arXiv subjects
Publications and source records attributed to Thor Wittich.
Over suitable base fields $k$ of characteristic not $2$, including algebraically closed ones, we construct universal abelian $\underline{\operatorname{K}}^{\operatorname{MW}}_2$-coverings for complements of closed embeddings $\mathbb{A}^1 \hookrightarrow \mathbb{A}^3$. Using these, we obtain a rectifiability invariant of such embeddings by lifting knot-theoretic ideas to algebraic geometry via motivic homotopy theory.
In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_\infty, E_1)$-Hopf algebra spectrum whose $\pi_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups $\widetilde K_{2n}\big(\mathcal{P}^{S^{2k+1}}_{O(2k+2)}\big)$ are nonzero for all nonnegative integers $n$ and $k$.
We show that algebra objects in model categories can be transferred to algebra objects in $\infty$-categories, without any cofibrancy or fibrancy assumptions on the algebra. We furthermore show under some mild extra assumptions that this correspondence extends to commutative bialgebras and to commutative Hopf algebras.
For all positive integers $n$ and all homotopy modules $M_*$, we define certain operations $\underline{\operatorname{K}}^{\operatorname{MW}}_n \rightarrow M_*$ and show that these generate the $M_*(k)$-module of all (in general non-additive) operations $\underline{\operatorname{K}}^{\operatorname{MW}}_n \rightarrow M_*$ in a suitable sense, if $M_*$ is $\mathbb{N}$-graded and has a ring structure. This also allows us to explicitly compute the abelian group $\operatorname{Op}(\underline{\operatorname{K}}^{\operatorname{MW}}_n,\underline{\operatorname{K}}^{\operatorname{MW}}_m)$ and all operations between related theories such as Milnor, Witt and Milnor-Witt K-theory.