Dalian notes on rational Pontryagin classes

TL;DR

This paper discusses the behavior of rational Pontryagin classes in fiber bundles with 2n-dimensional Euclidean fibers, highlighting their potential to be nonzero in unexpectedly high cohomology degrees, contrasting with vector bundle cases.

Contribution

It reveals that rational Pontryagin classes can be nonzero in high cohomology degrees for certain fiber bundles, unlike their behavior in vector bundles.

Findings

Pontryagin classes can be nonzero in high cohomology degrees

Contrast with Pontryagin classes of vector bundles

Highlights differences in topological properties

Abstract

The rational Pontryagin classes, evaluated on fiber bundles where the fiber is a 2n-dimensional euclidean space, can be nonzero in cohomology dimensions much greater than 4n. This makes a striking contrast with the Pontryagin classes of vector bundles.