Let be a closed orientable surface of genus . (Below we will occasionally write , omitting the genus.) Then its Euler characteristic is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be , roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.
Posts Tagged ‘Euler characteristic’
is a fundamental invariant of , and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups , but actually one can work on a more intuitive level characterized by the following slogan:
The Euler characteristic is a homotopy-invariant generalization of cardinality.
More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:
- Cardinality: .
- Homotopy invariance: If , then .
- Inclusion-exclusion: Suppose is the union of two subcomplexes whose intersection is a subcomplex of both and . Then .
Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.