TheoremProved

Classification of closed surfaces

Statement

Every compact connected surface without boundary is homeomorphic to exactly one of: the sphere $S^2$ ; a connected sum of $g \geq 1$ tori (an orientable genus- $g$ surface, $\chi = 2-2g$ ); or a connected sum of $k \geq 1$ copies of the projective plane $\mathbb{RP}^2$ (a non-orientable surface, $\chi = 2-k$ ). No two surfaces on this list are homeomorphic to each other.

Why is it true?

However twisted or crumpled a closed surface looks, cutting it up and reassembling it topologically always yields either a plain sphere, a sphere with some number of handles glued on (donuts stacked on a ball - a torus, a two-holed torus, and so on), or a sphere with some number of Möbius-band twists sewn in (which can't be embedded in ordinary 3D space without self-intersection, like the Klein bottle or the projective plane).

Proof sketch

Triangulate the surface, then cut it open along a spanning tree of the triangulation's dual graph to obtain a single polygon whose edges are identified in pairs according to a word in the edge labels (such as $aba^{-1}b^{-1}$ for the torus). A sequence of elementary cut-and-paste moves on this polygon (relabelling, cutting along a diagonal and regluing) reduces any such word to one of the standard forms $a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}b_g^{-1}$ or $a_1a_1\cdots a_ka_k$ , which identify the surface as a genus- $g$ orientable surface or a connected sum of $k$ projective planes.

Topics that use this theorem

Euler characteristic

Related theorems

Euler's polyhedron formula Gauss–Bonnet theorem Hairy ball theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

William S. Massey (1991). A Basic Course in Algebraic Topology
George K. Francis, Jeffrey R. Weeks (1999). Conway's ZIP Proof