Appears in collection : Riemannian Geometry Past, Present and Future: an homage to Marcel Berger

Filling a curve with an oriented surface can sometimes be "cheaper by the dozen".For example, L. C. Young constructed a smooth curve drawn on a projective plane in R^n which is only about 1.5 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. This phenomenon is based on the nonorientability of the projective plane; we will define a new invariant that quantifies the nonorientability of a manifold in R^n and use it to answer L. C. Young's question.