Homotopy Group - Relative Homotopy Groups

Relative Homotopy Groups

There are also relative homotopy groups π_n(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.