Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space

Thomas Francis Banchoff and Frank Albert Farris

Vol. 161 (1993), No. 1, 1–24

DOI: 10.2140/pjm.1993.161.1

Abstract

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.

Mathematical Subject Classification 2000

Primary: 57R20

Secondary: 57R42

Milestones

Received: 19 September 1991

Published: 1 November 1993