Homotopy fiber

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 $f \colon A \to B$.

In particular, given such a map, define $E_f$ to be the set of pairs $(a,p)$ where $a \in A$ and $p \colon [0,1] \to B$ is a path such that $p(0) = f(a)$. We give $E_f$ a topology by giving it the subspace topology as a subset of $A \times B^I$ (where $B^I$ is the space of paths in $B$ which as a function space has the compact-open topology). Then the map $E_f \to B$ 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 $p \colon [0,1] \to B$ 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

• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0, http://www.math.cornell.edu/~hatcher/AT/ATpage.html .