arXiv subjects
Publications and source records attributed to Nondas E. Kechagias.
A non-connected neither of finite type Hopf algebra $\mathcal{F}_{0}$ is defined over $\mathbb{Z}/ 2\mathbb{Z}$ and its hom dual turns out to be a tensor product of polynomial algebras. Certain quotient Hopf algebras include the Steenrod and Dyer-Lashof algebras. This setting provides a map between the Steenrod coalgebra and a direct limit of Dyer-Lashof coalgebras.
The notion of atomicity defined by Cohen, Moore and Neisendorfer is studied for the Dickson algebras. Not any ring of invariants respects this property. It depends on the property of the Dickson algebra that given any monomial $d$ there exists a sequence of Steenrod operations $P(Γ, d) $ such that $P(Γ, d) d$ becomes a $p$-th power of the top Dickson algebra generator.
In this work we compute the induced transfer map: $$\barτ^\ast: \func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast: H^\ast (Σ_{p^n}) \to H^\ast(V))$$ in $\func{mod}p$-cohomology. Here $Σ_{p^{n}}$ is the symmetric group acting on an $n$-dimensional $\mathbb F_p$ vector space $V$, $G=Σ_{p^{n},p}$ a $p$-Sylow subgroup, $Σ_{p^{n-1}}\int Σ_{p}$, or $Σ_{p}\int Σ_{p^{n-1}}$. Some answers are given by natural invariants which are related to certain parabolic subgroups. We also compute a free module basis for certain rings of invariants over the classical Dickson algebra. This provides a computation of the image of the appropriate restriction map. Finally, if $ ξ:\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast}: H^\ast(Σ_{p^n}) \to H^\ast(V)) $ is the natural epimorphism, then we prove that $\barτ^\ast=ξ$ in the ideal generated by the top Dickson algebra generator.
This work deals with Adem relations in the Dyer-Lashof algebra from a modular invariant point of view. The main result is to provide an algorithm which has two effects: Firstly, to calculate the hom-dual of an element in the Dyer-Lashof algebra; and secondly, to find the image of a non-admissible element after applying Adem relations. The advantage of our method is that one has to deal with polynomials instead of homology operations. A moderate explanation of the complexity of Adem relations is given.