Higher Order π Groups, Induced Homomorphism

Induced Homomorphism

If w is a continuous function from the space Y into Z, then w induces a group homomorphism from πn(Y) into πn(Z). This is a generalization of the related theorem on fundamental groups, and the proof is essentially the same, assuming you use sphere concatenation as the definition of πn. Another proof drills down to the fundamental group of the n-1 order loop spaces of Y and Z, but you don't want to go there.

By composing continuous maps, w takes each image of Sn in Y to an image of Sn in Z. Also, a homotopy on Sn in Y can be composed with w to create a homotopy on Sn in Z. Homotopic images remain homotopic, and w carries homotopy classes to homotopy classes.

Let h = fg, as described in the previous section. Using the same function μ, wrapping one sphere around two tangent spheres, build wh as the concatenation of wf and wg. Within Z, this is the concatenation of the images wf and wg. It is also w applied to fg. The induced map respects the action of the group, and defines a group homomorphism from πn(Y) into πn(Z).

We're a short putt away from proving πn is a functor from the category of topological spaces into the category of abelian groups. Show that the composition of continuous functions induces the composition of the corresponding group homomorphisms, and you're there. As we showed with π1, this is a straighforward consequence of the associative property of function composition. The image of a sphere, followed by one function, and then another, is the same as the image of the sphere acted upon by the composition of the two functions taken together.

Induced by a Covering Space

The projection p from a total space S~ down to a base space S is continuous, and induces a group homomorphism on πn, as described above. When n > 1, the homomorphism becomes an isomorphism.

Let D be a simply connected domain, such as Sn for n > 1. Thus D satisfies our lifting criteria. Every image of D has a lift, and the map induced by p is surjective.

Since D cross I is also simply connected, a homotopy on D can be lifted as well. If f and g are homotopic images of D in S, lift the entire homotopy to show f~ and g~ are homotopic. Therefore, images that are not homotopic in S~ cannot map onto images that are homotopic in S. The induced map on homotopy classes is injective.

When D = Sn, for n > 1, the induced map on πn is an isomorphism.

As you recall, the map on π1 is a monomorphism, but not an isomorphism. Surjective breaks down, because S1 is not simply connected. Injective remains, but a different proof is required. Loops are treated as paths, and homotopic paths lift to homotopic paths. The homotopy carries the endpoint along, a continuous image into the fiber of the base point, which is a discrete set. The image is constant, and the end of the path sits at the base point, upstairs and down. Nonhomotopic loops upstairs cannot project to homotopic loops downstairs, hence p induces a monomorphism on π1.