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Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The $k$th and $n-k$th Betti numbers, $b_k$ and $b_{n-k}$ of a closed orientable n-manifold are equal. $$b_k = b_{n-k}.$$

From Wikipedia, it says: The cohomology concept was at that time about 40 years from being clarified. In Poincaré 1895 paper Analysis Situs, Poincaré tried to prove the theorem using topological intersection theory, which he had invented. Criticism of his work by Poul Heegaard led him to realize that his proof was seriously flawed. In the first two complements to Analysis Situs, Poincaré gave a new proof in terms of dual triangulations.

Poincaré duality did not take on its modern form until the advent of cohomology in the 1930s, when Eduard Čech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.

Questions:

  1. What was the flaw of Poincaré 1895 proof in Analysis Situs, in Heegaard view and in the modern view?

  2. The first two complements to Analysis Situs, Poincaré gave a new proof in terms of dual triangulations. What is the new proof about? Was the additional new proof correct, in Heegaard view and in the modern view?

  3. How is Poincaré's new proof different from the Čech and Whitney's the cup and cap products and cohomology take?

wonderich
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  • Dieudonne covers (1) in great detail in his history textbook. Perhaps the largest flaw in his initial approach was that to Poincare, what he called homology at that time is what we would call singular oriented bordism now. And that does not satisfy the kind of duality he was claiming. This is what caused him to switch to simplicial homology, and then his version of Poincare Duality was true. As Lee mentions, Poincare worked purely in homology, cohomology did not exist. – Ryan Budney Jul 20 '23 at 15:42
  • While people understood cohomology as a concept early-on, it wasn't until Cech and Whitney's cup product that people paid any attention to the idea. – Ryan Budney Jul 20 '23 at 15:42
  • Re (3) Poincare's proof used the same map as cap product with orientation class, but he did not have a cap product. i.e. the general formalism of intersection theory came later. – Ryan Budney Jul 20 '23 at 15:49

1 Answers1

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I have no idea about question 1.

But, regarding question 2, the new proof, which is still taught in topology textbooks, goes like this.

Let $M$ be an closed, connected, oriented $n$-manifold, which we assume to have a triangulation. In addition we assume that this triangulation has a dual cell decomposition as follows:

  • One $0$-cell for each $n$ dimensional simplex, the former at the center of the latter.
  • One $1$-cell for each $n-1$ dimensional simplex, crossing at each others center.
  • One $2$-cell for each $n-2$ dimensional simplex, crossing at each others center.

and so on until

  • One $n-1$ cell for each $1$-dimensional simplex, crossing at each others center.
  • One $n$-cell for each $0$-dimensional simplex, the latter at the center of the former.

Now consider the chain complexes of these two decomposition (I'll use $\mathbb R$-coefficients). Notice: the chain complex of the dual cell decomposition is the linear dual of the chain complex of the given triangulation (i.e. the dual in the sense of linear algebra). From this it follows that the $n-k^{\text{th}}$ homology of the dual cell decomposition is isomorphic to the $k^{\text{th}}$ homology of the given triangulation. But, homology of $M$ in any dimension is independent of the cell decomposition. Therefore, the $n-k^{\text{th}}$ homomology of $M$ is isomorphic to the $k^{\text{th}}$ homology of $M$.

Regarding question 3, the relation of this proof with later proofs is basically that Poincare's advances drove the entire field of algebraic topology. Here's a simplistic view of how this happened. Formalizing Poincare's proof led to the development of simplicial homology theory, and then of singular homology theory in order to make the proof of topological invariance of simplicial homology more transparent (and as a bonus, singular homology applies to many more spaces than simplicial homology). The theory of cohomology was developed in order to formalize concepts of linear duality. Cup and cap products come in when you want to formalize the "crossing at each others center" concept, in the context of manifolds (and, as a bonus, cup and cap products are defined in general, not just for manifolds). At all steps, the idea was to make develop tools in more and more and more generality and abstraction, for more and more powerful applications.

Lee Mosher
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