Lefschetz formulae for modest vector bundles

Abstract

A number of results are known which relate the characteristic numbers on a differentiable manifold X to the differentiable actions of the circle group S1on X. Many of these results are consequences of the Atiyah–Singer index theorem (2). In particular, an idea due to Hirzebruch sometimes permits one to deduce that certain characters which occur in the index theorem are in fact constant. Examples of this procedure can be found on page 594 of(2), on page 46 of(4), and on pages 21–22 of(1). Our aim in this paper is to make a systematic investigation of such results, and it is this aim which led us naturally to the concept of a modest vector bundle.