On the cobordisms of M\"obius circles

Abstract

The boundary of a M\"obius manifold carries a canonical M\"obius structure. This enables one to define the cobordism group of $n$-dimensional (closed) M\"obius manifolds. The purpose of this note is to show that the cobordism group of M\"obius circles is zero, i.e., every M\"obius circle bounds a M\"obius surface. We also complete N. Kuiper's classification of projective structures on $S^1$ (we show that there are in fact two series of projective circles with parabolic holonomy, and not one).