composition operation

The composition operations give the action of the homotopy groups of spheres on the homotopy groups of an arbitrary pointed space.

Let XX be a pointed space then, of course, its r thr^{th}-homotopy group π r(X)\pi_r(X) can be defined as the group of pointed homotopy classes of pointed maps from S rS^r to XX. In particular π k(S r)\pi_k(S^r) consists of classes of maps from S kS^k to S rS^r. If απ k(S r)\alpha\in \pi_k(S^r) and is represented by a:S kS ra: S^k \to S^r, and ϕπ r(X)\phi\in \pi_r(X), represented by f:S rXf: S^r\to X then the composite fa:S kXf\circ a: S^k \to X so represents a class ϕαπ k(X)\phi\cdot \alpha\in \pi_k(X).

This composition operation is well defined. It forms one of the primary homotopy operations. An abstraction of this is a component part of the definitional structure of a Pi-algebra.

Created on November 7, 2010 at 13:56:54. See the history of this page for a list of all contributions to it.