THE TRANSVERSE ORIENTATION CLASS OF CERTAIN SINGULAR FOLIATIONS

In the 1980s, Connes defined the transverse orientation class of a transversely oriented foliation and showed it is a non-torsion element in K-theory. We will examine a special family of singular foliations for which a transverse orientation class can also be defined. Naturally, the holonomy groupoid of a singular foliation (studied by Androulidakis and Skandalis, Debord and others) will play an important role. The presence of "continuous holonomy" will bring the Connes-Thom isomorphism and its counterpart in cyclic cohomology (Elliott-Natsume-Nest) into the picture as well.