In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f : A → B {\displaystyle f:A\to B} . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups ⋯ → π n + 1 ( B ) → π n ( Hofiber ( f ) ) → π n ( A ) → π n ( B ) → ⋯ {\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots } Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle C ( f ) ∙ [ − 1 ] → A ∙ → B ∙ → [ + 1 ] {\displaystyle C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]} } gives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.
Construction The homotopy fiber has a simple description for a continuous map f : A → B {\displaystyle f:A\to B} . If we replace f {\displaystyle f} by a fibration, then the homotopy fiber is simply the fiber of the replacement fibration. We recall this construction of replacing a map by a fibration: Given such a map, we can replace it with a fibration by defining the mapping path space E f {\displaystyle E_{f}} to be the set of pairs ( a , γ ) {\displaystyle (a,\gamma )} where a ∈ A {\displaystyle a\in A} and γ : I → B {\displaystyle \gamma :I\to B} (for I = [ 0 , 1 ] {\displaystyle I=[0,1]} ) a path such that γ ( 0 ) = f ( a ) {\displaystyle \gamma (0)=f(a)} . We give E f {\displaystyle E_{f}} a topology by giving it the subspace topology as a subset of A × B I {\displaystyle A\times B^{I}} (where B I {\displaystyle B^{I}} is the space of paths in B {\displaystyle B} which as a function space has the compact-open topology). Then the map E f → B {\displaystyle E_{f}\to B} given by ( a , γ ) ↦ γ ( 1 ) {\displaystyle (a,\gamma )\mapsto \gamma (1)} is a fibration. Furthermore, E f {\displaystyle E_{f}} is homotopy equivalent to A {\displaystyle A} as follows: Embed A {\displaystyle A} as a subspace of E f {\displaystyle E_{f}} by a ↦ γ a {\displaystyle a\mapsto \gamma _{a}} where γ a {\displaystyle \gamma _{a}} is the constant path at f ( a ) {\displaystyle f(a)} . Then E f {\displaystyle E_{f}} deformation retracts to this subspace by contracting the paths.
… excerpt ends here. Continue reading the full article.
