Let $(X, x_0)$ be a pointed space. Then we can define the homotopy groups $\pi_i(X, x_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi_1(X, x_0)$ acts on each of the higher groups $\pi_i(X, x_0)$, and that this action generalizes to the Whitehead Products which are maps

$$ \pi_p(X, x_0) \times \pi_q(X, x_0) \to \pi_{p+q -1}(X, x_0).$$

The details are given in the wikipedia article I linked to above. Together the Whitehead products turn the graded group $\pi_*(X, x_0)$ (for $* > 0$) into a graded (quasi-) Lie algebra over $\mathbb{Z}$, where the grading is shifted so that $\pi_i(X, x_0)$ is in degree $(i-1)$. Well, it is a little funny since the bottom group is not necessarily abelian.

This is all well and good, but what if we don't want to pick base points? Is there a similar algebraic gadget in that situation?

If we don't pick base points, then it seems natural to consider the fundamental groupoid $\Pi_1(X)$. Then the different homotopy groups of $X$ at different base points can be assembled into local systems on $X$. That is for each $i \geq 2$ we have a functor,

$$\pi_i: \Pi_1 X \to AB$$

where $AB$ is the category of abelian groups. This already incorporates the action of $\pi_1$ on the higher homotopy groups but does it in a way which doesn't depend on the choice of base point.

Question: Can we enhance these local systems with a structure which generalizes the Whitehead product, and if so what precisely is this extra structure?

notby the Lie bracket: If $g \in pi_1$ and $m \in \pi_k$, then (up to a sign) $[g,m] = gm - m$. (For instance, the usual action of $\pi_1$ on itself is by conjugation so this recovers $[g,h]=g h g^{-1} h^{-1}$.) Using this last formula, I think one gets that the Whitehead algebraisa Lie algebra in $\pi_1$-modules. $\endgroup$ – Anatoly Preygel Feb 20 '10 at 21:28