Path space fibration
In algebraic topology, the path space fibration over a based space [math]\displaystyle{ (X, *) }[/math][1] is a fibration of the form[2]
- [math]\displaystyle{ \Omega X \hookrightarrow PX \overset{\chi \mapsto \chi(1)}\to X }[/math]
where
- [math]\displaystyle{ PX }[/math] is the based path space of X; that is, [math]\displaystyle{ PX = \{ f\colon I \to X \mid f \ \text{continuous}, f(0) = * \} }[/math] equipped with the compact-open topology.
- [math]\displaystyle{ \Omega X }[/math] is the fiber of [math]\displaystyle{ \chi \mapsto \chi(1) }[/math] over the base point of X; thus it is the loop space of X.
The free path space of X, that is, [math]\displaystyle{ \operatorname{Map}(I, X) = X^I }[/math], consists of all maps from I to X that may not preserve the base points, and the fibration [math]\displaystyle{ X^I \to X }[/math] given by, say, [math]\displaystyle{ \chi \mapsto \chi(1) }[/math], is called the free path space fibration.
The path space fibration can be understood to be dual to the mapping cone.[clarification needed] The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.
Mapping path space
If [math]\displaystyle{ f\colon X\to Y }[/math] is any map, then the mapping path space [math]\displaystyle{ P_f }[/math] of [math]\displaystyle{ f }[/math] is the pullback of the fibration [math]\displaystyle{ Y^I \to Y, \, \chi \mapsto \chi(1) }[/math] along [math]\displaystyle{ f }[/math]. (A mapping path space satisfies the universal property that is dual to that of a mapping cylinder, which is a push-out. Because of this, a mapping path space is also called a mapping cocylinder.[3])
Since a fibration pulls back to a fibration, if Y is based, one has the fibration
- [math]\displaystyle{ F_f \hookrightarrow P_f \overset{p}\to Y }[/math]
where [math]\displaystyle{ p(x, \chi) = \chi(0) }[/math] and [math]\displaystyle{ F_f }[/math] is the homotopy fiber, the pullback of the fibration [math]\displaystyle{ PY \overset{\chi \mapsto \chi(1)}{\longrightarrow} Y }[/math] along [math]\displaystyle{ f }[/math].
Note also [math]\displaystyle{ f }[/math] is the composition
- [math]\displaystyle{ X \overset{\phi}\to P_f \overset{p}\to Y }[/math]
where the first map [math]\displaystyle{ \phi }[/math] sends x to [math]\displaystyle{ (x, c_{f(x)}) }[/math]; here [math]\displaystyle{ c_{f(x)} }[/math] denotes the constant path with value [math]\displaystyle{ f(x) }[/math]. Clearly, [math]\displaystyle{ \phi }[/math] is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
If [math]\displaystyle{ f }[/math] is a fibration to begin with, then the map [math]\displaystyle{ \phi\colon X \to P_f }[/math] is a fiber-homotopy equivalence and, consequently,[4] the fibers of [math]\displaystyle{ f }[/math] over the path-component of the base point are homotopy equivalent to the homotopy fiber [math]\displaystyle{ F_f }[/math] of [math]\displaystyle{ f }[/math].
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 [math]\displaystyle{ \alpha, \beta }[/math] such that [math]\displaystyle{ \alpha(1) = \beta(0) }[/math] is the path [math]\displaystyle{ \beta \cdot \alpha\colon I \to X }[/math] given by:
- [math]\displaystyle{ (\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} }[/math].
This product, in general, fails to be associative on the nose: [math]\displaystyle{ (\gamma \cdot \beta) \cdot \alpha \ne \gamma \cdot (\beta \cdot \alpha) }[/math], as seen directly. One solution to this failure is to pass to homotopy classes: one has [math]\displaystyle{ [(\gamma \cdot \beta) \cdot \alpha] = [\gamma \cdot (\beta \cdot \alpha)] }[/math]. Another solution is to work with paths of arbitrary lengths, leading to the notions of Moore's path space and Moore's path space fibration, described below.[5] (A more sophisticated solution is to rethink composition: work with an arbitrary family of compositions; see the introduction of Lurie's paper,[6] leading to the notion of an operad.)
Given a based space [math]\displaystyle{ (X, *) }[/math], we let
- [math]\displaystyle{ P' X = \{ f\colon [0, r] \to X \mid r \ge 0, f(0) = * \}. }[/math]
An element f of this set has a unique extension [math]\displaystyle{ \widetilde{f} }[/math] to the interval [math]\displaystyle{ [0, \infty) }[/math] such that [math]\displaystyle{ \widetilde{f}(t) = f(r),\, t \ge r }[/math]. Thus, the set can be identified as a subspace of [math]\displaystyle{ \operatorname{Map}([0, \infty), X) }[/math]. The resulting space is called the Moore path space of X, after John Coleman Moore, who introduced the concept. Then, just as before, there is a fibration, Moore's path space fibration:
- [math]\displaystyle{ \Omega' X \hookrightarrow P'X \overset{p}\to X }[/math]
where p sends each [math]\displaystyle{ f: [0, r] \to X }[/math] to [math]\displaystyle{ f(r) }[/math] and [math]\displaystyle{ \Omega' X = p^{-1}(*) }[/math] is the fiber. It turns out that [math]\displaystyle{ \Omega X }[/math] and [math]\displaystyle{ \Omega' X }[/math] are homotopy equivalent.
Now, we define the product map
- [math]\displaystyle{ \mu: P' X \times \Omega' X \to P' X }[/math]
by: for [math]\displaystyle{ f\colon [0, r] \to X }[/math] and [math]\displaystyle{ g\colon [0, s] \to X }[/math],
- [math]\displaystyle{ \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} }[/math].
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, [math]\displaystyle{ p: P'X \to X }[/math] is an Ω'X-fibration.[7]
Notes
- ↑ Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces.
- ↑ Davis & Kirk 2001, Theorem 6.15. 2.
- ↑ Davis & Kirk 2001, § 6.8.
- ↑ using the change of fiber
- ↑ Whitehead 1978, Ch. III, § 2.
- ↑ Lurie, Jacob (October 30, 2009). "Derived Algebraic Geometry VI: E[k-Algebras"]. http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf.
- ↑ Let G = Ω'X and P = P'X. That G preserves the fibers is clear. To see, for each γ in P, the map [math]\displaystyle{ G \to p^{-1}(p(\gamma)),\, g \mapsto \gamma g }[/math] 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.
- [math]\displaystyle{ f: p^{-1}(b) \to q^{-1}(b) }[/math] is a weak equivalence for some b in B.
- [math]\displaystyle{ f: p^{-1}(b) \to q^{-1}(b) }[/math] is a weak equivalence for every b in B.
We apply the lemma with [math]\displaystyle{ B = I, D = I \times G, E = I \times_X P, f(t, g) = (t, \alpha(t) g) }[/math] where α is a path in P and I → X is t → the end-point of α(t). Since [math]\displaystyle{ p^{-1}(p(\gamma)) = G }[/math] 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
- Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Graduate Studies in Mathematics. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf.
- May, J. Peter (1999). A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. pp. x+243. ISBN 0-226-51182-0. http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf.
- Whitehead, George W. (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. https://books.google.com/books?id=wlrvAAAAMAAJ.
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