Relative Characteristic Classes

Abstract

In the proof, we shall make use of a not quite classical form of Whitney duality, involving Stiefel-Whitney characteristic classes which have to be considered as relative cohomology classes. Since these slightly generalized characteristic classes may have some interest for themselves, the present paper is divided into two parts as follows. In Part I, an attempt is made to give a systematic treatment of relative characteristic classes. Beside Stiefel-Whitney classes, relative Chern and Pontryagin characteristic classes will also be considered. It will be seen that most of the properties of the usual characteristic classes may be adapted to hold for the relative classes. In particular, the relative classes satisfy a generalized Whitney duality theorem and Wu's theorem [16] remains true if suitably stated. The fact that Wu's theorem may be extended to the case of a manifold with boundary was communicated to me by R. Thom and was the starting point of the proof of Lemma (1. 2). According to R. Thom, this extension of Wu's theorem was first known to H. Cartan, who proved it using (4D)-cohomology (unpublished). For our purpose, it will be sufficient to reduce (by Lemma (6. 1)) the extended Wu's theorem to the ordinary one, thus avoiding ('1)-cohomology. The proof of the generalized Whitney duality wvill be based on the interpretation of the relative characteristic classes as symmetric functions. The original author's proof was very cumbersome and will be omitted. The proof given here is due to A. Borel and is reproduced with his permission.