Cohomology Groups Research Articles

Nonliquid cellular states: Gluing gauge-higher-symmetry-breaking versus gauge-higher-symmetry-extension interfacial defects

The existence of quantum nonliquid states and fracton orders, both gapped and gapless states, challenges our understanding of phases of entangled matter. We generalize the cellular topological states to liquid or nonliquid cellular states. We propose a mechanism to construct more general non-Abelian states by gluing gauge-symmetry-breaking versus gauge-symmetry-extension interfaces as extended defects in a cellular network, including defects of higher symmetries, in any dimension. Our approach also naturally incorporates the anyonic particle/string condensations and composite string (related to particle-string or p-string)/membrane condensations. This approach shows gluing the familiar extended topological quantum field theory or conformal field theory data via topology, geometry, and renormalization consistency criteria (via certain modified group cohomology or cobordism theory data) in a tensor network can still guide us to analyze the nonliquid states. (Part of the Abelian construction can be understood from the $K$-matrix Chern-Simons theory approach and the coupled-layer-by-junction constructions.) This approach may also lead us toward a unifying framework for quantum systems of both higher symmetries and subsystem/subdimensional symmetries.

Projective free algebras of bounded holomorphic functions on infinitely connected domains

The algebra H ∞ ( D ) H^\infty (D) of bounded holomorphic functions on D ⊂ C D\subset \mathbb {C} is projective free for a wide class of infinitely connected domains. In particular, for such D D every rectangular left-invertible matrix with entries in H ∞ ( D ) H^\infty (D) can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of H ∞ ( D ) H^\infty (D) asserting that its covering dimension is 2 2 and the second Čech cohomology group is trivial.

A counter-example to the Andreotti--Grauert conjecture

In 1962, Andreotti and Grauert showed that every $q$-completecomplex space $X$ is cohomologically $q$-complete, that is for every coherent analytic sheaf${\mathcal{F}}$ on $X$, the cohomology group $H^{p}(X, {\mathcal{F}})$ vanishes if $p\geq q$.Since then the question whether the reciprocal statements of these theorems are true have been subject to extensive studies, where more specific assumptions have been added. Until now it is not known if these two conditions are equivalent. Using test cohomology classes, it was shown however that if $X$ is a Stein manifold and, if $D\subset X$ is an open subset which has $C^{2}$ boundary such that $H^{p}(D, {\mathcal{O}}_{D})=0$ for all $p\geq q$, then $D$ is $q$-complete.The aim of the present article is to give a counterexample tothe conjecture posed in $1962$ by Andreotti and Grauert [1]to show that a~cohomologically $q$-complete space is not necessarily $q$-complete.More precisely, we show that there exist for each $n\geq 3$ open subsets$\Omega\subset\mathbb{C}^{n}$ such that for every ${\mathcal{F}}\in coh(\Omega)$,the cohomology groups $H^{p}(\Omega, {\mathcal{F}})$ vanish for all $p\geq n-1$but $\Omega$ is not $(n-1)$-complete.

A Lefschetz hyperplane theorem for non-Archimedean Jacobians

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let J be the Jacobian of a curve X, and let $$W_{d} \subset J$$ be the locus of effective divisor classes of degree d. We show that the pair $$(J^{an},W_d^{an})$$ is d-connected, and thus in particular the inclusion of the analytification of the theta divisor $$\Theta ^{an}$$ into $$J^{an}$$ satisfies a Lefschetz hyperplane theorem for $${\mathbb {Z}}$$ -cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrary characteristics and allowing arbitrary singularities on the base, of a result of Brown and Foster for the homotopy type of analytic projective bundles.

A Categorification for the Signed Chromatic Polynomial

By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which categorifies the signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.

Skew braces from Rota–Baxter operators: a cohomological characterisation and some examples

Rota–Baxter operators for groups were recently introduced by Guo, Lang, and Sheng. Bardakov and Gubarev showed that with each Rota–Baxter operator, one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group $${{\,\mathrm{Aut}\,}}(G)$$ that are defined by a functional equation. For the skew braces obtained from a Rota–Baxter operator, the corresponding gamma functions take values in the inner automorphism group $${{\,\mathrm{Inn}\,}}(G)$$ of G. In this paper, we give a characterisation of the gamma functions on a group G, with values in $${{\,\mathrm{Inn}\,}}(G)$$ , that come from a Rota–Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota–Baxter operator. For gamma functions that can be obtained from a Rota–Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits.

Common splitting fields of symbol algebras

We study the common splitting fields of symbol algebras of degree \(p^m\) over fields F of \({\text {char}}(F)=p\). We first show that if any finite number of such algebras share a degree \(p^m\) simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees \(p^{m_0},\dots ,p^{m_t}\) share a cyclic splitting field of degree \(p^{m_0+\dots +m_t}\). This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol’s to bound the symbol length of classes in \({\text {Br}}_{p^m}(F)\) whose symbol length when embedded into \({\text {Br}}_{p^{m+1}}(F)\) is 2 for \(p\in \{2,3\}\). We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

The cohomology class of the mod 4 braid group

The mod 4 braid group, [Formula: see text], is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, [Formula: see text]. Kordek and Margalit proved [Formula: see text] is an extension of the symmetric group by [Formula: see text]. For [Formula: see text], we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying [Formula: see text]. We show this cocycle is the mod 2 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show that it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for [Formula: see text].

Cohomology of Leibniz triple systems with derivations

In this paper, we introduce the notion of a LeibtsDer pair, i.e., a Leibniz triple system with a derivation. We define a representation of a LeibtsDer pair and the corresponding cohomology theory. We prove that a LeibtsDer pair is rigid if the HD3(L,L)=0, and a deformation of order n is extensible if its obstruction class (Ob(μt,Dt)i3,Ob(μt,Dt)1)=∂i3(μn+1,Dn+1)(i=1,2). We also show that the central extensions of a LeibtsDer pair can be classified by the third cohomology group.

Relatively hyperbolic groups with free abelian second cohomology

Suppose G is a finitely presented group that is hyperbolic relative to P a finite collection of finitely presented proper subgroups of G. Our main theorem states that if the second cohomology group H2(P,ZP) is free abelian for each P∈P, then H2(G,ZG) is free abelian. The problem reduces to the case when G and the members of P are 1-ended. In that case, we prove that a “Cusped Space” for this pair has semistable fundamental group at ∞. This provides a starting point in our proof of the main theorem. Finally we prove a proper simplicial approximation result for maps of [0,∞)×[0,1]. This result and our cusped space result are primary components in the proof of our main theorem. Both the cusped space result and the simplicial approximation result have applications beyond this article.

Topological and Geometrical Properties of k-Symplectic Structures

We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic manifolds in a way analogous to symplectic manifolds which are all orientable.

Gauge symmetry of unimodular gravity in Hamiltonian formalism

We work out the description of the gauge symmetry of unimodular gravity in the constrained Hamiltonian formalism. In particular, we demonstrate how the transversality conditions restricting the diffeomorphism parameters emerge from the algebra of the Hamiltonian constraints. The alternative form is long known as parametrizing the volume preserving diffeomorphisms by unrestricted two-forms instead of the transverse vector fields. This gauge symmetry is reducible. We work out the Hamiltonian description of this form of unimodular gravity (UG) gauge symmetry. Becchi-Rouet-Stora-Tyutin--Batalin-Fradkin-Vilkovisky (BFV-BRST) Hamiltonian formalism is constructed for both forms of the UG gauge symmetry. These two BRST complexes have a subtle inequivalence: Their BRST cohomology groups are not isomorphic. In particular, for the first complex, which is related to the restricted gauge parameters, the cosmological constant does not correspond to any nontrivial BRST cocycle, while for the alternative complex it does. In the wording of physics, this means $\mathrm{\ensuremath{\Lambda}}$ is a fixed parameter defined by the field asymptotics rather than the physical observable from the standpoint of the first complex. The second formalism views $\mathrm{\ensuremath{\Lambda}}$ as the observable with unrestricted initial data.

A D-modules approach on the equations of the Rees algebra

Let I⊂R=𝔽[x1,x2] be a height two ideal minimally generated by three homogeneous polynomials of the same degree d, where 𝔽 is a field of characteristic zero. We use the theory of D-modules to deduce information about the defining equations of the Rees algebra of I. Let 𝒦 be the kernel of the canonical map α: Sym(I)→ Rees(I) from the symmetric algebra of I onto the Rees algebra of I. We prove that 𝒦 can be described as the solution set of a system of differential equations, that the whole bigraded structure of 𝒦 is characterized by the integral roots of certain b-functions, and that certain de Rham cohomology groups can give partial information about 𝒦.

On the growth of even K-groups of rings of integers in p-adic Lie extensions

Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficients in $\mathbb{Z}_p(i)$ for $i\geq 2$. We should mention that this latter approach is possible thanks to the Quillen-Lichtenbaum Conjecture which is now known to be valid by the works of Rost-Voevodsky. We also note that the approach allows us to work with more general $p$-adic Lie extensions that do not necessarily contain the cyclotomic $\mathbb{Z}_p$-extension, where the Kummer theoretical approach does not apply. Along the way, we study the torsionness of the second Iwasawa cohomology groups with coefficients in $\mathbb{Z}_p(i)$ for $i\geq 2$. Finally, we give examples of $p$-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial $\mu$-invariants.

Motivic Action on Coherent Cohomology of Hilbert Modular Varieties

Abstract We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and over ${\mathbb {C}}$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark’s conjecture on special values of Artin $L$-functions and by numerical evidence in base change cases.

On certain Hochschild cohomology groups for the small quantum group

We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for g=sl3, we obtain the H•(u,C)=C[N]-module structure of HH•(uλ).

On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let G be a semisimple and simply connected algebraic group G over an algebraically closed field of characteristic p>h, where h is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of G by G1 and g, respectively. First, we calculate the restricted cohomology of g with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of g is equivalent to the corresponding cohomology of G1, we describe them as the cohomology of G1 in terms of the cohomology for G1 with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms Hn(G1,V)≅Hn(G,V) and Hn(g,V)≅Hn(G,V), and a necessary condition for the isomorphism Hn(g,V)≅Hn(G1,V), where V is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of G, G1, and g with coefficients in the considered simple modules.

Lamplighters and the bounded cohomology of Thompson’s group

We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson’s group F. In particular, these groups are boundedly acyclic. Our method is ergodic and applies to “large” transformation groups where the Mather–Matsumoto–Morita method sometimes fails because not all are acyclic in the usual sense.

BOUNDED COHOMOLOGY AND BINATE GROUPS

AbstractA group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first nonamenable examples are the group of compactly supported homeomorphisms of$ {\mathbb {R}}^{n}$(Matsumoto–Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groupsF,T, andVis as simple as possible.

The third cohomology group of a monoid and admissible abstract kernels

We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].