Mathematics
The Fundamental Group and the Classification of Surfaces
Quick fact
The fundamental group of a surface is almost a complete classifier: every closed, connected surface is uniquely determined (up to homeomorphism) by its fundamental group—except for the so-called 'fake surfaces' whose fundamental groups coincide with the projective plane.
Why this is interesting
Take a balloon and a donut. You can stretch the balloon into a sphere, but you can never turn it into a donut without a hole. What is it that distinguishes these shapes so rigidly?