The Euler characteristic of product spaces

The Euler characteristic is a nifty function that pops up all over the place in math and geometry, and has all kinds of neat properties. It gives any topological space (think triangle meshes and simplicial complexes, their n-dimensional friends) an integer, and, more importantly, it is a topological invariant: if two spaces can be bent into one another, then they will have the same Euler characteristic. The traditional formula for the Euler characteristic is an alternating sum of simplices, so for a triangle mesh M, \chi(M) = V - E + F. If you have tetrahedra, pentatopes, etc. in your mesh, they would just keep alternating signs. The Euler characteristic kind of looks like the cardinality of a set. First, it obeys the straightforward property that \chi(A \cup B) = \chi(A) + \chi(B) - \chi(A \cap B) (keeping the intuition of triangle meshes: simply count up the simplices).

Amazingly, it also shares the following property of set cardinality: \chi(A \times B) = \chi(A) \chi(B). This formula makes sense for finite sets (using the cartesian product), but is sort of crazy for more general spaces because it’s easy to come up with spaces of negative Euler characteristic. If you dig up the proofs for this, however, they tend to be a lot more complicated and give very little intuition. So here’s a simple proof that works for cw-complexes. CW complexes are sort of like simplicial complexes, but we let things like “4-sided triangles” exist. We define the “d-simplices” (they’re actually called d-cells) by saying that 1) points are just like 0-d simplices and 2) n-d cells are bent “hypercubes of n dimensions” that must have a union of (n-1)-d cells as their boundaries.

For CW complexes, the Euler characteristic is computed by an alternating sum of d-cells in the same way as for simplicial complexes. No big surprise, since simplicial complexes are CW complexes. In addition, products of CW complexes are much easier to think about than products of simplicial complexes. For example, consider two spaces A and B each consisting of a single edge and two vertices. The product AxB should be, intuitively, a “square” with 4 vertices, 4 edges, and an internal face. A simplicial complex can only have 2-dimensional triangles, and so the calculation becomes a little more complicated.

Now let’s raise the abstraction level just a little bit. Instead of thinking about the Euler characteristic directly as an alternating sum, we will think of it as an evaluation of a specially crafted polynomial computed from the space. This polynomial is really pretty idiotic: for the case of the spaces A and B above, the polynomial will be P(A) = P(B) = 2 + 1.d, that is, the degree of the monomial represents the dimension, and the coefficient represents the number of cells of that dimension. The Euler characteristic, then, is found by simply evaluating this polynomial at value d=-1.

Now for the cool part. Look at the polynomial P(A \times B) = 4 + 4d + d^2. If you think about how to create the product space, you will see that you get one (a+b)-dimensional cell for every a-dimensional cell in A and every b-dimensional cell in B. The simplest way to express this is as a product between the two polynomials! So P(A \times B) = P(A) P(B), and since I hopefully convinced you that the Euler characteristic is obviously just an evaluation of P, it is “obvious” that \chi(A \times B) = \chi(A)\chi(B). So the “magic” comes from the rule used to create the product space. Really neat.

(Props to Jacob for giving me the original idea for this)

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