Homotopy fiber

Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber is part of a construction of associating to an arbitrary continuous function of topological spaces f colon A o B a fibration.

In particular, given such a map, define E_f to be the set of pairs (a,p) where a in A and p colon [0,1] o B is a path such that p(0) = f(a). We given E_f a topology by giving it the subspace topology as a subset of the function space B^I (which has the compact-open topology). Then the map E_f o B given by (a,p) mapsto p(1) is a fibration. Furthermore, E_f is homotopy equivalent to A as follows: Embed A as a subspace of E_f by a mapsto (a, p_a) where p_a is the constant path at f(a). Then E_f 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" F_f, which can be defined as the set of all (a, p) with a in A and p colon [0,1] o B a path such that p(0) = f(a) and p(1) = b_0, where b_0 in B is some fixed basepoint of B.

References

*citation| last=Hatcher |first= Allen |title=Algebraic Topology |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0.


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