Let $X$ be submanifold inside $Y$. Mod $2$ self-intersection number $I_2(X,X)$ has some strict constrains:
Since $X$ and $X$ need to have complementary dimension, $\operatorname{dim}X = \frac{1}{2}\operatorname{dim}Y$.
$X$ is a compact submanifold
$X$ is a closed submanifold
Since $I_2(X,X) \triangleq I_2(i,X) \triangleq$ the number of points in $i^{-1}(X): X \hookrightarrow Y$ modulo 2, where $i$ is the inclusion - hence $i^{-1}(X)$ just identity in the self-intersection case. So how can I count the number?
When $X \pitchfork X$, I couldn't think of any other example except for Möbius band. Is there any other example?
Thank you very much!