to Y\) the induced map. The solution to the extension problem in this case is shown to solve a particular integer equation arising from a bilinear form. The solvability of this and related integer equations are discussed and certain cases are proven undecidable by equivalence with versions of Hilbert’s tenth problem.