Miller Spaces and Spherical Resolvability of Finite Complexes

On the nonexistence of Smith-Toda complexes

Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that if p > 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller.

$\textrm{GL}_n$ 动机的几何分裂

A Geometric Splitting of the Motive of $\textrm{GL}_n$

该研究在Voevodsky的稳定动机复形张量三角范畴$\textbf{DM}(k,R)$框架下，构造了Miller酉群分裂中Thom空间的代数类似物，证明了域$k$上一般线性群概形$\textrm{GL}_n$的动机$M(\textrm{GL}_n)$存在对应的几何分裂。

Computability of finite simplicial complexes

The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, then any algorithm that semicomputes $X$ in some sense can be converted into an algorithm that fully computes $X$. In other words, the topological properties of $X$ enable one to derive full information about $X$ from partial information about $X$. In that case, we say that $X$ has computable type. Those results have been obtained by Miller, Iljazović, Sušić and oth…