Different way to view action of fundamental group on higher homotopy groups

Question

There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at the action on higher $\pi_n$. All of them have the flavor of the conjugation map, but are more geometric than algebraic, and in some cases work is needed to show the map is well defined. Here are a couple I have seen:

There is a homotopy equivalence $f : S^n \to S^n \vee I$. taking the basepoint of $S^n$ to the endpoint of the unit interval "far away" from $S^n$. Given a path $\alpha$ from $x_0$ to $x_1$, one can get a basepoint changing homomorphism $\pi_n(X,x_0) \to \pi_n(X,x_1)$ by taking $g : S^n \to X$ and mapping it to $(g \vee \alpha) \circ f$. If $\alpha$ is a loop this gives an action of $\pi_1$

Another way to proceed may be to look at elements of $\pi_n(X,x_0)$ as homotopy classes of maps $I^n \to X$ that send $\partial I^n$ to $x_0$. Then a base change homomorphism could be obtained by using a path $\alpha$ to define a map $I^n \cup (\partial I^n \times I) \to X$, which can be filled in to a map $I^{n+1} \to X$. Then the action would be to take the face opposite the original $I^n \subset I^{n+1}$.

These both define the same standard action of $\pi_1$ on $\pi_n$, but lose the algebraic flavor of the group action and instead have this stronger geometric feel, which can make working with the action a bit cumbersome. Are there other ways of looking at this action that are more algebraic?

Perhaps, can something be done wherein $\pi_0(Y)$ acts on $\pi_n(Y)$, where $Y$ is some sufficiently nice space like $\Omega X$, and does this coincide with the above defined actions? Is this a useful way of viewing the action?

Answer 1

If G is a topological group, then the group acts on itself by conjugation, and this action is base-point-preserving. In particular, for an element $g \in \pi_0(G)$ and a higher homotopy element $\alpha \in \pi_{n-1} G = [S^n, G]$, one can check that the conjugate $g \alpha g^{-1}$ is well-defined and defines an action of $\pi_0(G)$ on $\pi_{n-1} G$. The space G is weakly equivalent to the loop space of the classifying space BG, and under this equivalence the conjugation action is taken to the action of $\pi_1 BG$ on $\pi_n BG$.

(Unfortunately, this doesn't work directly for the conjugation action of the loop space on itself because it is not strictly basepoint-preserving; one needs to use that there is a natural homotopy from a loop $\gamma * e *\gamma^{-1}$ to $e$ to produce the action.)

Any path-connected based space X is weakly equivalent to the classifying space of a simplicial group G; specifically, the Kan loop group of a weakly equivalent simplicial set. Even more, there is a Quillen equivalence between the homotopy theories of spaces and simplicial groups.

(Kan's original paper can be found here: http://www.jstor.org/pss/1970006)

Answer 2

I doubt this would be considered less geometric than the actions in your question, but if $(X,x_0)$ has a universal cover with covering map $p:(\tilde X, \tilde x_0)\rightarrow (X,X_0)$, then $p$ induces isomorphisms $p_*:\pi_n(\tilde X, \tilde x_0)\rightarrow \pi_n(X,x_0)$ and $\tilde p: D\rightarrow \pi_1(X,x_0)$ where $D$ is the group of deck transformations (covering transformations) of $(\tilde X,\tilde x_0)$.

Now for $\alpha\in \pi_1(X,x_0)$ and $\rho\in\pi_n(X,x_0)$ let $\tilde \alpha$ be the preimage of $\alpha$ under $\tilde p$ and $\tilde\rho$ be the preimage of $\rho$ under $p_*$.

Then $\sigma=(\tilde\alpha)_*(\tilde\rho)$ is in $\pi_n(\tilde X, \tilde x_0)$ and we get $\alpha\cdot \rho$ as $p_*(\sigma)$.

Now that I think about it, this is at least as geometric as the actions in your question, but I like the picture better. In the picture, the universal cover not only unrolls the elements of $\pi_1$, but it also unrolls the action of the elements of $\pi_1$ on the maps of spheres.

Answer 3

I prefer to do a more general case, which is useful anyway, and to use fibrations of groupoids. For spaces $X,Y$ define the track groupoid $\pi_1 Y ^X$ to have objects the maps $X \to Y$ and arrows $f \to g$ the homotopy classes rel end maps of homotopies $f \simeq g$, with the usual composition of homotopies. If $i :A \to X$ is a cofibration then $$i^*: \pi_1 Y^X \to \pi_1 Y^A $$ is a fibration of groupoids (give the "obvious definition", but it first appeared in a paper of mine in 1970, J. Algebra, although it is a specialisation of an earlier definition for categories, which has a different purpose). Now for a fibration of groupoids $p: E \to B$ there is an operation of $B$ on the disjoint union of $\pi_0$ of the fibres. Applied to the above case, this gives an operation of $\pi_1 Y^A$ on homotopy classes $X \to Y$ relative to maps $A \to Y$.

You can find this in Chapter 7 of my book Topology and groupoids (2006), with applications to, for example, a gluing theorem for homotopy equivalences. (and in essence in the first 1968 edition).

These ideas were found by thinking about maps $(S^n,x) \to (Y,y)$, and then generalising first replacing $S^n$ by $X$ and then the point $x$ by a subspace $A$ and forgetting $y \in Y$.

Sorry to be so long in giving an answer to this but till June, 2011, I was busy with another writing job!