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?

  • $\begingroup$ What's wrong with just taking all higher homotopy groups as a functor $\oplus_{k>1}\pi_k:\Pi_1(X)\to $graded Lie algebras with $\pi_1$ actions? As you note, you can separate out the $\pi_1$ part. Also, could you explain why you don't want to pick base points? Whitehead products are obscure and tricky to calculate (except as commutators on $\pi_1$), and I'm curious about what context you are thinking about. $\endgroup$ – Paul Feb 20 '10 at 19:00
  • $\begingroup$ That was my first guess too. I played around with that a little, and the problem is that it didn't seem to give a functor to graded Lie algebras. The paths act by something which is more like a derivation. You can see this by looking at how $\pi_1$ acts. It acts by the Lie bracket [x,-]. I'm hoping someone who is more familiar with Whitehead product or graded Lie algebras will know what this structure is. $\endgroup$ – Chris Schommer-Pries Feb 20 '10 at 19:45
  • $\begingroup$ My motivation for this comes from trying to follow up on my previous MO questions: mathoverflow.net/questions/14266/… and mathoverflow.net/questions/430/… . Basically those questions led me to try to understand the homotopy of topological commutative monoids and I need a sufficiently rich algebraic invariant to help with this. It seems very important in that world that you can't simply reduce a single component and a single base point. This question is a step towards finding that invariant. $\endgroup$ – Chris Schommer-Pries Feb 20 '10 at 19:52
  • $\begingroup$ Chris, are you sure you that Paul's suggestion doesn't work? The action of $\pi_1$ on $\pi_k$ is not by 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 algebra is a Lie algebra in $\pi_1$-modules. $\endgroup$ – Anatoly Preygel Feb 20 '10 at 21:28
  • $\begingroup$ Rather, the formula shows that bracketing with elements of $\pi_1$ gives $\pi_1$-module morphisms. I think I have a conceptual reason why things should work, so I'll post that in an answer. $\endgroup$ – Anatoly Preygel Feb 20 '10 at 21:31

As I posted in my comment, I think Paul's suggestion does work. Here's a (sloppy) description of how I think things will work:

The local systems you describe can be obtained, by passing to homotopy groups, from a "local system of loop spaces" $$ \Omega: \Pi_{\leq \infty} X \to \Omega\mathbf{Spaces}$$ One can imagine that this corresponds under the Grothendieck construction to the free loop-space fibration $\Omega X \to LX \to X$. Alternatively, if we fix a basepoint and identify $X = BG$ for a simplicial group $G$, then this is just encoding the simplicial conjugation action of $G$ on itself.

Rather than think about (strangely-graded) Whitehead products, I prefer to think about (reasonably graded) Samelson products: We think of the structure (Whitehead product) on $\pi_{*+1} X $ as really being a structure (Samelson product) on $\pi_{*} \Omega X$. I claim that Samelson products give a functor $$ \pi_*: \Omega\mathbf{Spaces} \to \mathbf{grqLie} $$ so that composing with the above gives our desired "local system of graded (quasi-)Lie algebras".

For convenience, I'll replace loop spaces with (strict) simplicial groups. Then, the Samelson product comes from noticing that the commutator map $[,]: G^2 \to G$ is trivial if one of the factors is the identity, and so factors through a pointed map $[,]: G \wedge G \to G$. This pointed map goes on to induce the (quasi-)Lie structure on homotopy. A group homomorphism $H \to G$ preserves commutators and identities, and so induces a map $H \wedge H \to G \wedge G$ compatible with the brackets, so that this construction is indeed functorial.


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