Obstruction of $C_\infty$-algebra models and characteristic classes

TL;DR

This paper develops an obstruction-theoretical approach using simplicial methods to construct characteristic classes of fiber bundles, linking algebraic deformations to topological invariants like Euler and Morita-Miller-Mumford classes.

Contribution

It introduces a novel obstruction class for deformations of $C_ Infty$-algebra models and constructs characteristic maps, connecting algebraic deformations to geometric characteristic classes.

Findings

Derived the Euler class for sphere bundles.

Computed Morita-Miller-Mumford classes for higher genus fiber bundles.

Established a new algebraic framework for characteristic class construction.

Abstract

In this paper, we consider an obstruction-theoretical construction of characteristic classes of fiber bundles by simplicial method. We can get a certain obstruction class for a deformation of $C_\infty$-algebra models of fibers and a characteristic map from the exterior algebra of a vector space of derivations. Applying this construction for a surface bundle, we obtain the Euler class of a sphere bundle and the Morita-Miller-Mumford classes of a bundle with positive genus fiber.