Diffgeometer1 on Nostr: I have a related question on the self-intersection number. Suppose \(X\) is a closed ...

I have a related question on the self-intersection number. Suppose \(X\) is a closed oriented submanifold of a closed oriented manifold \(Y\) with \(2\dim X=\dim Y\). To compute the self-intersection number \(I(X,X)\), we homotope the inlcusion map \(\iota: X\hookrightarrow Y\) to a map \(f:X\rightarrow Y\) which is transversal to \(X\). The intersection \(X\cap f^{-1}(X)\) is now a finite number of points and \(I(X,X)\) is the sum of these points counted with sign, that is, if \(x\in X\cap f^{-1}(X)\), then we assign a \(+1\) if\[df_x(T_xX)\oplus T_xX=T_xY\]if the product orientation on the left agrees with the orientation on the right and \(-1\) otherwise. Is this right? As an example,consider \(S^1\subset S^2\) with \(S^1\) identified with the equator. To compute the self-intersection number of the circle \(S^1\), we homotope the inclusion \(\iota: S^1\hookrightarrow S^2\) to the map \(f: S^1\rightarrow S^2\) which rotates the equator \(S^1\) by \(90^\circ\) clockwise. Then \(S^1\cap f^{-1}(S^1)\) intersect at two antipodal points. One point has a \(+1\) intersection, the other point has a \(-1\) intersection. So the self-intersection \(I(S^1,S^1)=1+(-1)=0\), which makes sense since the self-intersection is suppose to coincide with the Euler characteristic. Is this right?