Title: 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.