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.