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1993 (Publication #73)

Tangential and Normal Euler Numbers, Complex Points, and Singularities of Projections for Oriented Surfaces in Four-Space (with Frank Farris)

Pacific Journal of Mathematics Vol. 161 No. 1. (1993) pp. 1-24.


This article gives a geometric proof of the following. Given a compact two-dimensional surface immersed smoothly in R^4 (or C^2), the sum of the Euler number of M, the Euler number of its normal bundle, and the algebraic number of isolated complex points is zero. Although this theorem has been examined and proven in previous papers, the techniques required do not illuminate the underlying geometry.