Actions of the fundamental group - Revision history

Vipul: 1 revision

2008-05-11T19:31:26Z

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Vipul at 16:43, 3 December 2007

2007-12-03T16:43:54Z

← Older revision		Revision as of 16:43, 3 December 2007
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	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.		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:

			* For a [[topological space with Abelian fundamental group]], the action of the fundamental group on itself is trivial. Thus there is a canonical identification of the fundamental groups at all points.

	==Action on higher homotopy groups==		==Action on higher homotopy groups==
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	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.		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 ~~associated~~ 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.		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===

	~~In particular~~, ~~for~~ a [[~~simply connected~~ space]], ~~there is~~ a ~~canonical identification~~ of ~~higher homotopy groups at all basepoints~~.		Given any topological space <math>(Y,y_0)</math> where <math>y_0</math> is a [[nondegenerate point]] in <math>Y</math> (viz its inclusion is a cofibration) we can mimic the above action to get an action of the fundamental group <math>\pi_1(X,x_0)</math> on the ''set'' of based maps <math>[(Y,y_0);(X,x_0)]</math>. Note that this is now only a group action on a set, because <math>(Y,y_0)</math> has no additional structure. If <math>(Y,y_0)</math> has the structure of a [[H-space]] with identity element <math>y_0</math>, 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.

Vipul: Ho

2007-11-02T23:21:50Z

New page

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

For this article <math>(X,x_0)</math> is the [[based topological space]] and <math>G = \pi_1(X, x_0)</math> is its [[fundamental group]].

==Action on itself==

<math>G</math> acts on itself by [[inner automorphism]]s. 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:

{{fillin}}

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.

==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 associated 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.

In particular, for a [[simply connected space]], there is a canonical identification of higher homotopy groups at all basepoints.