Self-intersection formula in complex cobordism ($i^{\ast }{i}_{!}(a)=a{e}_{MU}({\nu }_i)$)

Restricting the tubular-neighborhood collapse of an embedding back to its source is the normal bundle's zero section. The Thom isomorphism in complex cobordism therefore makes pullback after the Gysin map multiplication by the Euler class in a generalized cohomology theory. For an actual complex normal bundle this is its top Chern class in complex cobordism.