At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is *algorithmic*. But I've never come across a reference claiming this.

Is there a precise algorithm written down anywhere in the literature? Is there one in folklore, and if so what are the run-time estimates? Presumably they're pretty bad since nobody seems to ever mention them.

Are there any families for which there are better algorithms, say for the stable homotopy groups of spheres? or $\pi_k S^2$ ?

edit: I asked Francis Sergeraert a few questions related to his project. Apparently it's still an open question as to whether or not there is an exponential run-time algorithm to compute $\pi_k S^2$.

not even the author themselves considered the method practical (back in the days). – Peter Heinig Aug 3 '17 at 15:53