On the construction and topological invariance of the Pontryagin classes

TL;DR

This paper presents an alternative proof of Novikov's theorem on the topological invariance of rational Pontryagin classes for vector bundles, using sheaves and algebraic L-theory instead of transversality and torus tricks.

Contribution

It introduces a novel algebraic approach to proving the topological invariance of Pontryagin classes, avoiding traditional geometric techniques.

Findings

Constructs rational Pontryagin classes using sheaves and algebraic L-theory.

Provides an alternative proof of Novikov's theorem.

Shows topological invariance without transversality or torus tricks.

Abstract

We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber R^n. This amounts to an alternative proof of Novikov's theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided.