We prove that the degree $k$ unramified cohomology $H_{\text{nr}}^k(X,\mathbb{Q}∕\mathbb{Z})$ of a smooth complex projective variety $X$ with small ${\text{CH}}_0(X)$ has a filtration of length $[k∕2]$, whose first piece is the torsion part of the quotient of $H^{k+1}(X,\mathbb{Z})$ by its coniveau $2$ subgroup, and whose next graded piece is controlled by the Griffiths group ${\text{Griff}}^{k∕2+1}(X)$ when $k$ is even and is related to the higher Chow group ${\text{CH}}^{(k+3)∕2}(X,1)$ when $k$ is odd. The first piece is a generalization of the Artin–Mumford invariant ($k=2$) and the Colliot-Thélène–Voisin invariant ($k=3$). We also give an analogous result for the $\mathcal{\mathscr{H}}$-cohomology $H^{d-k}(X,{\mathcal{\mathscr{H}}}^d(\mathbb{Q}∕\mathbb{Z}))$, $d=\text{dim }X$.

We state the Paschke–Higson duality theorem for a transformation groupoid. Our proof relies on an equivariant localized and norm-controlled version of the Pimsner–Popa–Voiculescu theorem. The main consequence is the existence of a Higson–Roe exact sequence, involving the Baum–Connes assembly map for such a groupoid.

We prove that every stably projectionless separable simple amenable $C^{\ast }$-algebra in the UCT class has rationally generalized tracial rank one. Following Elliott’s earlier work, we show that the Elliott invariant of any finite separable simple $C^{\ast }$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group (which unifies the unital and nonunital, as well as stably projectionless cases). We also show that, for any given such invariant set, there is a finite separable simple $C^{\ast }$-algebra whose Elliott invariant is the given set, a refinement of the range theorem of Elliott. In the stably projectionless case, modified model $C^{\ast }$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features.

We show that the sheaf of ${\mathbb{𝔸}}^1$-connected components of a Nisnevich sheaf of sets and its universal ${\mathbb{𝔸}}^1$-invariant quotient (obtained by iterating the ${\mathbb{𝔸}}^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of ${\mathbb{𝔸}}^1$-connected components of any space. Given any natural number $n$, we construct an ${\mathbb{𝔸}}^1$-connected space on which the iterations of the naive ${\mathbb{𝔸}}^1$-connected components construction do not stabilize before the $n$-th stage.

We initiate the study of real $C^{\ast }$-algebras associated to higher-rank graphs $\mathrm{\Lambda }$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $\mathcal{𝒞}\mathcal{ℛ}$ $K$-theory of $C_{ℝ}^{\ast }(\mathrm{\Lambda },\gamma )$ for any involution $\gamma$ on $\mathrm{\Lambda }$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $\mathrm{\Lambda }$ and the involution $\gamma$. We provide a complete description of $K^{CR}(C_{ℝ}^{\ast }(\mathrm{\Lambda },\gamma ))$ for several examples of higher-rank graphs $\mathrm{\Lambda }$ with involution.