Computing the Euler characteristic of a manifold with boundary

Abstract:

Given a stable map, f, from an even-dimensional, compact manifold with boundary, M, into ${\mathbb {R}^2}$, the Euler characteristic of $M,\chi (M)$, is expressed in terms of projectivized rotation numbers of f restricted to $S(f)$, the singular curves of f, and of f restricted to $S(f|\partial M)$, the singular curves of f restricted to the boundary of M.