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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.

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Look up the Poincare polynomial $p_X(t)$. It is still multiplicative, by the Kunneth formula. The Euler characteristic is $\chi_X=p_X(-1)$. We have $p_{S^1\times S^1}(t)=(t+1)^2$ but $p_{S^1\times [0,1]}(t)=(t+1)$, with the former having a double root at $t=-1$.

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