Path space fibration

In algebraic topology, the path space fibration over a pointed space $(X, *)$ Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces. is a fibration of the form{{harvnb|Davis|Kirk|2001|loc=Theorem 6.15. 2.}}

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

where

$PX$ is the based path space of the pointed space $(X, *)$ ; that is, $PX = \{ f\colon I \to X \mid f \ \text{continuous}, f(0) = * \}$ equipped with the compact-open topology.
$\Omega X$ is the fiber of $\chi \mapsto \chi(1)$ over the base point of $(X, *)$ ; thus it is the loop space of $(X, *)$ .

The free path space of X, that is, $\operatorname{Map}(I, X) = X^I$ , consists of all maps from I to X that do not necessarily begin at a base point, and the fibration $X^I \to X$ given by, say, $\chi \mapsto \chi(1)$ , is called the free path space fibration.

The path space fibration can be understood to be dual to the mapping cone.{{clarify|more precise meaning|date=August 2022}} The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.

Mapping path space

If $f\colon X\to Y$ is any map, then the mapping path space $P_f$ of $f$ is the pullback of the fibration $Y^I \to Y, \, \chi \mapsto \chi(1)$ along $f$ . (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.{{harvnb|Davis|Kirk|2001|loc=§ 6.8.}})

Since a fibration pulls back 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 the fibration $PY \overset{\chi \mapsto \chi(1)}{\longrightarrow} Y$ along $f$ .

Note also $f$ is the composition

: $X \overset{\phi}\to P_f \overset{p}\to Y$

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

If $f$ is a fibration to begin with, then the map $\phi\colon X \to P_f$ is a fiber-homotopy equivalence and, consequently,using the change of fiber the fibers of $f$ over the path-component of the base point are homotopy equivalent to the homotopy fiber $F_f$ of $f$ .

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 $\alpha, \beta$ such that $\alpha(1) = \beta(0)$ is the path $\beta \cdot \alpha\colon I \to X$ 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: $(\gamma \cdot \beta) \cdot \alpha \ne \gamma \cdot (\beta \cdot \alpha)$ , as seen directly. One solution to this failure is to pass to homotopy classes: one has $[(\gamma \cdot \beta) \cdot \alpha] = [\gamma \cdot (\beta \cdot \alpha)]$ . 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.{{harvnb|Whitehead|1978|loc=Ch. III, § 2.}} (A more sophisticated solution is to rethink composition: work with an arbitrary family of compositions; see the introduction of Lurie's paper,{{cite web|first=Jacob|last=Lurie|authorlink=Jacob Lurie|url=http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf|title=Derived Algebraic Geometry VI: E[k]-Algebras|date=October 30, 2009}} leading to the notion of an operad.)

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

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

An element f of this set has a 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 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:

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

where p sends each $f: [0, r] \to X$ 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 $f\colon [0, r] \to X$ and $g\colon [0, s] \to X$ ,

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

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.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:

{{math_theorem|name=Lemma|math_statement=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.)

Notes

References

{{cite book|first1=James F. |last1=Davis|first2= Paul|last2= Kirk|url=http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf|title= Lecture Notes in Algebraic Topology|series=Graduate Studies in Mathematics|volume= 35|publisher=American Mathematical Society|location= Providence, RI|year= 2001|pages= xvi+367|isbn=0-8218-2160-1 |mr=1841974|doi=10.1090/gsm/035}}
{{cite book|last=May|first=J. Peter|authorlink=J. Peter May| url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A Concise Course in Algebraic Topology|series=Chicago Lectures in Mathematics|publisher= University of Chicago Press|location= Chicago, IL|year= 1999|pages= x+243|isbn=0-226-51182-0|mr=1702278}}
{{cite book|first=George W.|last= Whitehead|authorlink=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ|edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag|location=New York-Berlin|isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}

Category:Algebraic topology

Category:Homotopy theory