Generalized Grassmann invariant-redrawn

TL;DR

This paper explores how diagrams called Peiffer diagrams represent elements of third homology groups and relates them to algebraic K-theory, introducing new visual methods and connections to quiver representations.

Contribution

It introduces a novel diagrammatic approach to represent elements of $H_3G$ and links these pictures to deformation classes in algebraic K-theory, with new methods for visualizing roots.

Findings

Pictures represent elements of $H_3G$ for any group G.

A specific element of order 16 in $K_3(bZ)$ is constructed.

New visual techniques relate pictures to quiver representations and root systems.

Abstract

This is my old unpublished paper called "The generalized Grassmann invariant". It shows how "pictures" also known as "Peiffer diagrams" represent elements of $H_3G$ for any group $G$ and shows that $K_3(\mathbb Z [G])$ is isomorphic to a group of deformation classes of pictures for the Steinberg group of $\mathbb Z[G]$. A picture representing an element of order $16$ in $K_3(\mathbb Z)\cong \mathbb Z_{48}$ is also constructed. In this updated version of the paper, we modify only the pictures and leave the text more or less unchanged. We also added an Appendix to explain the new pictures using representations of quivers and root systems of type $A_n$. Often, some roots are missing in the Morse pictures. We give two ideas to replace these roots. One uses "ghost handle slides" to obtain a standard picture. The second idea uses the (real) Cartan subalgebra $H$ to obtain a "relative" picture for a torsion class and adds "ghost modules" which are directly related to the generalized Grassmann invariant. Additions and changes are in blue except the pictures are black with colored ghosts.