Actions of the fundamental group

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This article lists the various typical actions of the fundamental group.

For this article (X,x_0) is the based topological space and G = \pi_1(X, x_0) is its fundamental group.

Action on itself

G acts on itself by inner automorphisms. The motivation behind considering this action is as follows. If we look at the fundamental group at a different basepoint, and if that basepoint is in the same path component as the starting basepoint, then a path between the two basepoints can be used to define an isomorphism between the fundamental groups, as follows:

Fill this in later

However, the specific isomorphism we get depends on the homotopy class of path that we pick. Picking two different homotopy classes of paths gives two different isomorphisms, and the composite of one with the inverse of the other, gives an inner automorphism of the fundamental group at either basepoint.

The following are some easy facts:

Action on higher homotopy groups

Given two different basepoints, we can use a path between them to achieve an isomorphism of the higher homotopy groups at the basepoints as well. To do this we use the fact that the inclusion of a point in a sphere is a cofibration, and hence a homotopy of maps at the basepoint (which is just the path) gives a homotopy of maps from the whole sphere.

It turns out that homotopic paths give the same isomorphisms, so isomorphisms are parametrized by homotopy classes of paths between the points. In particular, any two isomorphisms differ by an automorphism which is parametrized by a homotopy class of loops at the point. This associates to every element of the fundamental group an automorphism of every higher homotopy group. The association defines a group action of the fundamental group on higher homotopy groups.

Some facts:

  • A simple space is a path-connected space where the fundamental group acts trivially on all homotopy groups. For simple spaces, there is a canonical identification of the homotopy groups at all points.
  • A simply connected space is a path-connected space with trivial fundamental group; simply connected spaces are in particular simple.

Action on homotopy classes of maps

Given any topological space (Y,y_0) where y_0 is a nondegenerate point in Y (viz its inclusion is a cofibration) we can mimic the above action to get an action of the fundamental group \pi_1(X,x_0) on the set of based maps [(Y,y_0);(X,x_0)]. Note that this is now only a group action on a set, because (Y,y_0) has no additional structure. If (Y,y_0) has the structure of a H-space with identity element y_0, we can see that the set of based maps also gets a multiplicative structure, and the action of the fundamental group preserves that multiplicative structure.