Algebraic Topology · Problem 2.2.20

The Euler characteristic is multiplicative because the product of two CW complexes has a natural cell structure: each cell in $X \times Y$ is a product of a cell from $X$ and a cell from $Y$. The key insight is that if $X$ has $c_p$ cells of dimension $p$ and $Y$ has $d_q$ cells of dimension $q$, then $X \times Y$ has $c_p d_q$ cells of dimension $p+q$. Reorganizing this double sum by total dimension and factoring reveals that $\chi(X \times Y)$ splits into a product of the individual Euler characteristics.

The Euler characteristic is multiplicative because the product of two CW complexes has a natural cell structure: each cell in $X \times Y$ is a product of a cell from $X$ and a cell from $Y$. The key insight is that if $X$ has $c_p$ cells of dimension $p$ and $Y$ has $d_q$ cells of dimension $q$, then $X \times Y$ has $c_p d_q$ cells of dimension $p+q$. Reorganizing this double sum by total dimension and factoring reveals that $\chi(X \times Y)$ splits into a product of the individual Euler characteristics.