Path space fibration

In algebraic topology, the path space fibration over a based space (X, *) is a fibration of the form

$\Omega X\hookrightarrow PX{\overset {\chi \mapsto \chi (1)}{\to }}X$

where

• $PX=\operatorname {Map} (I,X)=\{f:I\to X|f(0)=*\}$ is the space called the path space of X.
• $\Omega X$ is the fiber of $\chi \mapsto \chi (1)$ over the base point of X; thus it is the loop space of X.

The space $X^{I}$ consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration $X^{I}\to X$ given by, say, $\chi \mapsto \chi (1)$, is called the free path space fibration.

Mapping path space

If ƒ:X→Y is any map, then the mapping path space P_ƒ of ƒ is the pullback of $Y^{I}\to Y,\,\chi \mapsto \chi (1)$ along ƒ. Since a fibration pullbacks to a fibration, if Y is based, one has the fibration

$F_{f}\hookrightarrow P_{f}{\overset {p}{\to }}Y$

where $p(x,\chi )=\chi (0)$ and $F_{f}$ is the homotopy fiber, the pullback of $PY{\overset {\chi \mapsto \chi (1)}{\to }}Y$ along ƒ.

Note also ƒ is the composition

$X{\overset {\phi }{\to }}P_{f}{\overset {p}{\to }}Y$

where the first map φ sends x to $(x,c_{f(x)})$, $c_{f(x)}$ the constant path with value ƒ(x). Clearly, φ is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.

If ƒ is a fibration to begin with, then $\phi :X\to P_{f}$ is a fiber-homotopy equivalence and, consequently, the fibers of f over the path-component of the base point are homotopy equivalent to the homotopy fiber $F_{f}$ of ƒ.

Moore's path space

By definition, a path in a space X is a map from the unit interval I to X. Again by definition, the product of two paths α, β such that α(1) = β(0) is the path β · α: I → X given by:

$(\beta \cdot \alpha )(t)={\begin{cases}\alpha (2t)&{\text{if }}0\leq t\leq 1/2\\\beta (2t-1)&{\text{if }}1/2\leq t\leq 1\\\end{cases}}$.

This product, in general, fails to be associative on the nose: (γ · β) · α ≠ γ · (β · α), as seen directly. One solution to this failure is to pass to homotopy classes: one has [(γ · β) · α ] = [γ · (β · α)]. Another solution is to work with paths of arbitrary length, leading to the notions of Moore's path space and Moore's path space fibration.

Given a based space (X, *), we let

$P'X=\{f:[0,r]\to X|r\geq 0,f(0)=*\}.$

An element f of this set has the unique extension ${\widetilde {f}}$ to the interval $[0,\infty )$ such that ${\widetilde {f}}(t)=f(r),\,t\geq r$. Thus, the set can be identified as a subspace of $\operatorname {Map} ([0,\infty ),X)$. The resulting space is called Moore's path space of X. Then, just as before, there is a fibration, Moore's path space fibration:

$\Omega 'X\hookrightarrow P'X{\overset {p}{\to }}X$

where p sends each f: [0, r] → X to f(r) and $\Omega 'X=p^{-1}(*)$ is the fiber. It turns out that $\Omega X$ and $\Omega 'X$ are homotopy equivalent.

Now, we define the product map:

$\mu :P'X\times \Omega 'X\to P'X$

by: for $f:[0,r]\to X$ and $g:[0,s]\to X$,

$\mu (g,f)(t)={\begin{cases}f(t)&{\text{if }}0\leq t\leq r\\g(t-r)&{\text{if }}r\leq t\leq s+r\\\end{cases}}$.

This product is manifestly associative. In particular, with μ restricted to Ω'X × Ω'X, we have that Ω'X is a topological monoid (in the category of all spaces). Moreover, this monoid Ω'X acts on P'X through the original μ. In fact, $p:P'X\to X$ is a Ω'X-fibration.

References

• James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology
• May, J. A Concise Course in Algebraic Topology
• George William Whitehead (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508. Retrieved September 6, 2011.