Open Collections

Smooth manifolds with prescribed rational cohomology ring

Banff International Research Station for Mathematical Innovation and Discovery

Given a rational Poincare duality algebra A, is there a manifold M whose rational cohomology ring realizes A? The Hirzebruch signature formula provides an obstruction to the existence of such manifold. In the case of A=Q[x]/<x^3>, a realizing smooth manifold M^n (called a rational projective plane) could only exist in dimensions n=8(2^a+2^b). Sullivan's rational surgery realization theorem provides the necessary and sufficient condition to the existence; the problem boiled down to finding possible Pontryagin numbers satisfying a set of integrality conditions, which can be reduced to a Diophantine equation in our case. Similar techniques can be applied to study the realization of rational Octonionic projective spaces. This is joint work with Jim Fowler.

Mathematics; Manifolds and cell complexes; Algebraic topology

video/mp4

Author affiliation: Indiana University

2017-02-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60386

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/