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Oliver Röndigs

Publications and source records attributed to Oliver Röndigs.

At least 19 recordsLinked to original sources
arXiv2026

Slices of the special linear algebraic cobordism spectrum

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

math.AT↗
arXiv2025

Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$

The theme of this paper is to compute hermitian $K$-groups in terms of the recently developed theory of Milnor-Witt motivic cohomology. Our approach makes use of the very effective slice spectral sequence within the motivic stable homotopy category, which we analyze in detail for base schemes of arithmetic interest. We show a Grothendieck-Riemann-Roch theorem, determine the map between Milnor-Witt and hermitian $K$-theory up to degree five for all fields, and compute the hermitian $K$-groups and the higher Witt-groups of the ring of integers.

math.AG↗
arXiv2025

A cellular absolute motivic ring spectrum representing Hermitian K-theory

In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in the thesis of the first author via the geometry of orthogonal and hyperbolic Grassmannians over any scheme coincides with the motivic ring spectrum constructed recently by Calmès, Harpaz, and Nardin.

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arXiv2025

The motivic Adams conjecture

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety of maximal tori in a general linear group, which turns out to be not stably A1-connected. We also show that the higher motivic stable stems are of bounded torsion.

math.KT↗
arXiv2024

The Grothendieck ring of varieties and algebraic K-theory of spaces

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the Grothendieck ring of varieties over F.

math.KT↗
arXiv2022

Endomorphisms of the projective plane and the image of the Suslin-Hurewicz map

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel-Voevodsky motivic stable homotopy category than in Voevodsky's derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root. A byproduct of the computations is a proof of Suslin's conjecture on the Suslin-Hurewicz homomorphism from Quillen to Milnor K-theory in degree four, based on work of Asok, Fasel, and Williams.

math.AG↗
arXiv2019

Remarks on motivic Moore spectra

The term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an understanding of some Toda brackets in the motivic stable homotopy groups of spheres.

math.AT↗
arXiv2019

The homotopy groups of the η-periodic motivic sphere spectrum

We compute the homotopy groups of the η-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the η-periodic slice spectral sequence over Q determines the η-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a "connective Witt-theoretic J-spectrum" in η-periodic motivic homotopy theory.

math.AT↗
arXiv2018

Hermitian $K$-theory, Dedekind $ζ$-functions, and quadratic forms over rings of integers in number fields

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind $ζ$-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic $K$-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.

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arXiv2018

The first stable homotopy groups of motivic spheres

We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.

math.AT↗
arXiv2018

On the $η$-inverted sphere

It is shown that the first and second homotopy groups of the $η$-inverted sphere spectrum over a field of characteristic not two are zero. A cell presentation of higher Witt theory is given as well, at least over the complex numbers.

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arXiv2017

On very effective hermitian $K$-theory

We argue that the very effective cover of hermitian $K$-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological $K$-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.

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arXiv2017

Gigantic random simplicial complexes

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.

math.CO↗
arXiv2017

Slices of hermitian K-theory and Milnor's conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

math.KT↗