Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8)

TL;DR

This paper constructs geometric maps to demonstrate how certain cohomology classes for Spin(7) and Spin(8) Lie groups relate to complex cobordism, revealing nontrivial kernel elements linked to octonion symmetries.

Contribution

It provides a geometric construction showing the third cohomology generator is not in the image of the Thom morphism, highlighting new insights into the cohomology of Spin groups.

Findings

The eight-fold of the generator is in the image of the Thom morphism.

The generator itself is not in the image, indicating a nontrivial kernel element.

The construction uses octonion symmetries to achieve these results.

Abstract

By constructing concrete complex-oriented maps we show that the eight-fold of the generator of the third integral cohomology of the spin groups Spin(7) and Spin(8) is in the image of the Thom morphism from complex cobordism to singular cohomology, while the generator itself is not in the image. We thereby give a geometric construction for a nontrivial class in the kernel of the differential Thom morphism of Hopkins and Singer for the Lie groups Spin(7) and Spin(8). The construction exploits the special symmetries of the octonions.