Foams with flat connections and algebraic K-theory

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 Rⁿ⁺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.