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Difference between revisions of "Mapping cylinder"

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''cylindrical construction''
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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.
 
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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====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>

Revision as of 08:22, 10 April 2018

cylindrical construction

A construction associating with every continuous mapping of topological spaces the topological space that is obtained from the topological sum (disjoint union) by the identification , . The space is called the mapping cylinder of , the subspace is a deformation retract of . The imbedding has the property that the composite coincides with (here is the natural retraction of onto ). The mapping 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 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=43106
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article