On the cokernel of the Thom morphism for compact Lie groups

TL;DR

This paper thoroughly analyzes the failure of the Thom morphism's surjectivity for compact Lie groups, revealing algebraic structures and constructing explicit examples of non-trivial kernel elements in differential cobordism.

Contribution

It provides a complete description of the Thom morphism's failure for compact Lie groups and constructs explicit non-trivial kernel elements using geometric and algebraic methods.

Findings

Identifies conditions for non-surjectivity of the Thom morphism.

Constructs explicit non-trivial elements in the kernel of differential Thom morphism.

Analyzes the interplay of torsion and non-torsion in cohomology and cobordism rings.

Abstract

We give a complete description of the potential failure of the surjectivity of the Thom morphism from complex cobordism to integral cohomology for compact Lie groups via a detailed study of the Atiyah-Hirzebruch spectral sequence and the action of the Steenrod algebra. We show how the failure of the surjectivity of the topological Thom morphism can be used to find examples of non-trivial elements in the kernel of the induced differential Thom morphism from differential cobordism of Hopkins and Singer to differential cohomology. These arguments are based on the particular algebraic structure and interplay of the torsion and non-torsion parts of the cohomology and cobordism rings of a given compact Lie group. We then use the geometry of special orthogonal groups to construct concrete cobordism classes in the non-trivial part of the kernel of the differential Thom morphism.