The normal Euler class and singularities of projections for polyhedral surfaces in 4-space

This paper defines the Normal Euler Number and the Normal Euler Class for polyhedral surfaces in 4-space by means of singularities of projections into hyperplanes. There exist polyhedral analogues of nearly all of Whitneyʹs theorems on Normal Euler Classes of surfaces smoothly immersed in 4-space. However, although the Normal Euler Number of a smooth embedding of the real projective plane in 4-space must be plus or minus 2, the Normal Euler Number of a (locally knotted) polyhedral embedding of the real projective plane can be any integer congruent to 2 modulo 4.