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Concept
Homotopy fiber
Summary
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 . 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 groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.
The homotopy fiber has a simple description for a continuous map . If we replace 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 to be the set of pairs where and (for ) 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 fiberwhich can be defined as the set of all with and a path such that and for some fixed basepoint . A consequence of this definition is that if two points of are in the same path connected component, then their homotopy fibers are homotopy equivalent.
Another way to construct the homotopy fiber of a map is to consider the homotopy limitpg 21 of the diagramthis is because computing the homotopy limit amounts to finding the pullback of the diagramwhere the vertical map is the source and target map of a path , soThis means the homotopy limit is in the collection of mapswhich is exactly the homotopy fiber as defined above.
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