17
$\begingroup$

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ into unit circles?). But it has recently been proved that the same is not true of Möbius strips:

Frolkina, Olga D. "Pairwise disjoint Moebius bands in space." Journal of Knot Theory and Its Ramifications 27, no. 09 (2018): 1842005. Journal link.

Melikhov, Sergey A. "A note on O. Frolkina's paper 'Pairwise disjoint Moebius bands in space'." arXiv:1810.04089 (2018).

(The first paper proves it for tame subsets, the second removes the tame restriction.) My question is:

Q. Does this new result imply that $\mathbb{R}^4$, or $\mathbb{R}^n$, $n \ge 5$, cannot be partitioned into Klein bottles, congruent or otherwise?


          KlienBot
          Image from Sensei's Library.


$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.