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Difference between revisions of "Path space"

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The space $E$ of the fibre space $(E,p,X)$, which is called the path fibre space. Here $X$ is a [[Path-connected space|path-connected space]] with a distinguished point $*$, $E$ is the set of paths (cf. [[Path|Path]]) in $X$ starting from $*$ and $p$ is the mapping associating to each path its end-point. Moreover, $E$ is considered to have the compact-open topology. The fibre of this fibre space (which is a [[Serre fibration|Serre fibration]]) is the [[Loop space|loop space]] $\Omega X$ — the set of all loops (cf. [[Loop|Loop]]) in $X$ at $*$. A path space can be contracted within itself to a point, so the homotopy groups $\pi_n(E)=0$, and the homotopy sequence of the path fibre space degenerates into the so-called Hurewicz isomorphisms:
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The space $E$ of the fibre space $(E,p,X)$, which is called the path fibre space. Here $X$ is a [[path-connected space]] with a distinguished point $*$, $E$ is the set of [[path]]s in $X$ starting from $*$ and $p$ is the mapping associating to each path its end-point. Moreover, $E$ is considered to have the compact-open topology. The fibre of this fibre space (which is a [[Serre fibration]]) is the [[Loop space|loop space]] $\Omega X$ — the set of all loops (cf. [[Loop (in topology)]]) in $X$ at $*$. A path space can be contracted within itself to a point, so the homotopy groups $\pi_n(E)=0$, and the homotopy sequence of the path fibre space degenerates into the so-called Hurewicz isomorphisms:
  
 
$$\pi_n(\Omega X)\approx\pi_{n+1}(X).$$
 
$$\pi_n(\Omega X)\approx\pi_{n+1}(X).$$

Latest revision as of 20:06, 29 October 2016


The space $E$ of the fibre space $(E,p,X)$, which is called the path fibre space. Here $X$ is a path-connected space with a distinguished point $*$, $E$ is the set of paths in $X$ starting from $*$ and $p$ is the mapping associating to each path its end-point. Moreover, $E$ is considered to have the compact-open topology. The fibre of this fibre space (which is a Serre fibration) is the loop space $\Omega X$ — the set of all loops (cf. Loop (in topology)) in $X$ at $*$. A path space can be contracted within itself to a point, so the homotopy groups $\pi_n(E)=0$, and the homotopy sequence of the path fibre space degenerates into the so-called Hurewicz isomorphisms:

$$\pi_n(\Omega X)\approx\pi_{n+1}(X).$$


Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 75ff, 99ff
How to Cite This Entry:
Path space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Path_space&oldid=31871
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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