Is the Euler characteristc defined wrong? If not, why not?

Question

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$

(where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ then we get a genuine "homomorphism":

$$\chi'(S_0\# S_1) = \chi'(S_0)+\chi'(S_1)$$

So it is natural to ask whether or not $$\chi' = 2-V+E-F$$ is the "correct" Euler characterstic.

Obviously, smart people have thought about this before. What is the verdict? Is $\chi'$ more fundamental than $\chi$, and if not, why not?

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Discussion. I think its curious that $\chi' = 1-V+E-F+1$. If we can interpret that two hanging $1$'s, that would motivate $\chi'$ on geometric grounds.

Answer 1

The usual Euler characteristic is absolutely the correct choice, at least for closed manifolds. There are lots of reasons to believe this:

1. $\chi(X \sqcup Y) = \chi(X) + \chi(Y)$, where $\sqcup$ is the disjoint union. (This is the coproduct in the category of, say, topological spaces.) Under some conditions we even have the "inclusion-exclusion" formula $\chi(X \cup Y) = \chi(X) + \chi(Y) - \chi(X \cap Y)$.
2. $\chi(X \times Y) = \chi(X) \chi(Y)$. So we even get a "ring homomorphism."
3. If $X \to Y$ is an $n$-fold covering map, then $\chi(X) = n \chi(Y)$.
4. For closed surfaces, $\chi(X)$ naturally appears in the Gauss-Bonnet theorem. This is part of a major division in the theory of surfaces between the surfaces that admit Riemannian metrics of constant negative curvature, constant zero curvature, and constant positive curvature, which by the Gauss-Bonnet theorem corresponds to having negative Euler characteristic, zero Euler characteristic, and positive Euler characteristic. See also the uniformization theorem. In higher dimensions we have the Chern-Gauss-Bonnet theorem.
5. $\chi(X)$ also naturally appears in the Poincaré–Hopf theorem, which among other things singles out closed smooth manifolds of Euler characteristic $0$ among all closed smooth manifolds: these are precisely the ones which admit nonvanishing vector fields.
6. A slight generalization of the Euler characteristic, the Lefschetz trace $L(f)$ of an endomorphism $f : X \to X$, naturally appears in the Lefschetz fixed point theorem. The Euler characteristic turns out to be the Lefschetz trace $L(\text{id}_X)$ of the identity. Compare to the statement that the dimension of a vector space is the trace of its identity; there is a way to formalize the sense in which the Euler characteristic of a closed manifold is its "homotopy dimension" or "homotopy cardinality." See also this blog post, although I think I stated something incorrect in the introduction.

The connected sum is a less natural operation to consider than you might believe. Most importantly, you need to choose where to cut little holes out of your manifolds (and the choice matters e.g. if your manifolds are not connected). It should be thought of not as an operation on manifolds but as an operation on manifolds after you've chosen little holes to cut out of them, where it just becomes composition (gluing) of cobordisms. And it's this extra cutting step that alters the behavior of the Euler characteristic from the homomorphism property you might expect: with suitable hypotheses, cutting a hole out of a surface decreases its Euler characteristic by $1$.

For closed connected surfaces, the alternative number you define is the first Betti number $b_1 = \dim H_1(X, \mathbb{Q})$ of the surface. Its relevance to taking connected sums is that

$$H_1(X \# Y, \mathbb{Q}) \cong H_1(X, \mathbb{Q}) \oplus H_1(X, \mathbb{Q})$$

but don't read too much into this: the analogous statements for $H_0$ and $H_2$ are both false.

Answer 2

If you did that, you'd just move the $2$ to other places. For example, you'd have state the Poincaré-Hopf theorem as follows:

Theorem. If $S$ is a surface and $\chi'(S)\neq2$, then there is no nonzero tangent vector field to $S$.