Computational complexity of computing homotopy groups of spheres

Question

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 $\pi_k S^2$.

Answer 1

Francis Sergeraert and his coworkers have implemented his effective algebraic topology theory in a program named Kenzo. It seems capable of computing any $\pi_n(S^k)$ (in fact homotopy groups of any simply connected finite CW complex), although I don't know how far it is feasible. For instance $\pi_6 S^3$ is computed in about 30 seconds. In a 2002 paper, they mention other algorithms by Rolf Schön and by Justin Smith, not implemented at that time.

Answer 2

It is shown by D. J. Anick in The computation of rational homotopy groups is #℘-hard. Computers in geometry and topology, Proc. Conf., Chicago/Ill. 1986, Lect. Notes Pure Appl. Math. 114, 1–56, 1989. that, well, the computation of rational homotopy groups is #p-hard.

Answer 3

Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group. (This simplicial group has the homotopy type of the based loop space of $S^n$, so its homotopy groups compute the homotopy groups of $S^n$ shifted by one degree.)

For each simplicial degree $k$ define $N_k(S^n) \subset G_k(S^n)$ to be the intersection of the kernels of all but the last face maps $d_i: G_k(S^n) \to G_{k-1}(S^n)$. Then the last face map is a homomorphism $d_k: N_k(S^n) \to N_{k-1}(S^n)$. Moreover, the simplicial identities show $d_kd_{k+1}$ has constant value $1$, so we get a non-abelian chain complex of free groups. Its "homology," by a result of Kan, computes $\pi_*(S^n)$.

To get an algorithm for computing this homology, recall that the proof of the Nielsen-Schrier theorem gives a system of generators for the subgroup of any free group. So we obtain a system of generators for $N_k(S^n)$ as well as a system of generators for the image of $d_{k+1}$. So in principle we obtain a method for computing the homotopy groups of spheres.

In Kan's paper, $\pi_3(S^2)$ is computed in this way, and it takes several pages––so it's not a very good algorithm!

Answer 4

Weinberger's Computers, rigidity, and moduli: the large-scale fractal geometry of Riemannian moduli space contains several apparently useful references on pages 93-4, in the notes section of the chapter on designer homology spheres (which you may also find of interest). Weinberger mentions "the algorithmic nature of simply connected homotopy theory" and cites the paper of Brown that Mike mentioned before going on to cite Sullivan's "Infinitesimal computations in topology." Pub. Math. IHÉS, 47 269 (1977), Griffiths and Morgan's Rational Homotopy Theory and Differential Forms, Halperin's "Lectures on minimal models." Mém. Soc. Math. France, Sér. 2, 9-10 1 (1983), and Dwyer's "Tame Homotopy Theory." Topology 18 321 (1979).

The practical upshot of these later references seems to be the calculation of $\pi_k(S^n) \otimes \mathbb{Q}$, or in the case of tame homotopy theory the analogous object involving a finite number of primes (which number increases with dimension).

Answer 5

There is the paper of R. V. Mikhailov and J. Wu, http://arxiv.org/abs/1108.3055. They construct a group whose center is an unstable homotopy group of either a sphere or a Moore space. So now it seems we could apply our algorithmic understanding of computing centers of groups, which might not be much or might be a lot, to unstable homotopy groups.

I would imagine this would be easier to work into an algorithm, perhaps this has already been done. However, I am always unsure about these things, sometimes the word problem is hiding in the shadows.

Answer 6

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.)

Answer 7

Keita Allen put together a complexity analysis of Brown's algorithm, cited above.

The answer is the order of the computation of $\pi_i(|N|)$ for $1 < i \leq n$ is

$$O\left( n^2 \left[ \left( |N_{max}| + \prod_{j<n} |\pi_j(|N|)^{j + n \choose n} \right)^3 + n { 2n \choose n } \prod_{j<n} |\pi_j(|N|)^{j+n \choose n} \right] \right)$$

where $N$ is a simplicial set, $N_{max}$ is the level of the simplicial set $N$ with maximal order out of levels $2$ through $n+2$.