Is there a refinement of Euler characteristic that distinguishes between the torus $S^1 \times S^1$ and the cylinder $S^1 \times [0,1]$?

(The intuition here is that $\chi$ is multiplicative, so that $\chi(S^1 \times S^1) = \chi(S^1) \times \chi(S^1) = 0 \times 0$, which “vanishes twice”.)

Relatedly, is there a setting in which the Euler characteristic emerges as an eigenvalue, or more broadly as a root of an equation? This would provide a sense in which 0 might occur with multiplicity greater than 1.