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

An Illustrated Handbook