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I'm reading J.Dieudonné - The history of algebraic and differential topology

Here is what written in page 4-5

From this beginning, the evolution of homology went throught a series of steps, the description of which constitutes Part I of this book. We summarize them below, not necessarily in chronological order

$\textbf{I}$ Poincaré had already proved that a subdivision of every cell into smaller cells gives the same homology for the subdivided triangulations. This allows one to only consider euclidean simplicial complexes in which every cell is a rectilinear simplex contained in some $\mathbb{R}^N$ with large $N$.

$\textbf{II} \cdots$

I don't get what exactly the line $\textbf{I}$ means. Why does the fact that subdivision does not affect homology lets us to only consider euclidean simplicial complexes? What does cell mean here?

So say, we define homology using $n$-gons with fixed orientations. (There is a natural way to define this as we do to define the usual simplicial homology) Does $\textbf{I}$ mean that this homology and the usual simplicial homology induced by simplices coincide, and that Poincaré proved this?

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Right, exactly. On page 4, Dieudonné refers to "a triangulation $T$ consisting of cells homeomorphic to convex polyhedra". So Poincaré originally considered cell structures where the cells were arbitrary convex polyhedra, not just simplices. But you can subdivide such a cell structure to get one using only simplices, and Poincaré showed that this does not change the homology.

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