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Difference between revisions of "Loop space"

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The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608601.png" /> of all loops (cf. [[Loop (in topology)|Loop (in topology)]]) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608602.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608603.png" /> with distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608604.png" />, endowed with the compact-open topology. A loop space is a fibre in the [[Serre fibration|Serre fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608605.png" /> over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608606.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060860/l0608607.png" /> is the [[Path space|path space]]).
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The space $\Omega X$ of all loops (cf. [[Loop (in topology)|Loop (in topology)]]) at the point $*$ in a topological space $X$ with distinguished point $*$, endowed with the compact-open topology. A loop space is a fibre in the [[Serre fibration|Serre fibration]] $(E,p,X)$ over the space $X$ (here $E$ is the [[Path space|path space]]).
  
  

Latest revision as of 16:00, 19 April 2014

The space $\Omega X$ of all loops (cf. Loop (in topology)) at the point $*$ in a topological space $X$ with distinguished point $*$, endowed with the compact-open topology. A loop space is a fibre in the Serre fibration $(E,p,X)$ over the space $X$ (here $E$ is the path space).


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References

[a1] J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
How to Cite This Entry:
Loop space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loop_space&oldid=31872
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article