K-Theory and Homology Papers of the Week

We prove Lurie's proposed description of left rigid symmetric monoidal $(\infty,n)$-categories in terms of purely $\infty$-categorical data.

We study how Koszul duality relates two kinds of boundary constructions associated with a smooth category and a proper category.

We provide a novel, homotopy theoretic model for the motivic Eilenberg--Mac Lane spaces $K(Z_{(p)}(n), 2n) \in Spc(k)$ over fields of characteristic p. Using this, we determine their motives and, consequently, determine the motivic Steenrod algebra at the characteristic.

In this paper, we formulate Land-Tamme's $\boxtimes$-construction and $\odot$-construction for continuous exact monads on dualizable stable $\infty$-categories, and establish the behaviour of localizing invariants on pushouts of left modules over continuous exact monads on dualizable stable $\infty$-categories.

We construct higher additive noncommutative motives of stable $(\infty,n)$-categories, extending the additive motives of Blumberg, Gepner, and Tabuada and the secondary theory of Mazel-Gee and Stern.

As an application, we prove that the characteristic class Hodge index conjecture for singular varieties, formulated by Brasselet--Schürmann--Yokura, is preserved under taking symmetric products of complex projective varieties.

As a corollary, we establish a generalized Pontryagin-Thom isomorphism that gives a geometric interpretation of higher algebraic $K$-theory using cobordism classes of framed manifolds and stratified spaces decorated by categorical string diagrams, which we call lasagnas.

We extend the method of Van Woerden \cite{woerden} to totally real and CM number fields to obtain an upper bound for the number of $\GLo-$equivalence classes of perfect forms.

We give a negative answer to a question of Sánchez Saldaña--Velásquez and Lück asking whether the topological K-theory of the reduced group C*-algebra of a split crystallographic group with finite cyclic holonomy is always torsion-free.

In this paper we classify the algebraic K-theory of sets equipped with an action by a finite abelian monoid.

We establish homotopy invariance for connective continuous K-theory of affinoid Tate adic spaces of good reduction with respect to the algebraic affine line, distinct from the adic affine line as analytic stacks.

The aim of the series is to discretise the index of the Dirac operator.

The dimensions of its homology groups give meaningful numerical invariants of a category; we combine them with invariants coming from topological K-theory and Poincaré duality to prove equality of mixed Hodge numbers for singular complex projective Fourier--Mukai partners in several classes in dimensions two, three, and four, including normal surfaces and threefolds with rational singularities and equal irregularity. Beyond these applications, we construct Du Bois versions of topological cyclic homology of stable categories together with appropriate multiplicative structures, prove a Geller--Weibel-type result for the variants of negative cyclic homology we introduce, and give categorical characterizations of cohomological purity of proper varieties and the $\mathbb{Q}$-factoriality defect of complex projective threefolds with rational singularities.