arXiv subjects

Christian Nassau

Publications and source records attributed to Christian Nassau.

5 recordsLinked to original sources

arXiv2024

Formal groups over non-commutative rings

We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other. We show that this is the natural kind of formal group law for the needs of algebraic topology in the sense that a (possibly non-commutative) complex oriented ring spectrum is canonically equipped with just such a formal group law. The universal formal group law is carried by the Baker-Richter spectrum Mξ which plays a role analogous to MU in this non-commutative context. As suggested by previous work of Morava the Hopf algebra B of "formal diffeomorphisms of the non-commutative line" of Brouder, Frabetti and Krattenthaler is central to the theory developed here. In particular, we verify Morava's conjecture that there is a representation of the Drinfeld quantum-double D(B) through cohomology operations in Mξ.

math.AT↗

arXiv2020

Steenrod operations and A-module extensions

Explicit extensions representing cocycles $x \in Ext_{A}^{s,t}(F_2,F_2)$ are useful in calculating Steenrod operations $Sq^i : Ext_{A}^{s,t}(F_2,F_2) \longrightarrow Ext_{A}^{s+i,2t}(F_2,F_2)$ by a method devised by the second author. This can be used to identify explicit cocycles in the minimal resolutions produced by the first author's computer programs, and this information is useful in determining differentials in the Adams spectral sequence.

math.AT↗

arXiv2019

Computing a minimal resolution over the Steenrod algebra

We describe an algorithm that allows to compute a minimal resolution of the Steenrod algebra. The algorithm has built-in knowledge about vanishing lines for the cohomology of sub Hopf algebras of the Steenrod algebra which makes it both faster and more economical than the generic approach.

math.AT↗

arXiv2017

The Bruhat decomposition of real Grassmann manifolds

We study the Grassmann manifold $G_k$ of all $k$-dimensional subspaces of ${\mathbb R}^n$. The Cartan embedding $G_k\subset O(n)$ realizes $G_k$ as a subspace of $Sl_n({\mathbb R})$ and we study the decomposition $G_k=\coprod_w (BwB\cap G_k)$ inherited from the classical Bruhat decomposition. We prove that this defines a CW structure on $G_k$ and determine the incidence numbers between cells.

math.AG↗

arXiv2010

On the secondary Steenrod algebra

We introduce a new model for the secondary Steenrod algebra at the prime 2 which is both smaller and more accessible than the original construction of H.-J. Baues. We also explain how BP can be used to define a variant of the secondary Steenrod algebra at odd primes.

math.AT↗