The th homotopy group consists of based homotopy classes of maps . It is a group for and an abelian group for .
The rational homotopy group is obtained by tensoring a homotopy group with the rational numbers. It retains the free part and discards torsion.
For an odd-dimensional sphere , only is nonzero. For an even-dimensional sphere , the nonzero rational homotopy groups occur in degrees and , and both are isomorphic to .
If a path-connected space is -connected for , then its first potentially nonzero homotopy group maps isomorphically to homology:
A weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.
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In algebraic topology, a homotopy group is an important algebraic invariant that captures the topological structure of a space. The most common homotopy groups are the fundamental group and higher homotopy groups. 1. **Fundamental Group (\(\pi_1\))**: The fundamental group is the first homotopy group and provides a measure of the "loop structure" of a space.