Talk:Novikov conjecture

In the statements "Only one expression from the rational Pontryagin classes is homotopy invariant for simply connected manifolds of dimension 4k" and (later) "There are no other homotopy invariant expressions", the intention is of course to restrict to expressions of the form <f, [M]> and only to classify such expressions up to a scalar factor, as clearly multiplying by a scalar doesn't make any essential difference.

Prof. Novikov mentions that formulation of his conjecture arose out of discussions with A. Borel. It is perhaps worth mentioning that there is a related conjecture of Borel, which in turn was motivated by the Mostow rigidity theorem, that any two closed aspherical (i.e., having contractible universal covers) manifolds which are homotopy equivalent (or equivalently, have isomorphic fundamental groups) are in fact homeomorphic. This implies the Novikov conjecture for that particular fundamental group but is much stronger.

Prof. Novikov discusses the idea behind his proof of the topological invariance of rational Pontryagin classes. Readers can see an alternative proof of this theorem by M. Gromov in the long paper "Positive curvature, macroscopic dimension, spectral gaps and higher signatures", in Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), 1-213, Progr. Math., 132, Birkhäuser Boston, Boston, MA, 1996.