The fundamental group is ℤ₂

SO(3) has a strange topological property.

Its fundamental group is ℤ₂.

π₁(SO(3)) = ℤ₂

Meaning: there exist closed loops in SO(3) that cannot be continuously shrunk to a point.

A 360° rotation is one such loop. It does not contract to the identity.

A 720° rotation does. It can be continuously deformed back to no rotation at all.

This is the Dirac belt trick.

Twist a belt 360° and it stays twisted. Twist it 720° and it untwists.

You can demonstrate it on yourself.

A theorem of topology, not a physical assumption. Proven in any textbook on Lie groups.

The fundamental group being ℤ₂ — the two-element group {+1, −1} — means SO(3) has a unique double cover.

A group that maps 2-to-1 onto SO(3), with ℤ₂ as the kernel.