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Article
Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space
Mathematics and Computer Science
  • Thomas Banchoff
  • Frank A. Farris, Santa Clara University
Document Type
Article
Publication Date
11-1-1993
Publisher
Mathematical Sciences Publishers
Disciplines
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.
Comments
Reprinted with permission by Mathematical Sciences Publishers. http://doi.org/10.2140/pjm.1993.161.1
Citation Information
Banchoff, 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