TITLE: Computational complexity of computing homotopy groups of spheres
QUESTION [74 upvotes]: 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 $\pi_k S^2$.

REPLY [2 votes]: This is a bit aside from the question at hand, but I think it's worth making the observation.  Consider the function of two variables:
$$ (X,n) \mapsto \pi_n X.$$
As a function of $n$, the computational complexity is believed (for general $X$) to grow  exponentially.  But for fixed $n$, as a function of simply-connected $X$ (measured in terms of, say, number of simplices), the growth of the computational complexity is polynomial.  
(In fact, I'd guess that you could even specify the degree of the polynomial to be something like $n/c$ where $c$ is the connectivity of the $X$s.  I don't know if anyone has made that precise.)
So if you want to get in the game of making algorithms to compute homotopy groups, don't bother with high-dimensional spheres: it's a waste of effort.  Instead, compute low-dimensional homotopy groups of large spaces.   (This is basically what the "effective homotopy program" of Sergeraert, Rubio, Romero, and others, does.)