# Related constructions and facts

Some constructions related to the homotopy groups include:

• Relative homotopy groups: for $${n>1}$$, $${\pi_{n}\left(X,A,x_{0}\right)}$$ is defined to be all homotopy classes of maps $${\left(D^{n},\partial D^{n},s_{0}\right)\rightarrow\left(X,A,x_{0}\right)}$$ where $${s_{0}\in \partial D^{n}}$$ and $${x_{0}\in A\subset X}$$
• Stable homotopy group: $${\pi_{i}^{s}\left(X\right)}$$ is defined to be the group that the sequence $${\pi_{i}\left(X\right)\rightarrow\pi_{i+1}\left(SX\right)\rightarrow\pi_{i+2}\left(S^{2}X\right)\rightarrow\dotsb}$$ eventually arrives at, recalling the suspension $${SX}$$; a major unsolved problem in algebraic topology is computing the stable homotopy groups of the spheres
• n-connected space: indicates that $${\pi_{i}=0}$$ for $${i\leq n}$$; so $${0}$$-connected = path-connected, $${1}$$-connected = simply connected
• Action of $${\pi_{1}}$$ on $${\pi_{n}}$$: the homomorphism $${\pi_{1}\rightarrow\textrm{Aut}\left(\pi_{n}\right)}$$ defined by taking the basepoint of $${\pi_{n}}$$ around the loop defined by each element of $${\pi_{1}}$$
• n-simple space: indicates trivial action of $${\pi_{1}}$$ on $${\pi_{n}}$$; a simple (AKA abelian) space is $${n}$$-simple for all $${n\Rightarrow\pi_{1}}$$ is abelian
• Eilenberg-MacLane space: a space $${K\left(G,n\right)}$$ constructed to have one nontrivial $${\pi_{n}=G}$$, a rare case of $${\pi_{i}}$$ uniquely determining the homotopy type; for any connected cell complex $${X}$$, one can construct a $${K\left(G,n\right)}$$ such that the homotopy classes of maps from $${X}$$ to $${K\left(G,n\right)}$$ are isomorphic to $${H^{n}\left(X;G\right)}$$, thus turning homology groups into homotopy groups