The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-alternating sum of the ranks of the homology groups is an invariant. So, is there an intuitive reason why the Euler characteristic should be defined as an alternating sum instead of a non-alternating sum — aside from the fact that other theorems, such as Gauss-Bonnet, would break (or at least need to be re-worked)? Is this related to rank being additive? If so, then a historical question: what would have been the original motivation of Euler to consider an alternating sum?
Algebraic Topology – Intuitive Reason Why the Euler Characteristic is an Alternating Sum
algebraic-topologyhomology-cohomologyintuition
Best Answer
I don't know how intuitive this will be, but here is how I think of it. It is similar to what Travis mentioned in the comments. We need the Euler characteristic to not change when we change the triangulation. Consider the following triangulation of the plane.
This graph has 4 vertices, 5 edges, and 3 faces, which gives us the $4-5+3=2$ we expected. Now, if we were to pretend that we don't know how to compute the Euler characteristic, and just that we don't want it to change if we change the triangulation, we see that the new edge and the new vertex need to cancel each other out.
We also need this to work for the faces, so if we remove an edge, we will decrease the number of edges and decrease the number of faces. We still don't want it to change.
So, edges and faces also need to cancel each other out. The easiest way to get the Euler characteristic to remain invariant under these changes is to have the even dimensional count positively and the odd dimensions be negative. Like I said, this is how I think of it, hopefully it helps your intuition too.