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).

Solution 1:

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.

The demo I use is easily made at home and is shown by the figure

which comes from the Section "Motion" of a 1992 Royal Institution Friday Evening Discourse Out of Line.

Use two pieces of square board, mark an arrow on one, and join the top to the bottom with ribbon, preferably of different colours, and fastened at the corners by bulldog clips, in case it all gets too tangled. Rotate the top board through 360 degrees and the ribbons get tangled; another 360 degrees in the same direction, and you can untangle them, keeping the boards fixed. You can also involve a child, or student, from the audience in the whole thing. I usually bring in the old joke: "There is nothing however complicated which you cannot with sufficient effort make more complicated."

See also the Section on "Paths and knot spaces" or p.13 of this presentation What is and what should be higher dimensional group theory? for a knotty demonstration, which used string and a copper pentoil to demonstrate knot relations and a complicated loop which unties itself from the knot.