Vol. 161, No. 1, 1993

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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
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
Authors
Thomas Francis Banchoff
Frank Albert Farris