What are Homotopy rings good for?

Question

In his paper, Note on quasi-Lie rings, P. J. Hilton defines the (non-associative) Homotopy ring of a pointed space $X$ as$$\bigoplus_{n>1}\pi_n(X)$$ where the Whitehead product $\pi_m(X)\times\pi_n(X)\to\pi_{m+n-1}(X)$ is the product.

I suppose in a naïve fashion, one would hope for this to sprout some interesting structure in a similar way to cohomology rings. However it seems to be the case the non-associativity (stemming from the Whitehead product) kills this dream. As far as I can tell, no interesting ring theory can be done on this object, not to mention the fact that it is highly non-trivial for most $X$.

I failed to find anymore literature on this subject, partly due to terminology changing from one source to the next. I suppose the correct terminology would be to call this a quasi-Lie algebra, but this just doesn't have the same ring to it.

• Is there anything more to be said about this construction?
• Is it wrong to try to force a structure on homotopy groups in this way?
• Are there similar constructions that have the same feel as cohomology rings but for homotopy?

Answer 1

The rationalization of this ring can be understood in a very nice way, as follows. Suppose for simplicity that $X$ is simply connected. Then we can define its rational homotopy groups

$$\pi_n(X, \mathbb{Q}) \cong \pi_n(X) \otimes \mathbb{Q}.$$

What can we say about their structure? One observation is that $\pi_n(X) \cong \pi_{n-1}(\Omega X)$, and $\Omega X$ has a loop space structure. This means that the rational homology $H_{\bullet}(\Omega X, \mathbb{Q})$ has the structure of a graded Hopf algebra, with product given by the Pontryagin product. Furthermore, there is a rational Hurewicz map

$$\pi_{n+1}(X, \mathbb{Q}) \cong \pi_n(\Omega X, \mathbb{Q}) \to H_n(\Omega X, \mathbb{Q}).$$

Theorem 1: These maps are injective.

Theorem 2: The image of the rational Hurewicz map consists precisely of the primitive elements in the Hopf algebra $H_{\bullet}(\Omega X, \mathbb{Q})$.

As in the ungraded case, the primitive elements of a (graded) Hopf algebra naturally have the structure of a (graded) Lie algebra, under the commutator bracket. This is precisely (the rationalization of) the Whitehead bracket. Furthermore:

Theorem 3: $H_{\bullet}(\Omega X, \mathbb{Q})$ is the (graded) universal enveloping algebra of the (graded) Lie algebra $\pi_{\bullet}(\Omega X, \mathbb{Q}) \cong \pi_{\bullet-1}(X, \mathbb{Q})$.

This is one way to think about the shift in degree necessary to define the Whitehead bracket.

These theorems have many nice applications. For example, you can use them to compute the rational homotopy groups of spheres, by computing the rational homology groups of the based loop spaces of spheres.

There is also a nice connection to Koszul duality; the fact that the rational homotopy groups form a graded Lie algebra while the rational cohomology groups form a graded commutative ring reflects the fact that the Lie operad is Koszul dual to the commutative operad.

Answer 2

You may be interested in reading about $\Pi$-algebras, which are graded abelian groups possessing all the primary algebraic structure of $\pi_*(X)$, for $X$ a pointed and connected topological space. David Blanc and his collaborators have done a lot of work on this. The paper

Blanc, D.; Dwyer, W.G.; Goerss, P.G., The realization space of a $\Pi$-algebra: a moduli problem in algebraic topology, Topology 43, No. 4, 857-892 (2004). ZBL1054.55007.

looks quite friendly as an introduction. See also the nLab page Pi-algebras and the references therein.

Answer 3

The structure is just that of a graded Lie ring (homologically graded - to create the correct Koszul signs), once you shift degrees by 1. This structure is not at all exotic, you see it in Gerstenhaber algebras, Batalin-Vilkovisky algebras etc. In a way, in the context of deformation theory and homotopy theory it is a much more fundamental thing than a structure of an associative ring, so the tone of your question is quite surprising.