Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology (Q2884398)

Inspired by the Khovanov homology setting, the paper presents a general notion of bundle for coloured posets to which the authors associate independently a total coloured poset and a bicomplex. Under a technical assumption on the base of the bundle, the spectral sequence associated to the bicomplex is proven to converge from the homology of the base coloured by the homologies of its fibres to the homology of the total coloured poset.NEWLINENEWLINEA coloured poset (col-poset) is a partially ordered set \(P\) (with a unique maximal element) seen as a category, together with a covariant functor \(\mathcal{F}\) to the category of modules over a ring \(R\), this means there are \(R\)--modules \(\mathcal{F}(x)\) associated to every \(x\in P\) as well as coherent maps \(\mathcal{F}(x_1)\to\mathcal{F}(x_2)\) whenever \(x_1\leq x_2\). In [J. Algebra 322, No. 2, 429--448 (2009; Zbl 1229.05293)], the authors associated a chain complex to such a col-poset. Roughly speaking, a generator in degree \(k\in\mathbb{N}^*\) is an element \(\lambda\) of some \(\mathcal{F}(x_1)\) together with an ``history'' of length \(k\), that is an ordered sequence \(x_1\leq\cdots\leq x_k\) so that \(\lambda\) is simultaneously understood as its images in each \(\mathcal{F}(x_i)\). The differential is a signed sum over all the possibilities to forget one step of the ``history''.NEWLINENEWLINEIn this paper, the authors define a notion of bundle for col-posets, that is a covariant functor from a col-poset \(B\), called base, to the category of col-posets. Above each element of \(B\), there is hence a fibre which is a col-poset, and the set of the fibres inherits itself a partial order from the one of \(B\). To such a bundle, they associate a total col-poset defined as the union of all posets over \(B\), ordered, whenever two elements are from comparable fibres, by the order induced on images in the greatest fibre. Independently, the authors associate a bicomplex whose generators are elements in one of the \(R\)--modules which color elements of the fibres, together with a ``bi-history'', that is an ``history'' in the associated fibre, and an ``history'' of the whole fibre seen as an element of \(B\). Up to sign, the differentials are, in one direction, the differential within the fibre and, in the other direction, the differential within the base. Usual homological algebra associates then a total homology which is the homology of the chain complex obtained as the diagonal flattening of the bicomplex, and a spectral sequence which starts at the bicomplex, whose second page is the homology of the base coloured by the homologies of its fibres and which converges to the total homology.NEWLINENEWLINEThe main question adressed in the paper is whether, for a given bundle of col-posets, totality and homology commute, that is whether the total homology and the homology of the total col-poset coincide. It is answered positively under the assumption that the base is specially admissible. Whether the result holds without this assumption remains open.NEWLINENEWLINEThe paper contains many examples and is organized as follows. The first section presents bundles of col-posets and their associated total col-posets. The second section defines the bicomplex associated to a bundle of col-posets and states the spectral sequence which starts to it and converges to the homology of its diagonal flattening. Section 3 introduces admissible posets, which are posets with a unique maximal element and an element \(x_0\) just below it such that, for every element \(y\) below \(x_0\), the subposet of elements above \(y\) which are not below \(x_0\) has a unique minimal element; and specially admissible posets which is a recursive notion of admissible posets in the sense that the subposet of elements below \(x_0\) and its complementary are also both admissible. In section 4 and for a bundle of col-posets whose base is admissible (courtesy of \(x_0\)), the total homology and the homology of the total col-poset are shown to satisfy similar long exact sequences involving the whole bundle \(\xi\), its restriction \(\xi(x_0)\) to elements below \(x_0\) and its complementary \(\overline{\xi(x_0)}\). The proof starts with the short exact sequences associated to the inclusion of \(\xi(x_0)\) inside \(\xi\), and the (either kind of) chain complex for the associated quotient is shown, via an explicit chain map and under the admissibility assumption, to be quasi-isormorphic to the one of \(\overline{\xi(x_0)}\). The main theorem -- that is the homology of the total col-poset coincide with the total homology, so that the spectral sequence associated to the bicomplex converges to it -- is proven in section 5 by induction, using the above two long exact sequences and the 5-lemma. Most of the section is devoted to the well-definiteness of the explicit chain map given between the two long exact sequences. The last section is an application to Khovanov homology, stating that one can consider a link diagram, fix a subset of its crossings and put the Khovanov chain complexes of each resolution of the other crossings at the vertices of a boolean lattice whose edges correspond to Khovanov saddle maps. Since Khovanov's construction can be interpreted within the col-poset setting, this defines a bundle of col-posets and the associated spectral sequence converges from the hypercube of Khovanov homologies of partial resolutions to the Khovanov homology of the entire link. This provides, for instance, an alternative proof of Khovanov homology invariance under the Reidemeister I move.