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In mathematics, especially homotopy theory, the homotopy 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 → π n ] → π n → π n → ⋯ {\displaystyle \cdots \to \pi _{n+1}\to \pi _{n}]\to \pi _{n}\to \pi _{n}\to \cdots }
Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle
C ∙ → A ∙ → B ∙ → {\displaystyle C_{\bullet }\to A_{\bullet }\to B_{\bullet }\xrightarrow {} }