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Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class., 45 pages

We study the notion of Cartan-Eilenberg Gorenstein-injective $ m $-complexes. We show that a $ m $-complex $ G $ is Cartan-Eilenberg Gorenstein-injective if and only if $ G_n $, $ \mathrm{Z}_n^{t}(G) $, $ \mathrm{B}_n^{t}(G) $ and $ \mathrm{H}_n^{t}(G) $ are Gorenstein-injective modules for each $ n\in\mathbb{Z} $ and $ t = 1, 2, \ldots, m $. As an application, we show that an iteration of the procedure used to define the Cartan-Eilenberg Gorenstein-injective $ m $-complexes yields exactly the Cartan-Eilenberg Gorenstein-injective $ m $-complexes. Specifically, given a Cartan-Eilenberg exact sequence of Cartan-Eilenberg Gorenstein-injective $ m $-complexes \begin{document}$ \mathbb{G} = \cdots\rightarrow G^{-1}\rightarrow G^0\rightarrow G^1\rightarrow \cdots $\end{document} such that the functor $ \mathrm{Hom}_{\mathcal{C}_m({R})}(H, -) $ leave $ \mathbb{G} $ exact for each Cartan-Eilenberg Gorenstein-injective $ m $-complex $ H $, then $ \mathrm{Ker}(G^0\rightarrow G^1) $ is a Cartan-Eilenberg Gorenstein-injective $ m $-complex.

BCK-modules were introduced as an action of a BCK-algebra over an Abelian group. Homomorphisms of BCK-modules form an exact sequence which is called BCK-sequence. In this paper, we study homomorphisms of BCK-modules. We show that this homomorphisms have a module structure. Moreover, we show that sequences of Hom functors are BCK-sequences.

For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all of them with coefficients in $X$. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness., A typo in the degree of the torsion vector spaces of the first page of the spectral sequence is corrected. The degree where the exact long sequence ends at Theorem 6.5 is therefore updated. To appear in Bulletin of the London Mathematical Society

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen., V.3, the paper has been splitted into two papers, this is the first part on amalgamated free products

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the class of Lie groupoids corresponding to global quotient orbifolds with nonempty singular set, the fibration theorems do not hold. Secondly, we prove a short exact sequence of fundamental groups (called {\it pure orbifold braid groups}) of one of the configuration Lie groupoids of the Lie groupoid corresponding to the punctured complex plane with cone points. This shows the possibility of a quasifibration type Fadell-Neuwirth theorem for Lie groupoids. As consequences, first we see that the pure orbifold braid groups have poly-virtually free structure, which generalizes the classical braid group case. We also provide an explicit set of generators of the pure orbifold braid groups. Secondly, we prove that a class of affine and finite complex Artin groups are virtually poly-free, which partially answers the question if all Artin groups are virtually poly-free ([[4], Question 2]). Finally, combining this poly-virtually free structure and a recent result ([5]), we deduce the Farrell-Jones isomorphism conjecture for the above class of orbifold braid groups. This also implies the conjecture for the case of the affine Artin group of type $\tilde D_n$, which was left open in [[24], Problem]., Appeared in the Bulletin des Sciences Mathematiques

We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural question regarding composition series of finite tensor categories., amslatex, 30 pages

We study the behaviour of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We find a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. In the second part we will combine techniques of hereditary cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if $M$ is any module and $C$ is cotorsion, then $M$ will be also cotorsion. This will lead to some meaningful consequences in the category $\textrm{Ch}(R)$ of unbounded chain complexes and in Gorenstein homological algebra. For example we show that every acyclic complex of cotorsion modules has cotorsion cycles, and more generally, every map $F\to C$ where $C$ is a complex of cotorsion modules and $F$ is an acyclic complex of flat cycles, is null-homotopic. In other words, every complex of cotorsion modules is dg-cotorsion.

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to [math] . We find a basis for the depth-graded motivic double zeta values relative to [math] of even weight and a basis for the depth-graded motivic triple zeta values relative to [math] of odd weight. As an application of our main results, we prove Kaneko and Tasaka’s conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka’s conjecture in depth three. Finally, we formulate a conjecture which is related to sum odd multiple zeta values in higher depth.

In this article, we construct a natural group homomorphism $$ \psi: \text{Br}(f)\to H^{1}_{et}(S, f_{*}\mathcal{O}_{X}^{\times}/\mathcal{O}_{S}^{\times})$$ for a faithful affine map $f: X \to S$ of noetherian schemes. Here $\text{Br}(f)$ denotes the relative Brauer group of $f.$ We also discuss the behavior of Brauer groups for a subintegral map. Furthermore, we prove a relative version of Kummer's exact sequence., Comment: Some changes made due to referee report. Final version and accepted for publication

Given a fusion system $\mathcal{F}$ defined on a $p$-group $S$, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize $\mathcal{F}$. We study these models when $\mathcal{F}$ is a fusion system of a finite group $G$ and prove a theorem which relates the cohomology of an infinite group model $\pi$ to the cohomology of the group $G$. We show that for the groups $GL(n,2)$, where $n\geq 5$, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors $P\to \Theta(P)$ for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients., Comment: 23 pages

In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a $G$-crossed braided extension $\mathcal{C}\subseteq \mathcal{C}^{\times}_{G}$, we show that physical considerations require that a connected \'etale algebra $A\in \mathcal{C}$ admit a $G$-equivariant algebra structure for symmetry to be preserved under condensation of $A$. Given any categorical action $\underline{G}\rightarrow \underline{\sf Aut}_{\otimes}^{\sf br}(\mathcal{C})$ such that $g(A)\cong A$ for all $g\in G$, we show there is a short exact sequence whose splittings correspond to $G$-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of $A$. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of $\mathcal{C}^{\operatorname{loc}}_{A}$, and gauging this symmetry commutes with anyon condensation., Comment: 35 pages, comments welcome. To appear in Journal of High Energy Physics

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture, along with a boundary class, using Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $KR$-theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the "strong" and the "weak" invariants., v1: 60 pages; v2: References added, 61 pages; v3: Minor improvements, 60 pages

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion., 14 pages, 5 figures, to appear in European Journal of Mathematics

The affine group scheme of automorphisms of an evolution algebra that is equal to its square, is shown to lie in an exact sequence, such that the other terms depend solely on the directed graph associated to the algebra. As a consequence, the Lie algebra of derivations is shown to be trivial in characteristic 0 or 2, and to be abelian, with a precise description depending just on the graph, otherwise., 12 pages

The goal of this thesis is to classify all extensions where the kernel has order p^s and the cokernel has order p^t, p is a prime, and 1 ≤ s,t ≤ 2. We determine (up to weak congruence) the different combinations of kernel, cokernel and operators, and for each case, calculate the second cohomology group. By comparing resolutions, we get an explicit correspondence between the second cohomology group and the group of congruence classes of extensions. Using this construction, we determine (up to congruence) the extensions for the different combinations.

International audience; Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this work we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal. We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.

We study exact sequences of finite tensor categories of the form $\Rep G \to \C \to \D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $\Gamma$ and mutual actions by permutations $\rhd: \Gamma \times G \to G$ and $\lhd: \Gamma \times G \to \Gamma$ that make $(G, \Gamma)$ into matched pair of groups endowed with a natural crossed action on $\D$ such that $\C$ is equivalent to a certain associated crossed extension $\D^{(G, \Gamma)}$ of $\D$. Dually, we show that an exact sequence of finite tensor categories $\vect_G \to \C \to \D$ induces an $\Aut(G)$-grading on $\C$ whose neutral homogeneous component is a $(Z(G), \Gamma)$-crossed extension of a tensor subcategory of $\D$. As an application we prove that such extensions $\C$ of $\D$ are weakly group-theoretical fusion categories if and only if $\D$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical., Comment: 27 pages

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $\Gamma$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence for any finite-index subgroup of the full mapping class group. This theorem was announced in a 1990 preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author., Comment: 17 pages, one figure

Recent controversy focuses on the best method for delineating microbial taxa, based on either traditional operational taxonomic units (OTUs) or exact sequence variants (ESVs) of marker gene sequences. We sought to test if the binning approach (ESVs versus OTUs defined by >97% sequence similarity) affected the conclusions of a large field study. The dataset included sequences targeting all bacteria (16S) and fungi (ITS), across multiple environments diverging markedly in abiotic conditions, over three collection times. Despite quantitative differences in microbial richness, we found that all alpha- and beta-diversity metrics were highly positively correlated (r > 0.90) between samples analyzed with both approaches. Consequently, statistical inferences were nearly indistinguishable. Further, ESVs only moderately increased the genetic resolution of fungal and bacterial diversity (1.3 and 2.1 times OTU richness, respectively). We conclude that for broad-scale (e.g., all bacteria or all fungi) alpha- and beta-diversity analyses, OTU or ESV methods will often reveal similar ecological results. Thus, while there are good reasons to employ ESVs, we need not question the validity of results based on OTUs. There are advantages and disadvantages of both, and which binning method to employ will depend on question/data in hand.

Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ be a Thurston map and let $M(\tilde f)$ be its mapping class biset: isotopy classes rel $\tilde A$ of maps obtained by pre- and post-composing $\tilde f$ by the mapping class group of $(S^2,\tilde A)$. Let $A\subseteq\tilde A$ be an $\tilde f$-invariant subset, and let $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$ by fundamental groups of punctured spheres, $M(\tilde f)$ is an iterated extension of $M(f)$ by the dynamical biset of $f$. Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in $M(\tilde f)$ to that in $M(f)$. In case $\tilde f$ is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset $B(f)$ together with a "portrait of bisets" induced by $\tilde A$ is a complete conjugacy invariant of $\tilde f$. Along the way, we give a complete description of bisets of $(2,2,2,2)$-maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order $2$, and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps., v2 fixes some broken references

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface., 21 pages; minor changes; to appear in Izvestiya: Mathematics

Let 핐 be a fat point scheme in ℙn1 × … × ℙnk over a field K of characteristic zero. In this paper we introduce the multi-graded Kähler differential module for 핐 and we establish a short exact sequence of this module in terms of the thickening of 핐.

International audience; We study various exact sequences associated with a closed saturated subset in the space of units of a Lie groupoid: the corresponding exact sequence of groupoid C*-algebras, the associated index maps and full index maps. Moreover we study Connes-Thom type isomorphisms of Lie groupoid C*-algebras.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms., Comment: v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topology

We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions., 37 pages, 11 figures. Proof of naturality thoroughly revised. To appear in Quantum Topology

For odd $N\geq 5$, we establish a short exact sequence about motivic double zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. From this we classify all the relations among depth-graded motivic double zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight., Comment: 15 pages

We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a π 0 - π 1 exact sequence associated with a fractor between groupoids internal to a pointed exact category.

We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers.

Composite basis functions for pyramidal elements on the spaces $H^1(\Omega)$, $H(\mathrm{curl},\Omega)$, $H(\mathrm{div},\Omega)$ and $L^2(\Omega)$ are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart-Thomas-N\'ed\'elec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, ...) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements is successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size $h$ as one would obtain using purely hexahedral or purely tetrahedral partitions., Comment: 21 pages

In this article we prove exactness of the homotopy sequence of overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving a corresponding result for rigid analytic varieties in characteristic $0$, following dos Santos in the algebraic case. In characteristic $p$, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result., 31 pages, comments very welcome!

Recent advances have made it possible to analyze high-throughput marker-gene sequencing data without resorting to the customary construction of molecular operational taxonomic units (OTUs): clusters of sequencing reads that differ by less than a fixed dissimilarity threshold. New methods control errors sufficiently that sequence variants (SVs) can be resolved exactly, down to the level of single-nucleotide differences over the sequenced gene region. The benefits of finer taxonomic resolution are immediately apparent, and arguments for SV methods have focused on their improved resolution. Less obvious, but we believe more important, are the broad benefits deriving from the status of SVs as consistent labels with intrinsic biological meaning identified independently from a reference database. Here we discuss how those features grant SVs the combined advantages of closed-reference OTUs — including computational costs that scale linearly with study size, simple merging between independently processed datasets, and forward prediction — and of de novo OTUs — including accurate diversity measurement and applicability to communities lacking deep coverage in reference databases. We argue that the improvements in reusability, reproducibility and comprehensiveness are sufficiently great that SVs should replace OTUs as the standard unit of marker gene analysis and reporting.

In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We particularly study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all "KLQ" functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties., minor revisions