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Given a finite CW complex $X$ we define its Euler characteristic to be $\sum(-1)^k c_k$ where $c_k$ is the number of $k$-cells of $X$. It is not hard to show that the Euler characteristic of a CW complex $X$ is independent of the CW structure given to $X$.But what about the following question: Suppose $X$ and $Y$ are finite CW-complexes of the same dimension such that they have the same Euler characteristic. Are there subdivisions of $X$ and $Y$ such that the number of $k$-cells of $X$ under its new subdivision equals the number of $k$-cells of $Y$ under its new subdivision?

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Imagine you have a filled circle (one 0 cell, one 1 cell, and one 2 cell), then draw a point inside the circle. Connect that point to the original 0 cell. Then draw a 1-cell attached to the new point. The original 2 cell is now split into two 2 cells. More generally, you can split a k cell by adding cells of dimension lower than k, and without affecting the number of cells of dimension higher than k. You can see the answer if you imagine applying the process repeatedly to whichever space has more k-cells, starting at the dimension of X and Y.

This can fail if X or Y lacks any cells in some dimension, but it isn't hard to modify the method to get around that.

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