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Posts Tagged ‘Euler characteristic’

Let \Sigma_g be a closed orientable surface of genus g. (Below we will occasionally write \Sigma, omitting the genus.) Then its Euler characteristic \chi(\Sigma_g) = 2 - 2g 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 2 - 2g, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

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Let X be a finite CW complex with c_0 vertices, c_1 edges, and in general c_i different i-cells. The Euler characteristic

\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i

is a fundamental invariant of X, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula V - E + F = 2 = \chi(S^2) for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups H_i(X, \mathbb{Q}), 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:

  1. Cardinality: \chi(\text{pt}) = 1.
  2. Homotopy invariance: If X \sim Y, then \chi(X) = \chi(Y).
  3. Inclusion-exclusion: Suppose X is the union of two subcomplexes A, B whose intersection A \cap B is a subcomplex of both A and B. Then \chi(X) = \chi(A) + \chi(B) - \chi(A \cap B).

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.

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