Homology Groups Research Articles

Fundamental groups of James reduced products

It is well-known that if a based, path-connected topological space (X,e) is well-pointed, then the fundamental group of the James reduced product J(X) is naturally isomorphic to the first singular homology group H1(X). In this paper, we study the group π1(J(X),e) in much greater generality. If (X,e) is not well-pointed, then π1(J(X),e) may be a proper quotient of H1(X), which behaves in many ways as an infinitary abelianization of π1(X,e) for infinite products formed at the basepoint e. Our main result is that the canonical injection σ:X→J(X) induces a surjection σ#:π1(X,e)→π1(J(X),e) on fundamental groups for any path-connected Hausdorff space X. Equipped with this result and previous work on infinite commutativity in fundamental groups of monoids, we characterize the group π1(J(X),e) for a variety of examples.

(Co)homology of $$\Gamma $$-groups and $$\Gamma $$-homological algebra

This is a further investigation of our approach to group actions in homological algebra in the settings of homology of \(\Gamma \)-simplicial groups, particularly of \(\Gamma \)-equivariant homology and cohomology of \(\Gamma \)-groups. This new direction of homological algebra is called \(\Gamma \)-homological algebra. The abstract kernel of non-abelian extensions of groups, its relationship with the obstruction theory, and with the second group cohomology are extended to the case of non-abelian \(\Gamma \)-extensions of \(\Gamma \)-groups. We compute the rational \(\Gamma \)-equivariant (co)homology groups of finite cyclic \(\Gamma \)-groups. The isomorphism of the group of n-fold \(\Gamma \)-equivariant extensions of a \(\Gamma \)-group G by a \(G\,{\rtimes }\, \Gamma \)-module A with the \((n+1)\)th \(\Gamma \)-equivariant group cohomology of G with coefficients in A is proven. We define the \(\Gamma \)-equivariant Hochschild homology as the homology of the \(\Gamma \)-Hochschild complex when the action of the group \(\Gamma \) on the Hochschild complex is induced by its action on the basic ring. Important properties of the \(\Gamma \)-equivariant Hochschild homology related to Kähler differentials, Morita equivalence, and derived functors are established. Group (co)homology and \(\Gamma \)-equivariant group (co)homology of crossed \(\Gamma \)-modules are introduced and investigated by using relevant derived functors. Relations with extensions of crossed \(\Gamma \)-modules, in particular with relative extensions of group epimorphisms in the sense of Loday are established. Universal and \(\Gamma \)-equivariant universal central \(\Gamma \)-extensions of \(\Gamma \)-perfect crossed \(\Gamma \)-modules are constructed and relevant Hopf formulas are obtained. Finally, applications to the algebraic K-theory, the Galois theory of commutative rings, and the cohomological dimension of groups are given.

Variation of local fundamental group and local first homology group

In this paper, we discuss various results about variation of the local fundamental group of normal complex spaces. It is proved that the finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that finite Galois descent of upper semicontinuity of the local first homology group at a smooth germ is not true in general. We prove that the local fundamental groups of a normal complex algebraic variety are finite in number upto isomorphism.

Oriented Borel–Moore homologies of toric varieties

We generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties.

The contraction category of graphs

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan–Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan–Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups.

Alternating 3 different influenza vaccines for swine in Europe for a broader antibody response and protection

Heterologous prime-boost vaccination with experimental or commercial influenza vaccines has been successful in various animal species. In this study, we have examined the efficacy of alternating 3 different European commercial swine influenza A virus (swIAV) vaccines: the trivalent Respiporc® FLU3 (TIV), the bivalent GRIPORK® (BIV) and the monovalent Respiporc® FLUpan H1N1 (MOV). Five groups of 6 pigs each received 3 vaccinations at 4–6 week intervals in a homologous or heterologous prime-boost regimen. A sixth group served as a mock-vaccinated challenge control. Four weeks after the last vaccination, pigs were challenged intranasally with a European avian-like H1N1 (1C.2.1) swIAV, which was antigenically distinct from the vaccine strains. One heterologous prime-boost group (TIV–BIV–MOV) had higher hemagglutination inhibition (HI) and neuraminidase inhibition antibody responses against a panel of antigenically distinct H1N1, H1N2 and H3N2 IAVs than the other heterologous prime-boost group (BIV–TIV–MOV) and the homologous prime-boost groups (3xTIV; 3xBIV; 3xMOV). Group TIV–BIV–MOV had seroprotective HI titers (≥ 40) against 56% of the tested viruses compared to 33% in group BIV–TIV–MOV and 22–39% in the homologous prime-boost groups. Post-challenge, group TIV–BIV–MOV was the single group with significantly reduced virus titers in all respiratory samples compared to the challenge control group. Our results suggest that the use of different commercial swIAV vaccines for successive vaccinations may result in broader antibody responses and protection than the traditional, homologous prime-boost vaccination regimens. In addition, the order in which the different vaccines are administered seems to affect the breadth of the antibody response and protection.

Investigation of active site amino acid influence on carbon and chlorine isotope fractionation during reductive dechlorination.

Reductive dehalogenases (RDases) are corrinoid-dependent enzymes that reductively dehalogenate organohalides in respiratory processes. By comparing isotope effects in biotically catalyzed reactions to reference experiments with abiotic corrinoid catalysts, compound-specific isotope analysis (CSIA) has been shown to yield valuable insights into enzyme mechanisms and kinetics, including RDases. Here, we report isotopic fractionation (ε) during biotransformation of chloroform (CF) for carbon (εC = -1.52± 0.34‰) and chlorine (εCl = -1.84± 0.19‰), corresponding to a ΛC/Cl value of 1.13± 0.35. These results are highly suppressed compared to isotope effects observed both during CF biotransformation by another organism with a highly similar RDase (>95% sequence identity) at the amino acid level, and to those observed during abiotic dehalogenation of CF. Amino acid differences occur at four locations within the two different RDases' active sites, and this study examines whether these differences potentially affect the observed εC, εCl, and ΛC/Cl. Structural protein models approximating the locations of the residues elucidate possible controls on reaction mechanisms and/or substrate binding efficiency. These four locations are not conserved among other chloroalkane reducing RDases with high amino acid similarity (>90%), suggesting that these locations may be important in determining isotope fractionation within this homologous group of RDases.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 < 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

Hodge theory-based biomolecular data analysis

Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.

Annotation-free delineation of prokaryotic homology groups.

Phylogenomic studies of prokaryotic taxa often assume conserved marker genes are homologous across their length. However, processes such as horizontal gene transfer or gene duplication and loss may disrupt this homology by recombining only parts of genes, causing gene fission or fusion. We show using simulation that it is necessary to delineate homology groups in a set of bacterial genomes without relying on gene annotations to define the boundaries of homologous regions. To solve this problem, we have developed a graph-based algorithm to partition a set of bacterial genomes into Maximal Homologous Groups of sequences (MHGs) where each MHG is a maximal set of maximum-length sequences which are homologous across the entire sequence alignment. We applied our algorithm to a dataset of 19 Enterobacteriaceae species and found that MHGs cover much greater proportions of genomes than markers and, relatedly, are less biased in terms of the functions of the genes they cover. We zoomed in on the correlation between each individual marker and their overlapping MHGs, and show that few phylogenetic splits supported by the markers are supported by the MHGs while many marker-supported splits are contradicted by the MHGs. A comparison of the species tree inferred from marker genes with the species tree inferred from MHGs suggests that the increased bias and lack of genome coverage by markers causes incorrect inferences as to the overall relationship between bacterial taxa.

Comparative analysis between homotopy group and homology group

This paper seeks to demonstrate the relation between homology group and homotopy group. The result in this paper is a construction of the homology of a complex torus.

Morse Functions of G2/SO(4)

We construct ℤ2-perfect Morse functions of G2/SO(4) whose set of all critical points is a great antipodal set of G2/SO(4). In particular, we give the reason why the 2-number #2(G2/SO4)) matches the Betti number of the ℤ2-coefficient homology group of G2/SO(4).

Genomic insights into the recent chromosome reduction of autopolyploid sugarcane Saccharum spontaneum.

Saccharum spontaneum is a founding Saccharum species and exhibits wide variation in ploidy levels. We have assembled a high-quality autopolyploid genome of S. spontaneum Np-X (2n = 4x = 40) into 40 pseudochromosomes across 10 homologous groups, that better elucidates recent chromosome reduction and polyploidization that occurred circa 1.5 million years ago (Mya). One paleo-duplicated chromosomal pair in Saccharum, NpChr5 and NpChr8, underwent fission followed by fusion accompanied by centromeric split around 0.80 Mya. We inferred that Np-X, with x = 10, most likely represents the ancestral karyotype, from which x = 9 and x = 8 evolved. Resequencing of 102 S. spontaneum accessions revealed that S. spontaneum originated in northern India from an x = 10 ancestor, which then radiated into four major groups across the Indian subcontinent, China, and Southeast Asia. Our study suggests new directions for accelerating sugarcane improvement and expands our knowledge of the evolution of autopolyploids.

The second Betti number of doubly weighted homology groups of a certain pre Lie superalgebra

We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and showed the first Betti number is 0 when w = h. In this note, we show that the second Betti number is 0 when w = h.

Polychlorinated biphenyls (PCBs) in soils from typical paddy fields of China: Occurrence, influencing factors and human health risks

The contamination of paddy soils is of great concern since it links to human health via food supply. Limited knowledge is available on PCB residue characteristics and the associated health risks in paddy soils under various environmental conditions. In this study, a soil sampling campaign was conducted in three typical paddy fields, i.e., Sanjiang Plain (SP), Taihu Plain (TP) and Hani Terrace (HT), crossing a transect of 4000 km in China. The concentrations of 29 quantified PCBs varied from 58.6 to 1930 pg/g in paddy soils, with samples at TP showing the highest burden. Tri-CBs were the major homologue group at SP and HT, whereas hexa-CBs at TP. Altitude, temperature, soil organic matter content and soil conductivity well explained the variations in PCB concentrations among sites. The homologue profiles of soil PCBs followed the fractionation theory. In addition, soil conductivity was found to be negatively correlated to low-chlorinated PCBs and positively to high-chlorinated congeners. Furthermore, the toxicities of soil PCBs and the exposure risks through rice intake were estimated. Higher toxicity equivalent quantities and hazard indexes were found at SP than TP and HT, with over one third of the samples displaying health risks. The results of this work highlight the necessity to better understand the occurrence characteristics and the associated health risks of PCBs in soils of rice-growing regions.

Remote sensing image segmentation by combining manifold projection and persistent homology

This paper presents an image segmentation algorithm by combining Manifold Projection and Persistent Homology (MP_PH). First, for a given image, the spectral measures of each pixel and its neighbor pixels are modeled with Gaussian Probability Distribution Function (GPDF) in an exponential family fashion. The Riemannian manifold, i.e. the data sub-manifold for the pixel, is built by taking the parameters of the GPDF exponential family model as its coordinates to depict the statistical characteristics of the original image. By Legendre transformation, the data sub-manifold is transformed into a parameter sub-manifold to depict all possible segmentation results. Only points representing classes of current segmentation results are activated on the parameter sub-manifold. Then, simplicial complexes constructed from the original image are used to compute persistent homology. The optimal scale can be obtained from persistent homology to compute the optimal homology group generated by homology generators which are referred to some pixels belonging to the same class. Finally, the segmentation is performed by projecting points of the data sub-manifold belonging to the same homology generator to the nearest activated point of the parameter sub-manifold, and updating all the activated points according to the projection results. As a result, all the activated points tend to be optimal segmentation. The experiments for synthetic and real images show that the proposed algorithm has high segmentation accuracy.

Heterologous ChAdOx1 and Bnt162b2 vaccination induces strong neutralizing antibody responses against SARS-CoV-2 including delta variant with tolerable reactogenicity

ObjectivesWe assessed humoral responses and reactogenicity following the heterologous vaccination compared to the homologous vaccination groups. MethodsWe enrolled healthcare workers (HCWs) who were either vaccinated with ChAdOx1 followed by BNT162b2 (heterologous group) or 2 doses of ChAdOx1 (ChAdOx1 group) or BNT162b2 (BNT162b2 group). Immunogenicity was assessed by measuring antibody titers against receptor-binding domain (RBD) of SARS-CoV-2 spike protein in all participants and neutralizing antibody titer in 100 participants per group. Reactogenicity was evaluated by a questionnaire-based survey. ResultsWe enrolled 499 HCWs (ChAdOx1, n = 199; BNT162b2, n = 200; heterologous ChAdOx1/BNT162b2, n = 100). The geometric mean titer of anti–receptor-binding domain antibody at 14 days after the booster dose was significantly higher in the heterologous group (11 780.55 binding antibody unit (BAU)/mL [95% CI, 10 891.52–12 742.14]) than in the ChAdOx1 (1561.51 [95% CI, 1415.03–1723.15]) or BNT162b2 (2895.90 [95% CI, 2664.01–3147.98]) groups (both p < 0.001). The neutralizing antibody titer of the heterologous group (geometric mean ND50, 2367.74 [95% CI, 1970.03–2845.74]) was comparable to that of the BNT162b2 group (2118.63 [95% CI, 1755.88–2556.32]; p > 0.05) but higher than that of the ChAdOx1 group (391.77 [95% CI, 326.16–470.59]; p < 0.001). Compared with those against wild-type SARS-CoV-2, the geometric mean neutralizing antibody titers against the Delta variant at 14 days after the boosting were reduced by 3.0-fold in the heterologous group (geometric mean ND50, 872.01 [95% CI, 685.33–1109.54]), 4.0-fold in the BNT162b2 group (337.93 [95% CI, 262.78–434.57]), and 3.2-fold in the ChAdOx1 group (206.61 [95% CI, 144.05–296.34]). The local or systemic reactogenicity after the booster dose in the heterologous group was higher than that of the ChAdOx1 group but comparable to that of the BNT162b2 group. DiscussionHeterologous ChAdOx1 followed by BNT162b2 vaccination with a 12-week interval induced a robust humoral immune response against SARS-CoV-2, including the Delta variant, that was comparable to the homologous BNT162b2 vaccination and stronger than the homologous ChAdOx1 vaccination, with a tolerable reactogenicity profile.

Hydrophobic deep eutectic solvents as pseudo-stationary phases in capillary electrokinetic chromatography: An explorative study

Research into the application of deep eutectic solvents (DESs) in capillary electrophoresis (CE) is still in its infancy. Even so, a controversy has already arisen over the “true role” of DESs in CE. This is because the DESs employed previously were all hydrophilic, and it is therefore reasonable to speculate that they may be virtually completely dissociated into its initial components (hydrogen-bond acceptor (HBA) and hydrogen-bond donor (HBD)) in aqueous CE buffers. Since there seems to be no essential difference in CE performance between adding a DES and adding its two constituents separately (HBA and HBD) in a running buffer, the significance of using DESs in these CE systems has not been clearly established. In this context, this study established the first DES-based CE system in which a DES really worked as a single entity. To achieve this goal a hydrophobic DES (HDES), (−)-menthol:octanoic acid, was employed to act as a pseudo-stationary phase (PSP) in electrokinetic chromatography (EKC) mode. We found that the HDES is capable of forming stable aggregates (micro-droplets) which can act as a PSP in the running buffer. The novel HDES-type PSP exhibited impressive selectivities toward several groups of analogues, homologues, and isomers via hydrophobic mechanism. It can be used either solely for direct EKC separation, or in combination with traditional PSPs (e.g., cyclodextrins). As a proof-of-concept study, we have demonstrated that the synergistic effect generated in an “HDES + Hydroxypropyl-γ-cyclodextrin” dual-PSP system allowed an excellent separation of a simulated complex mixture consisting of 13 aromatic acids.

Frequency and Nuisance Level of Adverse Events in Individuals Receiving Homologous and Heterologous COVID-19 Booster Vaccine

This study aimed to compare the occurrence and nuisance of adverse events following administration of each COVID-19 vaccine dose between two groups: individuals given three doses of mRNA vaccine (homologous group, 3 × mRNA, n = 252) and those given two doses of adenoviral vector vaccine further boosted with mRNA vaccine (heterologous group, 2 × AZ + 1 × mRNA, n = 205). Although the studied groups differed significantly in the frequency and number of side effects after the first and second vaccine dose, no relevant differences were seen following the booster administration. Arm pain and fatigue were the most common effects, regardless of the vaccination group and vaccine dose. In the homologous group, female sex, lower BMI, and no history of regular influenza vaccination were associated with a higher frequency of side effects of a booster dose. In the heterologous group, the history of COVID-19 was associated with an increased number of side effects seen after a booster. In both groups, the number of side effects related to the first and second dose correlated with the number observed after administration of a booster dose. Individuals receiving a homologous booster reported a higher nuisance of side effects than the heterologous group. It was similar to the level reported after the second dose in both groups. The use of pharmaceuticals to counteract the side effects was more frequent after a first dose in the 2 × AZ + 1 × mRNA group, but higher after second dose in individuals receiving the 3 × mRNA vaccination scheme. The frequency of pharmaceutical use after a booster dose was similar in both groups (approx. 60%). Paracetamol was most frequently chosen, regardless of the group and vaccine dose. In addition, the vast majority of participants (93%) declared to accept future doses of the COVID-19 vaccine if their administration would be recommended. This study provides an overview of the response to homologous and heterologous mRNA vaccine booster dose that may be valuable in shaping accurate and honest communication with vaccinated individuals, especially in those regions which are yet to pursue booster strategies.

Decoupling Decorations on Moduli Spaces of Manifolds

AbstractWe study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.