Diffgeometer1 on Nostr: You’re right! We can just lift the equator vertically (say to 5 degrees latitude) ...

You’re right! We can just lift the equator vertically (say to 5 degrees latitude) and then there is no intersection. My understanding of intersection theory is very limited so I thought the homotoped circle would still have to intersect the original circle just transversally. So going forward, homotoping to no intersection (if possible) is also valid in the calculation abs immediately yields zero intersection. This must be the case also in the way the Poincare-Hopf index theorem is presented. If we have a non vanishing vector field \(X\) on a closed oriented manifold, the theorem gives the Euler characteristic to be zero since the vector field has no zeros let alone isolated ones and the sum of the indices of \(X\) is simply empty (there’s nothing to sum over) and that sum of indices is assigned a default value of zero.