Minimal algebraic complexes over $D_{4n}$

TL;DR

This paper proves that all dihedral groups of order 4n satisfy the D(2) property by demonstrating cancellation of free modules in a specific stable class, completing a key step in understanding their algebraic topology.

Contribution

It establishes the cancellation property of free modules over dihedral groups of order 4n in the stable class, advancing the understanding of their algebraic K-theory.

Findings

Cancellation of free modules holds in ()() over dihedral groups of order 4n.

Completes the proof that all dihedral groups satisfy the D(2) property.

Supports recent results on realizing k-invariants for these groups.

Abstract

We show that cancellation of free modules holds in the stable class $\Omega_3(\mathbb{Z})$ over dihedral groups of order $4n$. In light of a recent result on realizing $k$-invariants for these groups, this completes the proof that all all dihedral groups satisfy the D(2) property.