Cartesian factorization

(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)