Pontrjagin Classes, the Fundamental Group and some Problems of Stable Algebra¹

Abstract

This paper deals mainly with problems connected with topological and homotopic invariance of Pontrjagin classes and some closely related problems of algebraic topology and stable algebra. These questions have arisen from the authors’ papers on topological and homotopic invariance of rational Pontrjagin classes, based on the discovery of deep connections between characteristic classes and the fundamental group. There are a number of new stable algebraic problems connected with the diffeomorphism problem and Pontrjagin classes of nonsimply connected manifold (especially when π₁= Z × • • • × Z).

This paper had been written originally for a lecture at the Moscow International Congress (1966), but the subject of my ½ hour lecture was, in fact, changed (to cobordism theory). I include here some additional remarks on the latest developments of these ideas.