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A construction associating with every continuous mapping of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622901.png" /> the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622902.png" /> that is obtained from the topological sum (disjoint union) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622903.png" /> by the identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622905.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622906.png" /> is called the mapping cylinder of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622907.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622908.png" /> is a [[Deformation retract|deformation retract]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m0622909.png" />. The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229010.png" /> has the property that the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229011.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229012.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229013.png" /> is the natural retraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229014.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229015.png" />). The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229016.png" /> is a homotopy equivalence, and from the homotopy point of view every continuous mapping can be regarded as an imbedding and even as a [[Cofibration|cofibration]]. A similar assertion holds for a [[Serre fibration|Serre fibration]]. For any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062290/m06229017.png" /> the fibre and cofibre are defined up to a homotopy equivalence.
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''cylindrical construction''
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A construction associating with every continuous mapping of topological spaces  $  f:  X \rightarrow Y $
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the topological space $  I _ {f} \supset Y $
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that is obtained from the topological sum (disjoint union) $  X \times [ 0, 1] \amalg Y $
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by the identification $  x \times \{ 1 \} = f ( x) $,  
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$  x \in X $.  
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The space $  I _ {f} $
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is called the mapping cylinder of $  f $,  
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the subspace $  Y $
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is a [[Deformation retract|deformation retract]] of $  I _ {f} $.  
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The imbedding $  i:  X = X \times \{ 0 \} \subset  I _ {f} $
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has the property that the composite $  \pi \circ i: X \rightarrow Y $
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coincides with $  f $(
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here $  \pi $
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is the natural retraction of $  I _ {f} $
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onto $  Y $).  
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The mapping $  \pi : I _ {f} \rightarrow Y $
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is a homotopy equivalence, and from the homotopy point of view every continuous mapping can be regarded as an imbedding and even as a [[Cofibration|cofibration]]. A similar assertion holds for a [[Serre fibration|Serre fibration]]. For any continuous mapping $  f: X \rightarrow Y $
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the fibre and cofibre are defined up to a homotopy equivalence.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and their application in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and their application in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The literal translation from the Russian yields the phrase "cylindrical constructioncylindrical construction"  for the mapping cylinder. This phrase sometimes turns up in translations.
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The literal translation from the Russian yields the phrase "cylindrical construction"  for the mapping cylinder. This phrase sometimes turns up in translations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 22, 23</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 22, 23</TD></TR></table>

Latest revision as of 07:59, 6 June 2020


cylindrical construction

A construction associating with every continuous mapping of topological spaces $ f: X \rightarrow Y $ the topological space $ I _ {f} \supset Y $ that is obtained from the topological sum (disjoint union) $ X \times [ 0, 1] \amalg Y $ by the identification $ x \times \{ 1 \} = f ( x) $, $ x \in X $. The space $ I _ {f} $ is called the mapping cylinder of $ f $, the subspace $ Y $ is a deformation retract of $ I _ {f} $. The imbedding $ i: X = X \times \{ 0 \} \subset I _ {f} $ has the property that the composite $ \pi \circ i: X \rightarrow Y $ coincides with $ f $( here $ \pi $ is the natural retraction of $ I _ {f} $ onto $ Y $). The mapping $ \pi : I _ {f} \rightarrow Y $ is a homotopy equivalence, and from the homotopy point of view every continuous mapping can be regarded as an imbedding and even as a cofibration. A similar assertion holds for a Serre fibration. For any continuous mapping $ f: X \rightarrow Y $ the fibre and cofibre are defined up to a homotopy equivalence.

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[2] R.E. Mosher, M.C. Tangora, "Cohomology operations and their application in homotopy theory" , Harper & Row (1968)

Comments

The literal translation from the Russian yields the phrase "cylindrical construction" for the mapping cylinder. This phrase sometimes turns up in translations.

References

[a1] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 22, 23
How to Cite This Entry:
Mapping cylinder. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping_cylinder&oldid=18232
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article