I read Poincaré's paper called Analysis Situs. And here's the thing about chain complex.
(Page 104 in this file)
That being given, let ${ε}^q_{i,j}$ be a number which is equal to zero if ${a}^{q−1}_j$ is not in the boundary of ${a}^q_i$, $+1$ if ${a}^{q−1}_j$ is part of the boundary of ${a}^q_i$ and in a direct relation to ${a}^q_i$ and finally $−$1 if ${a}^{q−1}_j$ is a part of the boundary of ${a}^q_i$ but in an inverse relation to ${a}^q_i$. We agree to write the congruence $$\tag{3} {a}^q_i = \sum {ε}^q_{i,j},{a}^{q−1}_j$$ to express the boundary of the ${a}^q_i$. The set of congruences (3) relative to the different $v_p, v_{p−1},\cdots, v_0$ of $V$ constitute what may be called the schema of a polyhedron. Two questions may be posed:
- Given a schema, does there always exist a corresponding polyhedron?
- If two polyhedra have the same schema, are they homeomorphic?
Without dealing with these two questions for the moment, we seek some conditions a schema must satisfy in order to correspond to a polyhedron. Consider one of the $v_{p−1}$, ${a}^{p−1}_1$ for example; this manifold should separate two and only two of the $v_p$ from each other; so that among the numbers ${ε}^p_{i,1}$ we have one which is equal to $+1$ and one which is equal to $−1$, and all the others are zero. This is not all; consider any of the $v_q$, ${a}^q_i$ for example, and any of the $v_{q−2}$, $a^{q−2}_k$ for example. There are two possibilities: first, where ${a}^{q−2}_k$ does not belong to ${a}^q_i$, all the products $$\tag{4} {ε}^q_{i,j} {ε}^{q−1}_{j,k}$$ must be zero, for if $a^{q−1}_j$ does not belong to ${a}^q_i$ the first factor is zero; while if ${a}^{q−1}_j$ belongs to ${a}^q_i$ the manifold ${a}^{q−2}_k$ cannot belong to ${a}^{q−1}_j$ (otherwise it would belong to ${a}^q_i$, contrary to hypothesis) and the second factor must be zero.
I think this quote's ${\color{Red}{ε}}^q_{i,j} {\color{Red}{ε}}^{q−1}_{j,k} =0$ corresponds to ${\color{Red}∂{\color{Red}∂}}=0$. However, my question is, if the above idea is correct, I am curious about the questions and motivations that Poincaré thought of when he first thought of the formula that is the origin of $∂∂=0$. I would appreciate it if you could answer the motivation for $∂∂=0$ in the history of mathematics and Poincaré's questions that led to the emergence of $∂∂=0$.