On the Chern–Losik number of codimension 2 foliations

Yu. D. Efremenko

Novosibirsk State University, Novosibirsk, Russia

Abstract: We study the first Chern–Losik class of codimension 2 foliations on bundles over the circle whose structure group is a cyclic subgroup of the special linear group over the field of complex numbers. We introduce the notion of the Chern–Losik number for foliations having at least two hyperbolic leaves. It is shown that for matrices conjugate to diagonal matrices with distinct diagonal entries whose moduli are different from one, the Chern–Losik class of the corresponding foliation is nontrivial. In this case the value of the Chern–Losik number is uniquely determined by the diagonal entries. For matrices conjugate to diagonal matrices with distinct diagonal entries of modulus one, the Chern–Losik class is trivial. For matrices conjugate to the identity matrix, the Chern–Losik class is trivial, while for matrices conjugate to a Jordan block it is nontrivial.

Keywords: foliation, leaf space, characteristic classes of foliations, Gelfand–Fuchs cohomology, Chern–Losik class, dynamical systems, fixed points.

UDC: 514.7

MSC: 35R30

Received: 25.03.2025
Revised: 01.07.2025
Accepted: 15.07.2025

DOI: 10.33048/smzh.2025.66.506

 English version: Siberian Mathematical Journal, 2025, 66:5, 1158–1170