Simplicial Homotopy Theory, Link Homology and Khovanov Homology

Abstract

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diagrams to a category of simplicial objects such that up to looping or delooping, link diagrams related by Reidemeister moves will give rise to homotopy equivalent simplicial objects, and the homotopy groups of these objects will be equal to the link homology groups of the original link homology theory. The construction is independent of the particular link homology theory. A simplifying point in producing a homotopy simplicial object in relation to a chain complex occurs when the chain complex is itself derived (via face maps) from a simplicial object that satisfies the Kan extension condition. Under these circumstances one can use that simplicial object rather than apply the Dold-Kan functor to the chain complex. We will give examples of this situation in regard to Khovanov homology. We will investigate detailed working out of this correspondence in separate papers. The purpose of this note is to announce the basic relationships for using simplicial methods in this domain. Thus we do more than just quote the Dold-Kan Theorem. We give a review of simplicial theory and we point to specific constructions, particularly in relation to Khovanov homology, that can be used to make simplicial homotopy types directly.