Foams with flat connections and algebraic K-theory

TL;DR

This paper establishes a novel link between algebraic K-theory and foam cobordisms, representing algebraic structures through stratified manifolds with singularities and exploring their cobordism groups.

Contribution

It introduces a new geometric interpretation of algebraic K-theory via decorated foam cobordisms and proposes extensions to exact categories and higher dimensions.

Findings

First K-theory group corresponds to cobordism group of decorated 1-foams in the plane.

Expected generalization of the relation to higher K-theory groups and n-foams in higher dimensions.

Potential for new K-theory variants through modifications of foam embeddings and conditions.

Abstract

This paper proposes a connection between algebraic K-theory and foam cobordisms, where foams are stratified manifolds with singularities of a prescribed form. We consider $n$-dimensional foams equipped with a flat bundle of finitely-generated projective $R$-modules over each facet of the foam, together with gluing conditions along the subfoam of singular points. In a suitable sense which will become clear, a vertex (or the smallest stratum) of an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of vertices. We show that the first K-theory group of a ring $R$ can be identified with the cobordism group of decorated 1-foams embedded in the plane. A similar relation between the $n$-th algebraic K-theory group of a ring $R$ and the cobordism group of decorated $n$-foams embedded in $\mathbb{R}^{n+1}$ is expected for $n>1$. An analogous correspondence is proposed for arbitrary exact categories. Modifying the embedding and other conditions on the foams may lead to new flavors of K-theory groups.