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 $f : A \to B$ .

In particular, given such a map, define the mapping path space $E f$ to be the set of pairs $(a, p)$ where $a \in A$ and $p : [0,1] \to B$ 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 $(a, p) ⟼ p (1)$ is a fibration. Furthermore, $E f$ is homotopy equivalent to $A$ as follows: Embed $A$ as a subspace of $E f$ by $a ⟼ (a, p a)$ 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 $p : [0,1] \to B$ a path such that $p (0) = f (a)$ and $p (1) = b 0$ , where $b 0 \in B$ is some fixed basepoint of $B$ .

In the special case that the original map $f$ was a fibration with fiber $F$ , then the homotopy equivalence $A \to E f$ given above will be a map of fibrations over $B$ . 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 $F \to F f$ is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.

References

Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0 .

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Homotopy fiber

See also

References