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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 . In particular, given such a map, define to be the set of pairs where and is a path such that . We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology). Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at . Then deformation retracts to this subspace by contracting the paths. The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiber , which can be defined as the set of all with and a path such that and , where is some fixed basepoint of . In the special case that the original map was a fibration with fiber , then the homotopy equivalence given above will be a map of fibrations over . This will induce a morphism of their long exact sequences of homotopy groups, from which (by applying the Five Lemma) one can see that the map is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one. == See also == *Quasi-fibration 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homotopy fiber」の詳細全文を読む スポンサード リンク
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