A Classification of the Stable Type of $BG$

A geometric classification of the path components of the space of locally stable maps $S^3\to \mathbb{R}^4$

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extension of $f$ to a locally stable map of the $4$-disk into $\mathbb{R}^5$. Relations between the invariants are described, and it is proved that for any $σ$, $ν$, and $κ$ which satisfy these relations, there exists a map $f:S^3\to\mathbb{R}^4$ with $σ(f)=σ$, $ν(f)=ν$, and $κ(f)=κ$. I…

Koszul对偶性与稳定Weiss塔的分类

Koszul duality and a classification of stable Weiss towers

本文通过引入范畴的Koszul对偶性，证明Weiss演算中的导数构成相应Koszul对偶范畴上的右模，解决Arone--Ching与Espic猜想，并利用范畴Fourier变换对稳定Weiss塔进行分类。

Matrix problems and stable homotopy types of polyhedra

It is a survey of the results on the classification of stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author. The proofs are based on the technique of matrix problems (bimodule categories).

Adams-type maps are not stable under composition

We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of $\mathbb{E}_{\infty}$-algebras is a transfinite composition of Adams-type maps.

Torus actions on stable module categories, Picard groups, and localizing subcategories

Given an abelian $p$-group $G$ of rank $n$, we construct an action of the torus $\mathbb{T}^n$ on the stable module $\infty$-category of $G$-representations over a field of characteristic $p$. The homotopy fixed points are given by the $\infty$-category of module spectra over the Tate construction of the torus. The relationship thus obtained arises from a Galois extension in the sense of Rognes, with Galois group given by the torus. As one application, we give a homotopy-theoretic proof of Dade's classification of endotrivial modules for abelian $p$-groups. As another application, we give a sl…

Matrix problems, triangulated categories and stable homotopy types

We show how matrix problems (bimodule categories) can be used in studying triangulated categories. Then we apply the general technique to the classification of stable homotopy types of polyhedra, find out the "representation types" of such problems and give a description of stable homotopy types in finite and tame cases.

Modeling Stable One-Types

Classification of homotopy n-types has focused on developing algebraic categories which are equivalent to categories of n-types. We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable one-types. These include a model for the Postnikov one-truncation of the sphere spectrum, and for its action on the model of a stable one-type. We show that a bicategorical cokernel introduced by Vitale models the cofiber of a map between stable one-types, and apply this to develop an algebraic model for the Postnikov data of a stable one-type.

An sl_n stable homotopy type for matched diagrams

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To …