Relative Homotopy Groups
There are also relative homotopy groups πn(X,A) for a pair (X,A), where A is a subspace of X. The elements of such a group are homotopy classes of based maps Dn → X which carry the boundary Sn−1 into A. Two maps f, g are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy F : Dn × → X such that, for each p in Sn−1 and t in, the element F(p,t) is in A. The ordinary homotopy groups are the special case in which A is the base point.
These groups are abelian for but for form the top group of a crossed module with bottom group π1(A).
There is a long exact sequence of relative homotopy groups.
Read more about this topic: Homotopy Group
Famous quotes containing the words relative and/or groups:
“Are not all finite beings better pleased with motions relative than absolute?”
—Henry David Thoreau (1817–1862)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)