Cohomology of Split Group Extensions and Characteristic Classes

TL;DR

This paper investigates the characteristic classes arising from split group extensions and their relations, focusing on how these classes behave in the Lyndon-Hochschild-Serre spectral sequence when the lattice decomposes.

Contribution

It establishes relations between characteristic classes of a lattice extension and those of its G-sublattice components, enhancing understanding of spectral sequence obstructions.

Findings

Characteristic classes serve as obstructions in spectral sequence differentials.

Relations are derived between classes of a lattice and its G-sublattices.

Decomposition of lattices influences the structure of characteristic classes.

Abstract

There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.