Difference between revisions of "Cartesian factorization"
From Encyclopedia of Mathematics
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| − | A factorization of a space into a topological product. An important problem on non-trivial Cartesian factorizations concerns the cubes | + | 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 | + | 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. |
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| − | <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 | + | <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 $E_4$," ''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> |
Revision as of 12:01, 13 August 2014
(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 non-manifold and a line is $E_4$," Ann. of Math. , 70 (1959) pp. 399–412 |
| [a2] | R.J. Daverman, "Decompositions of manifolds" , Acad. Press (1986) |
How to Cite This Entry:
Cartesian factorization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian_factorization&oldid=32893
Cartesian factorization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian_factorization&oldid=32893
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article