Path space fibration

In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the path space fibration over a <a href="/facts/Pointed_space/V4DTgRCd">pointed space</a> 
 
 
 
 (
 X
 ,
 ∗
 )
 
 
 {\displaystyle (X,*)}
 
 is a <a href="/facts/Fibration/h51Cx7xU">fibration</a> of the form

Ω
        X
        ↪
        P
        X
        
          
            →
            
              χ
              ↦
              χ
              (
              1
              )
            
          
        
        X
      
    
    {\displaystyle \Omega X\hookrightarrow PX{\overset {\chi \mapsto \chi (1)}{\to }}X}

where

<ul><li>
 
 
 
 P
 X
 
 
 {\displaystyle PX}
 
 is the based <a href="/facts/Path_space_(algebraic_topology)/XeS9AWD2">path space</a> of the pointed space 
 
 
 
 (
 X
 ,
 ∗
 )
 
 
 {\displaystyle (X,*)}
 
; that is, 
 
 
 
 P
 X
 =
 {
 f
 :
 I
 →
 X
 ∣
 f
  
 
 continuous
 
 ,
 f
 (
 0
 )
 =
 ∗
 }
 
 
 {\displaystyle PX=\{f\colon I\to X\mid f\ {\text{continuous}},f(0)=*\}}
 
 equipped with the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>.</li>
<li>
 
 
 
 Ω
 X
 
 
 {\displaystyle \Omega X}
 
 is the fiber of 
 
 
 
 χ
 ↦
 χ
 (
 1
 )
 
 
 {\displaystyle \chi \mapsto \chi (1)}
 
 over the base point of 
 
 
 
 (
 X
 ,
 ∗
 )
 
 
 {\displaystyle (X,*)}
 
; thus it is the <a href="/facts/Loop_space/S5xkdCkx">loop space</a> of 
 
 
 
 (
 X
 ,
 ∗
 )
 
 
 {\displaystyle (X,*)}
 
.</li></ul>
The free path space of X, that is, 
 
 
 
 Map
 ⁡
 (
 I
 ,
 X
 )
 =
 
 X
 
 I
 
 
 
 
 {\displaystyle \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
 
 
 →
 X
 
 
 {\displaystyle X^{I}\to X}
 
 given by, say, 
 
 
 
 χ
 ↦
 χ
 (
 1
 )
 
 
 {\displaystyle \chi \mapsto \chi (1)}
 
, is called the free path space fibration.
The path space fibration can be understood to be dual to the <a href="/facts/Mapping_cone_(topology)/a32m1wAa">mapping cone</a>. The fiber of the based fibration is called the mapping fiber or, equivalently, the <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a>.

Path space fibration open-in-new

Path space fibration