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''(in topology)''
  
A factorization of a space into a topological product. An important problem on non-trivial Cartesian factorizations concerns the cubes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206101.png" /> and the Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206102.png" />. For instance, if a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206103.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206105.png" />, by identifying the points of an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206106.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206107.png" /> (cf. [[Wild imbedding|Wild imbedding]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c0206109.png" />. Any smooth compact contractible manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061010.png" /> is a factor of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061012.png" />. Any factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061014.png" />, is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061016.png" />.
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A factorization of a space into a topological product. An important problem on non-trivial Cartesian factorizations concerns the cubes $I^n$ and the Euclidean spaces $\mathbf R^n$. For instance, if a space $M$ is obtained from $\mathbf R^m$, $3\leq m<n$, by identifying the points of an arc $l\subset\mathbf R^m$ for which $\pi_1(\mathbf R^m\setminus l)\neq1$ (cf. [[Wild imbedding|Wild imbedding]]), then $M\times\mathbf R=\mathbf R^{m+1}$ and $M\times M=\mathbf R^{2m}$. Any smooth compact contractible manifold $M^m$ is a factor of an $I^n$, $n>m$. Any factor of $I^n$, $n<4$, is an $I^m$, $m<n$.
  
 
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Another famous example is Bing's  "Dog Bone"  decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061018.png" />-dimensional Euclidean space, its product with a line is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061019.png" />-dimensional Euclidean space.
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Another famous example is Bing's  "Dog Bone"  decomposition of $3$-dimensional Euclidean space, its product with a line is homeomorphic to $4$-dimensional Euclidean space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "The cartesian product of a certain non-manifold and a line is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020610/c02061020.png" />,"  ''Ann. of Math.'' , '''70'''  (1959)  pp. 399–412</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Daverman,  "Decompositions of manifolds" , Acad. Press  (1986)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "The cartesian product of a certain nonmanifold and a line is $E^4$,"  ''Ann. of Math.'' , '''70'''  (1959)  pp. 399–412. {{ZBL|0089.39501}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Daverman,  "Decompositions of manifolds" , Acad. Press  (1986). {{ZBL|0608.57002}}</TD></TR>
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Latest revision as of 14:08, 28 July 2021

(in topology)

A factorization of a space into a topological product. An important problem on non-trivial Cartesian factorizations concerns the cubes $I^n$ and the Euclidean spaces $\mathbf R^n$. For instance, if a space $M$ is obtained from $\mathbf R^m$, $3\leq m<n$, by identifying the points of an arc $l\subset\mathbf R^m$ for which $\pi_1(\mathbf R^m\setminus l)\neq1$ (cf. Wild imbedding), then $M\times\mathbf R=\mathbf R^{m+1}$ and $M\times M=\mathbf R^{2m}$. Any smooth compact contractible manifold $M^m$ is a factor of an $I^n$, $n>m$. Any factor of $I^n$, $n<4$, is an $I^m$, $m<n$.

References

[1] Itogi Nauk. Algebra. Topol. Geom. 1965 (1967) pp. 227; 243


Comments

Another famous example is Bing's "Dog Bone" decomposition of $3$-dimensional Euclidean space, its product with a line is homeomorphic to $4$-dimensional Euclidean space.

References

[a1] R.H. Bing, "The cartesian product of a certain nonmanifold and a line is $E^4$," Ann. of Math. , 70 (1959) pp. 399–412. Zbl 0089.39501
[a2] R.J. Daverman, "Decompositions of manifolds" , Acad. Press (1986). Zbl 0608.57002
How to Cite This Entry:
Cartesian factorization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian_factorization&oldid=13939
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article