Path space fibration

In algebraic topology, the path space fibration over a based space (X, *)^[1] 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,^[2] 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:

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

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.^[3]

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

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

An element f of this set has the unique extension $\widetilde{f}$ to the interval $[0, \infty)$ such that $\widetilde{f}(t) = f(r),\, t \ge r$. 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 $g: [0, s] \to X$,

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

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 an Ω'X-fibration.^[4]

Notes

1. Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Haudsorff spaces.
2. using the change of fiber
3. Whitehead 1979, Ch. III, § 2.
4. Let G = Ω'X and P = P'X. That G preserves the fibers is clear. To see, for each γ in P, the map $G \to p^{-1}(p(\gamma)),\, g \mapsto \gamma g$ is a weak equivalence, we can use the following lemma:
   Lemma — Let p: D → B, q: E → B be fibrations over an unbased space B, f: D → E a map over B. If B is path-connected, then the following are equivalent:

   • f is a weak equivalence.
   • $f: p^{-1}(b) \to q^{-1}(b)$ is a weak equivalence for some b in B.
   • $f: p^{-1}(b) \to q^{-1}(b)$ is a weak equivalence for every b in B.
   We apply the lemma with $B = I, D = I \times G, E = I \times_X P, f(t, g) = (t, \alpha(t) g)$ where α is a path in P and I → X is t → the end-point of α(t). Since $p^{-1}(p(\gamma)) = G$ if γ is the constant path, the claim follows from the lemma. (In a nutshell, the lemma follows from the long exact homotopy sequence and the five lemma.)

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 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508. Retrieved September 6, 2011.