Non-null framed bordant simple Lie groups

TL;DR

This paper revisits the classification of compact simple Lie groups with left invariant framings that are non-null framed bordant, offering a new verification approach to existing results using a refined product formula.

Contribution

It introduces a more ingenious verification formula for non-null framed bordantness, enhancing the understanding of the bordism classes of simple Lie groups.

Findings

Verification formula for non-null framed bordantness

Application to known classification results

Refined use of the decomposition formula

Abstract

Let $G$ be a compact simple Lie group equipped with the left invariant framing $L$. It is known that there are several groups $G$ such that $(G, L)$ is non-null framed bordant. Previously we gave an alternative proof of these results using the decomposition formula of its bordism class into a Kronecker product by E. Ossa. In this note we propose a verification formula by reconsidering it, through a little more ingenious in the use of this product formula, and try to apply it to the non-null bordantness results above.