Let’s start with Euler’s polyhedron formula. Vertices - Edges + Faces = 2 on a sphere.

Why is this? Well, take the sphere, considered as one big face, and draw all the vertices onto it. Then, draw the edges onto it, one by one. Each time you draw an edge connecting two vertices that are already connected, you split one face into two, increasing Faces by one. Each time you draw an edge connecting two vertices that are not already connected, you reduce the number of connected components of vertices by one. So, Edges = (Ending Faces - Starting Faces) + (Starting Components - Ending Components) = Faces - 1 + Vertices - 1, given that we start with one big face of the entire sphere and all the vertices as separate components, and end with all the faces we want and all vertices connected in one component. This gives us Euler’s formula, as desired. [Incidentally, just as we think of connected components of vertices connected by edges, we can dually think of faces as connected regions connected by ABSENCE of edges…]

More generally, though, the key fact is that for a connected solid “blob” (something with no holes, homotopy equivalent to a point), we should have that Vertices - Edges + Faces - Volumes + …, alternating over cells of each type of dimension, = 1. Why is this true for any blob? [Homology proof is below; TODO: Is there any better way to see this?]. We can define a blob as a space where every sphere of every dimension has some filling by a ball of one higher dimension (including the empty (-1)-sphere being filled by a point).

Note that this Euler characteristic is the unique additive function that assigns 1 to every blob. Why is that? Well, consider that a sphere of a given dimension consists of two hemispheric blobs intersecting in a sphere of lower dimension; thus, each sphere of a particular dimension must take value 2 - the value of a lower dimensional sphere. This terminates in the base case that the (-1)-sphere is empty and has value 0 [thus, the 0-sphere of two points has value 2, the 1-sphere of a circle has value 0, the 2-sphere of an ordinary sphere has value 2, and so on]. And then filling this sphere with an interior to make a solid blob, we find that interiors’ values alternate with dimension, from 1 for a point, -1 for the interior of an edge, +1 for the interior of a face, and so on.

Once we know that Euler characteristic is additive and assigns the same value 1 to every blob (and indeed, the same +1 or -1 to every interior based on its dimension), we see that refining a shape by splitting an edge up into a path or splitting a face up into multiple sub-faces, etc, doesn’t change its Euler characteristic (just replacing one blob by another more complicated blob). And any two topologically equivalent shapes have a common refinement (seen by laying both triangulations over each other), so topologically equivalent shapes have the same Euler characteristic. Indeed, a fortiori, any two homotopy equivalent shapes are obtained from each other just by some substitution of blobs for blobs, so homotopy equivalent shapes have the same Euler characteristic.

Purely combinatorially, we might also note that in a blob complex, not only is every sphere filled by a ball, but in fact, every chain whose boundary is zero, in the sense of reduced homology, is a linear combination of spheres and thus filled by a linear combination of balls. Thus, all cycles are exact; all homology groups are trivial. This is because we can first of all come up with a (weak) deformation retraction to a point, in the sense of picking some point P and some recursively defined assignment to all points, closed edges, closed faces, etc, Q of a ball r(Q) [possibly made of many sub-components] whose outer sphere consists of north pole P, south pole Q, and r(the outer sphere of Q), oriented suitably. [TODO: Describe this better]. From this, we get that r(any chain) is a chain whose boundary is said chain + r(boundary of said chain), so that r(any cycle) has as boundary said cycle. [Note here that it is important to keep in mind that we are using reduced homology at the lowest level; r(a point) will be an edge from that point to P, whose boundary will include P, thought of as r(the unique -1 cell).]

The fact that all homology groups are trivial gives us that Euler characteristic is 1. The key fact is that the alternating sum … + 0-cells - 1-cells + 2-cells - … is equivalent to the alternating sum … + B_0 - B_1 + B_2 - …, where B_n = the n-th Betti number = rank of the n-th homology group = rank(n cycles modulo boundaries of n + 1 chains) = rank(n cycles) - rank(n + 1 chains) + rank(n + 1 cycles) = rank(n cycles) - (n + 1)-cells + rank(n + 1 cycles). In the alternating sum, the first and third term here cancel each other out, leaving just the middle term, … + 0-cells - 1-cells + 2-cells - ….. In the conventional Euler characteristic, we take all points to have boundary zero. [We can also use reduced homology and introduce a single (-1)-cell, reducing Euler characteristic by one, corresponding to reducing B_0 by one or if there are no points instead increasing B_{-1} from zero to one.]. (This is another argument, the traditional one, that Euler characteristic in general is a topological invariant, since it depends only on homology groups, and indeed only on Betti numbers.)

N.B.: The individual Betti numbers can depend on what field or more generally Abelian group we use for coefficients in our homology, but the alternating sum gives the same Euler characteristic regardless. And in a blob, the Betti numbers will all be zero regardless, except perhaps at B_0 and B_{-1} as noted above. Actually, I believe that, when using a field, all that will matter for the Betti number (computed in a dimension over the field sense, rather than rank as an Abelian group simpliciter), is its characteristic.

Note that more generally, in the series -1 + 0-cells - 1-cells + 2-cells - .., evaluated on a blob complex, we find that the partial sums alternate between \(\leq 0\) and \(\geq 0\), as these partial sums correspond to negated-or-unnegated ranks of the cycles of each dimension. [TODO: Expand on this]. The fact that the sum overall comes to zero can be seen as a consequence of how the series eventually hits only zero terms, and yet the partial sums still must continue alternating in sign.

[TODO: Prove that a blob can always be reduced to a point by iteratively finding a cell of highest dimension, and some cell of one dimension less appearing only once on its boundary, and cancelling the two against each other to yield a smaller blob]

[TODO: See also see https://www.ics.uci.edu/~eppstein/junkyard/euler/euler.html for an unusual proof. Does this generalize to higher dimensions? Perhaps not (this invokes the Eulerian graph fact about how every cycle in the homology sense (zero boundary) is a combination of simple cycles in the graph-theoretic sense (subspaces homeomorphic to a circle; loops without repetition), which I think doesn’t generalize to higher dimensions). If it does, though, extract and present the core simple idea.]