For an étale groupoid, we define a pairing between the Crainic–Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is $C^*$-algebra theory. The Elliott invariant of a $C^*$-algebra is defined in terms of $K$-theory and traces; it is fundamental in the long-running programme to classify simple $C^*$-algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the $C^*$-algebraic Elliott invariant of the groupoid $C^*$-algebra: this includes irrational rotation algebras and the $C^*$-algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui’s HK conjecture for the relevant groupoids.

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups and use it for Milnor fibres of isolated complete intersection singularity (icis). We study the homology of images of mappings ft that arise as deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with n < p, and the behaviour of singularities (instabilities) in this context. We study two generalizations of the notion of image Milnor number µI given by Mond and give a workable way of compute them, in corank one, with Milnor numbers of icis. We also study two unexpected traits when $p \gt n+1$: stable perturbations with contractible image and homology of $\text{im} f_t$ in unexpected dimensions. We show that Houston’s conjecture, µI constant in a family implies excellency in Gaffney’s sense, is false, but we give a correct modification of the statement of the conjecture which we also prove.

In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.

What happens at the point of interchange between scholarly communities? We examine this question by investigating the case of growing ties between historical sociology and ethnography, two social scientific methods that once seemed to have little in common. Drawing on methodological writings by ethnographers and original interviews with practicing historical sociologists, we argue that these ties have been shaped by structural and methodological homologies between the two disciplines. Structurally, ethnography and historical sociology are similarly positioned in sociology more broadly, as enterprises with sometimes-tense relationships with dominant assumptions of the social sciences. Methodologically, both ethnographers and historical sociologists face the challenges of bounding the research process, navigating access to data, analyzing and retaining data while “in the field,” and overcoming cultural distance between themselves and the worlds they are studying. Taken together, these findings extend work in the sociology of science and knowledge and suggest some key conditions for intellectual efflorescence.

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$-algebras associated to them, in terms of the pair of original graphs.

The origin of carbon fixation is a fundamental question in astrobiology. While the Calvin cycle is the most active on the modern Earth, the reductive tricarboxylic acid (TCA) cycle (rTCA) pathway for carbon fixation has been proposed to have played an important role in early evolution. In this study, we examined the evolution of key enzymes in the rTCA, which are rare in extant organisms, occurring in a few groups of Bacteria and Archaea. We investigated one of the least common reactions of this pathway, cleavage of citrate into oxaloacetate and acetyl-CoA, which can be performed by either a two-enzyme system (CCS/CCL) or a single enzyme (ACL) that is assumed to be the result of fusion of the two active sites into a single polypeptide. For broader context, we also studied functionally diverged homologues of these enzymes, succinyl-CoA synthetase (SCS) and citrate synthase. Our phylogenetic analysis of these enzymes in Bacteria and Archaea shows that SCS, a homologue of CCS from distant bacterial taxa capable of citrate cleavage, are monophyletic, suggesting linked horizontal gene transfers of SCS and citrate cleavage enzymes. We also found evidence of the horizontal transfer of SCS from a clade of anaerobic Archaea (Archaeoglobi, Methanomicrobia or Crenarchaeota) to an ancestor of Cyanobacteria/Melainabacteria clade – both of which share a succinate semialdehyde shunt in their oxidative TCA cycles. We identified new bacterial and archaeal taxa for which complete rTCA cycles are theoretically possible, including Syntrophobacter, Desulfofundulus, Beggiatoa, Caldithrix, Ca. Acidulodesulfobacterales and Ca. Micrarchaeota. Finally, we propose a mechanism for syntrophically-regulated fluxes through oxidative and rTCA reactions in microbial communities particularly Haloarchaea-Nanohaloarchaea symbiosis and its implications for carbon fixation during retinal-based phototrophy and the Purple Earth hypothesis. We discuss how the inclusion of an ecological perspective in the studies of evolution of ancient metabolic pathways may be beneficial to understanding the origin of life.

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.

Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.