Can someone explain the difference (and hence the sense of studying) between the usual and K-theoretical Euler class? Most famous and for now the only difference is showed during computations of tensor product of two line bundles.
Namely $e^K(L_1\otimes L_2)\neq e^H(L_1\otimes L_2)$, where $e^K$ and $e^H$ are K-theory and cohomology Euler classes.
Any intuition or maybe an article?