A note on the Thom morphism for the classifying space of certain Lie groups and gauge groups

TL;DR

This paper characterizes the non-torsion generators outside the image of the Thom morphism for classifying spaces of certain Lie groups, and demonstrates non-surjectivity of the Thom morphism for gauge groups of E_7-bundles.

Contribution

It provides a complete description of the Thom morphism's image for exceptional Lie groups (except E_8) and shows its non-surjectivity for specific gauge groups, advancing understanding of cobordism and cohomology.

Findings

Identifies non-torsion generators not in the Thom morphism's image for exceptional Lie groups (excluding E_8)

Proves the Thom morphism is not surjective for the gauge group of an E_7-bundle over S^4

Detects nontrivial elements in the kernel of the reduced Thom morphism for Lie groups

Abstract

We give a complete description of which non-torsion generators are not in the image of the Thom morphism from complex cobordism to integral cohomology for the classifying space of exceptional Lie groups except for E_8. We then show that the Thom morphism is not surjective for the classifying space of the gauge group of a principal E_7-bundle over the four-dimensional sphere. We use the results to detect nontrivial elements in the kernel of the reduced Thom morphism for Lie groups and their classifying spaces.