View source for Cartesian factorization

{{TEX|done}}
''(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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  ''Itogi Nauk. Algebra. Topol. Geom. 1965''  (1967)  pp. 227; 243</TD></TR></table>



====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====
<table>
<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>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Daverman,   "Decompositions of manifolds" , Acad. Press  (1986). {{ZBL|0608.57002}}</TD></TR>
</table>