Spheres alternate Euler characteristic $0$ and $2$; orientable genus $g$ gives $2-2g$, while $\RP^2$ gives $1$ and the Klein bottle $0$.Suggest a slogan
The Euler characteristics of the following spaces are \[ \chi(S^n) = 1 + (-1)^n \quad (n\geq 0),\qquad \chi(\Sigma_g) = 2-2g \quad (g\geq 0),\qquad \chi(\RP^2) = 1,\qquad \chi(K) = 0, \] where \(\Sigma_g\) is the closed orientable surface of genus \(g\) and \(K\) is the Klein bottle [@Hat02]. In particular \(\chi(S^1) = 0\), \(\chi(S^2) = 2\), \(\chi(\Sigma_1) = 0\), \(\chi(\Sigma_2) = -2\) and \(\chi(\Sigma_3) = -4\).
What are the Euler characteristics of $S^n$, $\Sigma_g$, $\RP^2$ and the Klein bottle?