Narrow Results

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E₂-pages can be described in terms of groups arising from the action of a certain endomorphism on 𝔽₂-Khovanov homology. Some simple consequences are discussed.

In this note, we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author's work on Morse–Bott homology with A. Banyaga [A. Banyaga and D. E. Hurtubise, Morse–Bott homology, Trans. Amer. Math. Soc. (to appear), arXiv:math/0612316v2].

Suppose the ground field $𝔽$ is an algebraically closed field of characteristic zero. By means of spectral sequences, the computation of the first cohomology group of the model filiform Lie superalgebra $L_{n,m}$ with coefficients in the adjoint module is reduced to the computation of the first cohomology group of an Abel ideal and a one-dimensional subalgebra. Then, by calculating the outer derivations, the algebra structure of the first cohomology group of $L_{n,m}$ is completely characterized.

We consider the category of comodules over a smash coproduct coalgebra $C>⊲H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C>⊲H$-colinear, $H$-colinear and rational $C$-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

To help study the double suspension when localised at a prime p, Selick filtered Ω²S²ⁿ⁺¹ by H-spaces which geometrically realise a natural Hopf algebra filtration of H_*(Ω²S²ⁿ⁺¹;ℤ/p). Later, Gray showed that the fiber W_n of E² has an integral classifying space BW_n and there is a homotopy fibration . In this paper we correspondingly filter BW_n in a manner compatible with Selick's filtration and the homotopy fibration , study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S²ⁿ⁺¹. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BW_n.