Was homology influenced by Euler's polyhedron formula?

Question

First of all, Alama - Formal proofs and refutations contains information about Euler's Polyhedron Formula. If you look at pp. 47–49, you will see that Poincaré proved Euler's Polyhedron Formula, and the following information appears there.

The property of being a homology sphere is the property that $B_k = Z_k$, that the $k$-circuits are the bounding $k$-chains. The inclusion $B_k \subseteq Z_k$ says that $\partial_{k+1}\partial_k \equiv 0$.

From this, we can see that Poincaré's proof of Euler's Polyhedron Formula has some relation to the homology sphere.

And also if you read Lamb - A few of my favorite proofs: The Poincaré homology sphere,

Early topologists wanted to try to find ways of distinguishing spaces by finding invariants: numbers or other mathematical objects that could be assigned to each space. Ideally, two spaces with the same invariant would be the same space, and two spaces with different invariants would be different spaces. Poincaré came up with Betti numbers, which are informally a way to catalogue the holes of different dimensions in a space, and torsion coefficients, which sort of keep track of twistedness. In a 1900 paper, Poincaré conjectured that these Betti numbers and torsion coefficients (also known today as homology) could tell you whether or not a space was a sphere.

When Poincaré thought about homology, we can see that he was also thinking about spheres.

Of course, the book Euler's Gem p.254 also explains that homology originated from Riemann's connectivity and Betti's higher-dimensional generalizations, but while reading the papers and articles mentioned above, I wondered if Euler's Polyhedron Formula might have also contributed to the emergence of homology.

Answer 1

The topological proof of the Euler polyhedron formula is a special case of the general proof that the alternating sum of Betti numbers equals the alternating sum of numbers of simplices, which follows from the Hopf trace formula proved after 1925, after Hopf had learned from Noether to consider homology as abelian groups, while Poincaré was considering Betti numbers and torsion coefficients just as numerical invariants, not as algebraic objects. I guess that it is hard to avoid thinking of chains and homology as abelian groups (or better vector spaces) if one wants to prove the Hopf trace formula, so it is not clear to me how Poincaré might have gotten at a topological proof of the Euler polyhedron formula. Also topological invariance of Betti numbers was not known at the time. (But perhaps there is a simpler argument just for the 2-sphere?)

EDIT (after looking at Hopf’s paper) In Hopf’s paper (Göttingen 1928) he calls this formula the Euler-Poincaré formula, so apparently he is attributing it to Poincaré. The paper actually gives a proof of the more general Lefschetz fixed point formula. His acknowledgment in the introduction translates as follows: “My original proof of this generalization of the Euler-Poincaré formula could be made much more transparent and simple in the course of a lecture that I gave in summer 1928 in Göttingen by using group-theoretical notions under the influence of Miss E. Noether. In the following I present this changed proof.”