We qualify a claim made in 1, regarding the dimensions in which all orientable manifolds admit spinh structures, with a compactness assumption, and comment on when this assumption can be removed.
Abstract Goerss-Hopkins theory Pstrągowski, Piotr; VanKoughnett, Paul
Advances in mathematics (New York. 1965),
02/2022, Letnik:
395
Journal Article
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We present an abstract version of Goerss-Hopkins theory in the setting of a prestable ∞-category equipped with a suitable periodicity operator. In the case of the ∞-category of synthetic spectra, ...this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.
Given a height at most two Landweber exact $\mathbb {E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion ...admits a unique lift to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to \mathrm {tmf}_1 (n)$ for all $n\, {\geq}\, 2$.
The question of which manifolds are spin or spinc has a simple and complete answer. In this paper we address the same question for spinh manifolds, which are less studied but have appeared in ...geometry and physics in recent decades. We determine that the first obstruction to being spinh is the fifth integral Stiefel–Whitney class W5. Moreover, we show that every orientable manifold of dimension 7 and lower is spinh, and that there are orientable manifolds which are not spinh in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we show that there is no integer k such that every manifold embeds in a spin manifold with codimension k.
In this work, a theoretical approach based on the obstruction theory is proposed to estimate the diffusion characteristics associated with the natural and synthetic wound antimicrobials through ...polyethylene glycol (PEG) hydrogel matrix. The simulation outcomes have been compared with the free volume theory based model which has been reported in our earlier study. The influence of the skin layer on the predicted parameters is also investigated. The results show that the obstruction theory based model is able to distinguish between the diffusion kinetics of the natural and synthetic antimicrobials for the considered range of molecular weights and PEG matrix configuration. The diffusion time of the therapeutic formulations through the hydrogel is found to vary in the range of 2.7–7.2 h. The predicted time scale is comparable with the real-time treatment window which would aid in preventing the formation of biofilm in wounds. Gentamicin is observed to take 4.2 h to completely diffuse out of the hydrogel matrix, while its diffusion period is increased by 95 times in the presence of the 1 mm thick skin layer. The diffusivity values of the antimicrobial compounds estimated using the obstruction theory are noted to be consistently higher than that of the free volume theory. This study seems to be clinically significant as the computational approaches to analyze the diffusivity of drugs in the hydrogel matrices are essential for the therapeutic decision making during wound treatment.
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Synthetic hydrogels formed from poly(ethylene glycol) (PEG) are widely used to study how cells interact with their extracellular matrix. These in vivo-like 3D environments provide a basis for tissue ...engineering and cell therapies but also for research into fundamental biological questions and disease modeling. The physical properties of PEG hydrogels can be modulated to provide mechanical cues to encapsulated cells; however, the impact of changing hydrogel stiffness on the diffusivity of solutes to and from encapsulated cells has received only limited attention. This is particularly true in selectively cross-linked “tetra-PEG” hydrogels, whose design limits network inhomogeneities. Here, we used a combination of theoretical calculations, predictive modeling, and experimental measurements of hydrogel swelling, rheological behavior, and diffusion kinetics to characterize tetra-PEG hydrogels’ permissiveness to the diffusion of molecules of biologically relevant size as we changed polymer concentration, and thus hydrogel mechanical strength. Our models predict that hydrogel mesh size has little effect on the diffusivity of model molecules and instead predicts that diffusion rates are more highly dependent on solute size. Indeed, our model predicts that changes in hydrogel mesh size only begin to have a non-negligible impact on the concentration of a solute that diffuses out of hydrogels for the smallest mesh sizes and largest diffusing solutes. Experimental measurements characterizing the diffusion of fluorescein isothiocyanate (FITC)-labeled dextran molecules of known size aligned well with modeling predictions and suggest that doubling the polymer concentration from 2.5% (w/v) to 5% produces stiffer gels with faster gelling kinetics without affecting the diffusivity of solutes of biologically relevant size but that 10% hydrogels can slow their diffusion. Our findings provide confidence that the stiffness of tetra-PEG hydrogels can be modulated over a physiological range without significantly impacting the transport rates of solutes to and from encapsulated cells.
We set up a fibred categorical theory of obstruction and classification of morphisms that specialises to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane ...theory of group extensions. Further applications are provided to crossed extensions and crossed bimodule butterflies, with in particular a classification of non-abelian extensions of unital associative algebras in terms of Hochschild cohomology.
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