Combinatorial relative algebraic $K$-theory

TL;DR

This paper proves that a combinatorial presentation of higher relative algebraic K-groups, proposed in 2016, is isomorphic to the classical definition, supporting the conjecture's validity.

Contribution

It establishes an isomorphism between the combinatorial and classical higher relative algebraic K-groups, confirming a conjecture from 2016.

Findings

The combinatorial presentation matches the classical higher relative algebraic K-groups.

Provides evidence supporting the conjecture's correctness.

Bridges combinatorial and algebraic approaches in K-theory.

Abstract

In a preprint released in 2016, Daniel Grayson introduces a conjectural presentation of the (higher) relative algebraic $K$-groups using purely combinatorial means. In this paper, we will show that this presentation is isomorphic to the classically defined higher relative algebraic $K$-groups, and provide some reasons to believe the conjecture posed in this 2016 preprint to be true.