Question

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:

1. $\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$

2. $\mathbb{Z}\times ({B\Sigma_\infty})_+$, where $\Sigma_\infty$ is the group of automorphisms of a countable set which have finite support, and $+$ is the Quillen plus-construction.

3. The group completion of $B\left(\bigsqcup_n \Sigma_n\right)$, where $\Sigma_n$ is the symmetric group on $n$ letters, and $B(\sqcup_n \Sigma_n)$ is given the structure of a topological monoid via the block addition map $\Sigma_n\times \Sigma_m\to \Sigma_{n+m}$.

4. $\Omega|S^\bullet.\operatorname{FinSet}|$, where $S^\bullet$ is the Waldhausen $S$-construction, and $\operatorname{FinSet}$ is the category of pointed finite sets, given the structure of a Waldhausen category by declaring cofibrations to be injections and weak equivalences to be isomorphisms. I don't want to define this since it's complicated (for a reference, see Chapter IV of Weibel's K Book), but it should be thought as a homotopical version of the Grothendieck ring of finite sets, where addition is given by disjoint union and multiplication is given by the cartesian product. Clark Barwick's answer here makes this more precise.

Now, the homotopy groups of the first space are manifestly the stable homotopy groups of spheres; on the other hand, the last two spaces clearly encode some information about the combinatorics of finite sets. So my question is:

Is there a concrete combinatorial interpretation of the higher stable homotopy groups of spheres in terms of the combinatorics of finite sets or symmetric groups?

For example it is easy to see via (3) or (4) that $\pi_{0+k}(S^k)=\mathbb{Z}$ corresponds to the Grothendieck ring of finite sets. Similarly, (2), or with some theory (4), make it clear that $\pi_{1+k}(S^k)=\mathbb{Z}/2\mathbb{Z}$ corresponds to the abelianization of $\Sigma_n$ (via the sign homomorphism). I am interested in concrete interpretations of the higher stable homotopy groups in this style.

A good answer would be, for example, a direct combinatorial interpretation of $\pi_{2+k}(S^k)=\mathbb{Z}/2\mathbb{Z}$ and $\pi_{3+k}(S^k)=\mathbb{Z}/24\mathbb{Z}$; a not-so-good answer would be a statement like "the sphere spectrum is a categorification of the integers," which is not the sort of concrete thing I'm looking for.

EDIT: So with the exception of Jacob Lurie's comment on $\pi_{2+k}(S^k)$ below (interpreting it as the Schur multiplier $H_2(\Sigma_\infty, \mathbb{Z})$ of $\Sigma_\infty$), it seems like it might be too much to hope for any reasonably complete combinatorial interpretation of the stable homotopy groups. So I'd settle for something like the following: namely, a sequence of groups $G_n$ defined in some combinatorial way, and maps $f_n: G_n\to \pi_{n+k}(S^k)$ or $g_n: \pi_{n+k}(S^k)\to G_n$ such that

1. $f_n$ or $g_n$ are nontrivial for infinitely many $n$,

2. The maps are related in some way to the constructions 2-4 above, and

3. The $G_n$ are combinatorially interesting.

One such example is $G_n:=H_n(\Sigma_\infty, \mathbb{Z})$ with $g_n$ the Hurewicz map (whence the interpretation of $\pi_{2+k}(S^k)$). But even in this case, the combinatorial meaning is sort of mysterious (to me at least) for large $n$.

Answer 1

Combinatorial descriptions of homotopy groups of spheres have been studied in the unstable world rather than the stable world.

Jie Wu found a description of homotopy groups of $S^2$ in his thesis and published here. He described $\pi_n(S^2)$ as the center of a combinatorially defined group.

As is well known, there is an analogue of Barratt-Priddy-Quillen theorem for braid groups. By replacing symmetric groups by braid groups, we obtain $\Omega^2 S^2$. And it is natural to expect to have a combinatorial description of $\pi_*(S^2)$ in terms of braid groups. Such a description was obtained by Berrick, Cohen, Wong, and Wu in this paper. They described $\pi_*(S^2)$ as the homotopy groups of a $\Delta$-group (simplicial group without degeneracies) constructed from braid groups.

Recently a papaer by Mikhailov and Wu appeared, in which they extended Wu's description of $\pi_*(S^2)$ in his thesis.

Answer 2

The following construction is due to Jones and Westbury, in a paper titled "Algebraic $K$-Theory, homology spheres and the $\eta$-invariant" (it is a very nice paper).

Let $M$ be a homology $3$-sphere and $\rho:\pi_1 (M) \to GL_N(\mathbb{C})$ a representation. Then the Quillen plus construction gives a map $S^3 = M^+ \to (BGL_N(\mathbb{C})^+$, in other words, an element $[M,\rho] \in K_3 (\mathbb{C})$. On this group, there is the $e$-invariant $e:K_3 (\mathbb{C})\to \mathbb{C}/\mathbb{Z}$. Jones and Westbury give a formula for $e([M,\rho])$ in terms of the eigenvalues of $\rho$. If $M$ is the Poincare sphere, you get a complicated, but manageable formula.

Now replace $\rho$ by an action of $\pi_1 (M)$ on a finite set $X$. This then gives, By Barratt-Priddy-Quillen, an element of $\pi_{3}^{st}$. The Jones--Westbury formula tells you how to compute the e-invariant of the image of this element in $K_3 (\mathbb{C})$ under the map induced by $\Sigma_N \to GL_N(\mathbb{C})$. As the homomorphism $\pi_{3}^{st} \to K_3 (\mathbb{C})$ is injective, you do not loose information.

As a grad student, I played with these formulae in the similar situation of mapping class groups (instead of general linear or symmetric group). This gives elements in the homotopy of the plus-construction of the classifying space of the mapping class group and I was able to find a generator of $\pi_3 (B\Gamma^{+})=Z/24$ in this way (by an acion of the fundamental group of the Poincare sphere on a surface of rather small genus, see http://wwwmath.uni-muenster.de/mjm/vol3.html. I guess it is possible to get a generator of $\pi_{3}^{st}=Z/24$ by an action of the fundamental group of the Poincare sphere on a finite set.

Answer 3

There are papers by Eccles, Freedman, Koschorke, and Koschorke and Sanderson that date circa 1978-1982 which discuss the $n$th-stable stem as the bordism group of oriented codimension 1 immersions of closed $n$-manifolds in $(n+1)$-space. For example the generator of $\pi_1^s$ is represented by an immersed circle in the plane that forms the figure-$8$. The mod-2 number of double points is the invariant. The $2$nd stabe stem is represented by a fully twisted torus that is formed from the figure-$8$ times $[0,1]$, fully twisting one end and attaching to form a torus. This is the Lie framed torus (perturbed from $\mathbb{C}^2$) and projected into space.

The generator of ${\mathbb Z}/(24)$ can be given by the sphere eversion. (Shameless self-promotion) My upcoming book gives this explicitly, not as a generator, but you can compute the self-intersection invariants from the diagrams that I give. In general the Kahn-Priddy maps are the invariants constructed from the self-intersection sets. I have not done it, but it would be a nice exercise to understand your $3$rd point of view from these immersions.