# Refined Euler characteristic

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.

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 $$t=-1$$.