Tools of Unstable Homotopy Theory

2 Hilton–Milnor and James Splitting

It turns out that iterated Whitehead/Samelson products, along with the homotopy groups of spheres, account for all \(n\)-ary natural operations on homotopy groups with coefficients in some spaces \(X_1\dots X_n\). The precise result along these lines is the Hilton–Milnor theorem, which describes a decomposition of the free \(\EE _1\)-group on a wedge sum of connected spaces.

Theorem 2.1 (Hilton–Milnor). Let \(n\geq 1\), and \(X_i\) connected. There is a natural equivalence \(\prod _{w_I}\Omega \Sigma X^{\wedge w_I} \xrightarrow {\sim } \Omega \Sigma \vee _1^n X_i \), where \(w_I\) is a basis for the free Lie algebra³ on \(X_i, 1\leq i \leq n\).

³ Note that the free Lie algebra appearing Theorem 2.1 is alternating rather than antisymmetric, in that \([x,x] =0\). This does not mean however that \([x,x] = 0\) for Whitehead products, since it simply appears as a higher homotopy group of a sphere.

The map in the theorem is given by iterated Samelson products determined by the basis element of the free Lie algebra. The projections in the other direction, which are dependent on the ordering of the basis, are called the Hilton–Hopf invariants.

This result can be thought of as reflecting the nilpotence of \(\Omega \Sigma X\) as follows: A discrete group \(X\) has a lower central series with associated graded a Lie algebra. If \(X\) is nilpotent, we can choose lifts of the associated graded and get an equivalence (of sets) of \(X\) with the product of the terms in the associated graded. The associated graded of a free group is the free Lie algebra, but it is not true that a free group on more than one generator is nilpotent. But nevertheless, as long the space over which one takes the free group is connected, the product map from the associated graded is an equivalence, because the higher filtration terms become highly connected.

Two approachs to proving Theorem 2.1 are to either use Mather’s second cube theorem (see for example [sanathpeter]), or to use facts about free groups, and geometrically realize these to say something about free \(\EE _1\)-algebras. Ultimately these are proofs doing the same thing but the former is more axiomatic and so works in a bit more generality.

\(\Omega \Sigma \) commutes with geometric realizations, so whatever we say about it reduces to the case of discrete sets, in which case it coincides with the free group functor. It is a result of Milnor [milnorfk] that the free simplicial group on \(X\), denoted \(FX\), is a model for \(\Omega \Sigma X\). He used this model to prove the results below, but all one really needs is that the functor \(\Omega \Sigma \) preserves geometric realizations (which is model independent, and follows from Milnor’s result).

Lemma 2.2. There is a split exact sequence of groups

\[1 \to \Omega \Sigma (B\vee B\wedge \Omega \Sigma A) \to \Omega \Sigma (A\vee B) \to \Omega \Sigma A \to 1\]

Proof. The kernel of the projection map is the free group generated by \(b \in B\) and \([b,w]\), where \(w\) is a nontrivial word in \(FA\). The sequence obviously splits. □

Lemma 2.3. There is an equivalence of groups

\[\Omega \Sigma (B\wedge \Omega \Sigma A) \simeq \Omega \Sigma (B\wedge A \vee B\wedge A \wedge \Omega \Sigma A)\]

Proof. The free group generated by \([b,w]\) for \(w\) a nontrivial word in \(A\) is also freely generated by \([b,a]\) and \([[b,a],w]\) for \(w\) running over nontrivial words in \(A\). □

The lemma above, as a statement at the level of groups, can be delooped to get the James splitting

\[\Sigma (B\wedge \Omega \Sigma A) \simeq \Sigma (B\wedge A\vee B\wedge A\wedge \Omega \Sigma A)\]

Proposition 2.4. Let \(A\) be connected. Then there is a split exact sequence of \(\EE _1\)-algebras

\[1 \to \Omega \Sigma (\vee _0^{\infty } B\wedge A^{\wedge i}) \to \Omega \Sigma (A\vee B) \to \Omega \Sigma A \to 1\]

Proof. Apply Lemma 2.2 and Lemma 2.3 repeatedly, using the fact that \(A^{\wedge n} \wedge \Omega \Sigma A\) becomes highly connected. □

Similarly, the James splitting can be iterated to obtain:

Theorem 2.5. (James) If \(X\) is connected, there is an equivalence \(\vee _0^{\infty }\Sigma X^{\wedge i} \simeq \Sigma \Omega \Sigma X\).

Theorem 2.1 is a consequence of Proposition 2.4: one simply repeatedly applies the proposition, observing that the remaining factors become more and more highly connected. Tracing through the construction, we find that the equivalence is given by the product of iterated Samelson products where the order of the product is chosen by the order in which we have realized the equivalence.

The James splitting is very useful. For example, adjoint to the projection of \(\Sigma \Omega \Sigma X\) onto the factors are the James-Hopf maps \(H_n:\Omega \Sigma X \to \Omega \Sigma X^{\wedge n}\). It is these maps that are traditionally used in the construction of EHP sequences.