# Why do we only consider Euclidean simplicial complex?

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?

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.