Topological finiteness properties of monoids. Part 1: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

A Tandem Approach to Photoactivated Olefin Metathesis: Combining a Photoacid Generator with an Acid Activated Catalyst

Combining an acid activated precatalyst with a photoacid generator (PAG) in the presence of ultraviolet light resulted in a highly efficient catalyst for olefin metathesis. The tandem system of precatalyst and PAG was capable of both ring closing metathesis (RCM) and ring opening metathesis polymerization (ROMP) in good to excellent conversion. Mechanistic investigations revealed that the catalytically active species is very similar to that of other well-known Ru-based catalysts

Wind turbines and invisible technology: unarticulated reasons for local opposition to wind energy

Local opposition to wind turbines stems from concerns about environmental and economic damage, as well as conflicts between rural and urban residents. This essay goes beyond these considerations to explore the often-unarticulated explanations for animosity toward this energy technology. Originally, it posits that opposition to visually obvious turbines arises from the successful history of an electric utility system that made its product largely invisible in its manufacture and physical manifestation. The existence of conspicuous turbines, however, reminds observers that power generation requires difficult choices in a technology-based society. The system’s previous achievement in hiding infrastructural elements, in other words, sometimes works ironically to spur objections to wind turbines. Receiving little historical study, the concealed features of a system’s infrastructure often influence assessments of technologies. By revealing the previously invisible, this essay, which draws on research in history, landscape architecture, geography, and psychology, therefore provides insights for social scientists and policymakers

Ruthenium Olefin Metathesis Catalysts Bearing Carbohydrate-Based N-Heterocyclic Carbenes

Ru-based olefin metathesis catalysts containing carbohydrate-derived NHCs from glucose and galactose were synthesized and characterized by NMR spectroscopy. 2D-NMR spectroscopy revealed the presence of Ru−C (benzylidene) rotamers at room temperature, and the rate of rotation was measured using magnetization transfer and VT-NMR spectroscopy. The catalysts were found to be effective at ring-opening metathesis polymerization (ROMP), ring-closing metathesis (RCM), cross-metathesis (CM), and asymmetric ring-opening cross-metathesis (AROCM) and showed surprising selectivity in both CM and AROCM