Bundles of coloured posets and a Leray-Serre spectral sequence

A coloured poset is a poset with a unique maximal element that is equipped with a sheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a LeraySerre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link. Introduction Our motivation for studying coloured posets comes from a desire to understand better some of the structures arising in the Khovanov homology of links. One can ask if Khovanov’s “cube” construction can be placed in a broader context, hoping that this new perspective allows one to lever some advantage. In [2] we showed that this is possible using the framework of what we call coloured posets. This is something of a half-way house between the specifics of “cubes” and the full generality of sheaves of modules over small categories. We believe this is an appropriate place to study Khovanov homology as it provides enough generality to be useful, without losing sight of the starting point. The idea being that one can then bring the algebraic topology of coloured posets into the game. One direction of interest is a bundle theory for coloured posets which we initiate in this paper. Given a poset B with unique maximal element we define bundles of coloured posets over B. Roughly, we wish to capture the idea that a coloured poset may be decomposed as a family of coloured posets parametrized by another poset, much as a fibre bundle is a family of spaces parametrized by some base space. A bundle of coloured posets has a total coloured poset (E,F), and our central interest is in computing its homology. Our main result in this direction, proved as Theorem 1 in §5, is a Leray-Serre style spectral sequence for bundles where the base has a certain technical property that we call special admissibility: Main Theorem. Let ξ : B → CPR be a bundle of coloured posets with B finite and specially admissible, and let (E,F) be the associated total coloured poset. Then there is a spectral sequence E p,q = Hp(B,H fib q (ξ)) =⇒ H∗(E,F). It turns out that a number of naturally occurring posets are specially admissible, and importantly for applications to knot homology theories, this class includes the Boolean lattices. Given a knot diagram D together with k distinguished crossings c1, . . . , ck, one can obtain a new complex in the spirit of Khovanov: resolving each of the remaining l crossings gives another link diagram, and the l diagrams that result can be placed as the vertices of a hyper-cube (a Boolean lattice). Taking the unnormalised Khovanov homology of these diagrams and using induced maps along edges one obtains, by the construction of Khovanov, a triply graded complex whose homology we denote KH∗,∗,∗(D; c1, . . . , ck). BRENT EVERITT: Department of Mathematics, University of York, York YO10 5DD, United Kingdom. e-mail: [email protected]. PAUL TURNER: Maxwell Institute for Mathematical Sciences, Edinburgh EH14 4AS, United Kingdom and Département de mathématiques, Université de Fribourg, CH-1700 Fribourg, Switzerland. e-mail: [email protected]. ⋆ The first author thanks Finnur Larusson for many useful and stimulating discussions. He is also grateful to the Alpine Mathematical Institute, Haute-Savoie, France, and to the Institute for Geometry and its Applications, University of Adelaide, Australia. The second author thanks the University of Fribourg, Switzerland for their support. 2 Brent Everitt and Paul Turner Main Application. Let D be a link diagram and let c1, . . . , ck be a subset of the crossings of D. For each i, there is a spectral sequence E p,q = KHp,q,i(D; c1, . . . , ck) =⇒ KHp+q,i(D). The r-th differential in the spectral sequence has bidegree (−r, r − 1). The paper unfolds as follows. In the next section we define bundles of coloured posets, describe the associated total coloured poset, and give a number of examples. We also recollect facts from [2] about coloured posets and their homology. In §2 we start with a bundle of coloured posets and introduce a certain bi-complex which (by standard methods) gives rise to a spectral sequence converging to the homology of the total complex. Our task later will be to recognize this as the homology of the total coloured poset of the bundle we started with. In fact we can only do this for a restricted class of bases, the specially admissible ones, and we introduce these in §3. We give a number of examples, including the Boolean lattices. Following this, §4 introduces the technical tools needed to prove the main theorem, namely certain long exact sequences. In §5 we state and prove the main theorem. Finally in §6 we turn to Khovanov homology and construct a new spectral sequence which for a given link has a potentially computable E2-page, and which converges to the Khovanov homology of the link. 1. Bundles of coloured posets Recall from [2] that a coloured poset is a pair (P,F) consisting of a poset P having a unique maximal element 1P , and a covariant functor F : P → ModR, called the colouring. This may be regarded as a representation of P or as a sheaf of modules over P according to taste. A morphism of coloured posets (P1,F1) → (P2,F2) is a pair (f, τ) where f : P1 → P2 is a map of posets, and τ = {τx}x∈P1 is a collection of R-module homomorphisms τx : F1(x) → F2(f(x)). This pair must satisfy (i) f(x) = 1P2 if and only if x = 1P1 , and (ii) for all x ≤ y in P1, the following diagram commutes P1 P2