For a compact oriented smooth surface immersed in Euclidean four-space (thought of as complex two-space), the sum of the tangential and normal Euler numbers is equal to the algebraic number of points where the tangent plane is a complex line. This follows from the construction of an explicit homology between the zero-chains of complex points and the zero-chains of singular points of projections to lines and hyperplanes representing the tangential and normal Euler classes.
Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-spaceMathematics and Computer Science
PublisherMathematical Sciences Publishers
Citation InformationBanchoff, Thomas, and Frank Farris. "Tangential and Normal Euler Numbers, Complex Points, and Singularities of Projections for Oriented Surfaces in Four-space." Pacific Journal of Mathematics 161.1 (1993): 1-24