# Can torsion in the fundamental group happen in “the real world”

Suppose that $X$ is a CW-complex such that $\pi_1(X)$ has non-trivial torsion.

Does this imply that $X$ cannot be embedded in $\mathbb{R}^3$?

Intuitively, this seems like it should be clear, since (at least this is my intuition about this) a CW-complex embedded in $\mathbb{R}^3$ is something we could "build", and it seems absurd that we could take an actual object and wrap a string around it in such a way that it does not come off, but such that if we wrap it around several more times in the same way, then it does come off.

On the other hand, I have no idea how one would go about showing something like this (if it is even true. My intuition about CW-complexes might be flawed).

• – Seirios Jan 8 '14 at 9:43
• @Seirios Thank you for that link. – Tobias Kildetoft Jan 8 '14 at 9:45
• – Grigory M Jan 8 '14 at 10:45
• The answer is yes, provided the CW-complex has a regular neighbourhood. Basically you think through the possible 3-manifold regular neighbourhoods, and argue 3-manifolds with torsion fundamental group do not embed. See: mathoverflow.net/questions/4478/… – Ryan Budney Jan 8 '14 at 19:03

Replying to the question as stated, i.e. "the real world", rather than just subspaces of the plane or $3$-space, one should mention the fundamental group of the rotation group $SO(3)$ is $\mathbb Z_2$, and this is shown by what is known as "the Dirac String Trick". You can find lots of demos, as well as fine videos, by a web search. Dirac liked this because of its relation to the spin of the electron.