Tools of Unstable Homotopy Theory

1 Whitehead and Samelson products

A basic incarnation of this is the fact the associated graded of the lower central series of the homotopy groups of \(\Omega X\) forms a Lie algebra¹. The origin of this Lie algebra structure is exactly the same as for discrete groups: it comes from the commutator map.

Given a pointed space \(V\), we can take homotopy groups with coefficients in \(V\), i.e \(\pi _i^V(X) = \pi _i(X^V) = [\Sigma ^iV,X]\). In what follows, one may assume \(W,V = S^0\) to obtain statements about ordinary homotopy groups. Given \(x \in \pi _k^V(\Omega X), y \in \pi _n^W(\Omega X)\), the commutator map \(c(x,y) =xyx^{-1}y^{-1}\) gives a map²,

\[\Sigma ^{k}\Sigma ^n(V\wedge W) \xrightarrow {\sim }\Sigma ^kV\wedge \Sigma ^n W \to \Omega X\wedge \Omega X \xrightarrow {c} \Omega X\]

which we denote \(\langle x ,y\rangle \), and is called the Samelson product. The universal case of the Samelson product is when \(X = \Sigma (\Sigma ^kV\vee \Sigma ^n W)\), where it amounts to a map \(\Sigma ^{k}\Sigma ^n(V\wedge W) \to \Omega \Sigma (\Sigma ^kV \vee \Sigma ^nW)\).

This is adjoint to a map \(\Sigma \Sigma ^k\Sigma ^n(V\wedge W) \to \Sigma (\Sigma ^kV \vee \Sigma ^nW)\), and one sees that the composite

\[\Sigma \Sigma ^k\Sigma ^n(V\wedge W) \to \Sigma (\Sigma ^kV \vee \Sigma ^nW) \to \Sigma ^{k+1}{V}\times \Sigma ^{n+1}W\]

is canonically null, since the adjoint map is null since \(\Sigma ^kV\) and \(\Sigma ^nW\) canonically commute in \(\Omega \Sigma (\Sigma ^kV)\times \Omega \Sigma (\Sigma ^nW)\).

This nulhomotopy in fact gives rise to a cofibre sequence. To check this, it suffices to observe that we can assume \(n,k=0\), and \(V,W\) are finite sets, since the sequence commutes with sifted colimits. Then this amounts to the fact that the product \((\vee _1^lS^1)\times (\vee _1^mS^1)\) is obtained from \((\vee _1^lS^1)\vee (\vee _1^mS^1)\) by attaching 2-cells killing all commutators between the two parts.

The Whitehead product is up to a sign, adjoint to the Samelson product. Given maps \(x \in \pi _{k+1}^V(X),y \in \pi _{n+1}^W(X)\), the Whitehead product is the operation denoted \([x,y]\in \pi _{k+n+1}^{V\wedge W}(X)\) corepresented by the map

\[\Sigma ^{k}\Sigma \Sigma ^n(V\wedge W)\xrightarrow {\sim }\Sigma \Sigma ^k\Sigma ^n(V\wedge W) \to \Sigma (\Sigma ^kV \vee \Sigma ^nW)\]

where the second map is adjoint to the Samelson product.

On the level of homotopy classes, we have that \([x,y]\) is adjoint to \((-1)^{x+1}\langle x',y'\rangle \). We claim that the Whitehead product satisfies Lie algebra identities:

¹ if \(\pi _1\) acts trivially on \(\pi _n\), the lower central series is trivial, and so the Lie algebra structure is just on \(\pi _*\Omega X\).

² It is written as \(\Sigma ^k\Sigma ^n\) rather than \(\Sigma ^{k+n}\) to indicate the sign.

Lemma 1.1. The Whitehead product satisfies for \(x\in \pi _i^U(X),y \in \pi _j^V(X), z \in \pi _k^W(X)\) for \(i,j,k \geq 2\):
- 1. It is bilinear.
- 2. \([x,y] = (-1)^{|xy|}[y,x]\)
- 3. \([[x,y],z](-1)^{|xz|} + [[y,z],x](-1)^{|yx|} + [[z,x],y](-1)^{|zy|} = 0\)

Proof. We will check the most interesting relation, \((3)\). On the level of Samelson products, it corresponds to the identity:
- • \(\langle x,\langle y,z\rangle \rangle + \langle y,\langle z,x\rangle \rangle (-1)^{|x(y+z)|} + \langle z,\langle x ,y\rangle \rangle (-1)^{|z(x+y)|} = 0\)
This follows from the fact that in any group, \([x,[y,z]][y,[z,x]][z,[x,y]] = 1\) modulo commutators of length \(4\). Iterated Samelson products of length \(4\) in \(3\) variables are null, since they are factor through a map smashed with a diagonal map of the form \(S^n \xrightarrow {\Delta } S^n\wedge S^n\), which is null since since \(n\geq 1\). Thus the desired relation holds. □

When \(i,j,k\) are allowed to be \(1\), the relations are still satisfied, but only in the associated graded of the lower central series.

Note from the proof of Lemma 1.1 that there does not seem to be a canonical homotopy for the Jacobi identity.