Timeline for Why can't the Klein bottle embed in $\mathbb{R}^3$?
Current License: CC BY-SA 2.5
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| Mar 22, 2010 at 13:00 | comment | added | Don Stanley | This means that the linking number of the meridian of $M_1$ and $\partial M_2$ is twice the linking number with $M_2$. However we know that this linking number is odd by the link appearing theorem. Maybe you had something more elementary in mind? | |
| Mar 22, 2010 at 13:00 | comment | added | Don Stanley | This is very nice, and I read the proofs on Maehara and the Conway and Gordon paper it refers too. So the proof that $P^2$ doesn't embed in $\mathbb{R}^3$ is elementary. So thanks for the reference. However I don't understand your comment "therefore the two-cells must intersect". You can isotope the boundary link to something very close to the meridian of $M_2$, but which wraps around it twice. | |
| Mar 22, 2010 at 11:30 | history | answered | Gjergji Zaimi | CC BY-SA 2.5 |