Cohomology Groups Research Articles

Embedding Topological Local Group Cohomology in a Reduced Cohomology Topological Group}

Embedding Topological Local Group Cohomology in a Reduced Cohomology Topological Group}

Complexity of Supersymmetric Systems and the Cohomology Problem

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA-complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k-local Cohomology problem and showing that it is QMA1-hard and, for a large class of instances, is contained in QMA. We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1, and for a large class of instances is contained in BQP. In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l-dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.

Hyperbolicity and bounded-valued cohomology

We generalise a theorem of Gersten on surjectivity of the restriction map in ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^{\\infty }$$\\end{document}-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^{\\infty }$$\\end{document}-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type FP2(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$FP_2({{\\mathbb {Q}}})$$\\end{document} and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Martínez-Pedroza.

Congruence modules in higher codimension and zeta lines in Galois cohomology

This article builds on recent work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms.

Sections and Unirulings of Families over $\mathbb{P}^{1}$

AbstractWe consider morphisms $\pi : X \to \mathbb{P}^{1}$ of smooth projective varieties over $\mathbb{C}$. We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π.To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

Tautological characteristic classes II: the Witt class

AbstractLet K be an arbitrary infinite field. The cohomology group $$H^{2}(SL(2,K);H_2 SL(2,K))$$ H 2 ( S L ( 2 , K ) ; H 2 S L ( 2 , K ) ) contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in SL(2, K) it is useful to have classes stable under deformations (Fenchel-Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovàř. The Milnor-Wood inequality asserts that an $$SL(2,\textbf{R})$$ S L ( 2 , R ) -bundle over a surface of genus g admits a flat structure if and only if its Euler number is $$\le (g - 1)$$ ≤ ( g - 1 ) . We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.

Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

We study ’t Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the “anomaly cocycle” using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group H^3(G,U(1)) × H^2(G,U(1))H3(G,U(1))×H2(G,U(1)). The first factor captures the anomaly in the internal symmetry group G, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving G and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.

Reflection Vectors and Quantum Cohomology of Blowups

Let $X$ be a smooth projective variety with a semisimple quantum cohomology. It is known that the blowup $\operatorname{Bl}_{\rm pt}(X)$ of $X$ at one point also has semisimple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ is a reflectiongroup. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ depending only on the geometry of the exceptional divisor.

Calculations of the Euler characteristic of the Coxeter cohomology of symmetric groups

Calculations of the Euler characteristic of the Coxeter cohomology of symmetric groups

Off-shell fields and conserved currents

We study interactions of higher-spin massless fields φ with conserved currents multilinear in the off-shell matter fields ϕ. Specifically, we focus on the 3d case where a slight modification of the σ−-cohomology technique developed earlier is directly applicable to control nontriviality of the interaction vertices at the convention that the vertices removable by a local field redefinition of a higher-spin field or having schematic form F(φ)G(ϕ), where F(φ) is a gauge invariant field strength of a free higher-spin field, are called deformationally trivial. It is demonstrated how the σ−-cohomology approach can be applied to the analysis of nonlinear vertices. Generally, deformationally trivial vertices are not σ−-closed while the deformationally non-trivial ones must be in H(σ−). It is shown that, at least in the 3d case, the relevant cohomology group H1(σ−)=0 and, hence, no deformationally non-trivial off-shell vertices exist. On the other hand, there exists an infinite class of deformationally trivial vertices, that includes the vertex recently proposed for spin three. Our analysis goes beyond the higher-spin vertices allowing to show that, at least in three dimensions, nonlinear combinations of the off-shell scalar fields and their derivatives cannot obey non-trivial equations.

HKT Manifolds: Hodge Theory, Formality and Balanced Metrics

ABSTRACT Let $(M,I,J,K,\Omega)$ be a compact HKT manifold, and let us denote with $\partial$ the conjugate Dolbeault operator with respect to I, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$, where Λ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold, the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$ structure $(I,J,K,\Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.

On deformations of Rota–Baxter algebra morphisms

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

A decomposition theorem for 0-cycles and applications to class field theory

We describe the abelian {e}tale fundamental group with modulus in terms of 0-cycles on a class of smooth quasi-projective varieties over finite fields which may not admit smooth compactifications. Using a limit process, this yields a cycle-theoretic analogue of the $K$-theoretic reciprocity theorem of Kato-Saito for such varieties. The proof is based on a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme in terms of the Chow groups with and without modulus of the scheme. This also extends the decomposition theorem of Binda-Krishna to smooth schemes over imperfect fields.

Maximality Properties of Generalized Springer Representations of SO (N, ℂ)

Abstract Let $C$ be a unipotent class of $G=\textrm{SO}(N,\mathbb{C})$, $\mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $\rho (C,\mathcal{E})$ appears in the top cohomology of some variety. Let $\bar \rho (C,\mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $\rho (C,\mathcal{E})$ appears in $\bar \rho (C,\mathcal{E})$ with multiplicity $1$ and that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $\bar \rho (C,\mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $\bar \rho (C,\mathcal{E})$ (resp. $\textrm{sgn}\otimes \bar \rho (C,\mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $\rho ^{\textrm{max}}$ (resp. “minimal” subrepresentation $\textrm{sgn}\otimes \rho ^{\textrm{max}}$), where $\textrm{sgn}$ is the sign representation. These are analogues of results for $\textrm{Sp}(2n,\mathbb{C})$ by Waldspurger.

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

AbstractIn this paper, we investigate the rigidity and vanishing properties of compact submanifolds with free boundary of arbitrary codimension in the unit ball. We first show that a minimal submanifold with free boundary in the unit ball satisfying a pointwise or integral curvature pinching condition on the second fundamental form is a flat equatorial disk. Then we prove a vanishing theorem for cohomology groups for submanifolds with free boundary in the unit ball under an integral curvature pinching condition on the trace‐free second fundamental form.

Cohomology and geometry of Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} of Deligne–Lusztig varieties X(w) for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}, for all elements w in the Weyl group. As a consequence, we obtain the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}. Moreover, using our result for X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} and a spectral sequence associated to a stratification of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}, we deduce the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}. Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.

Cohomology and homotopy of Lie triple systems

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems.

Representations and formal deformations of Leibniz H-pseudoalgebras

In this paper, we first give some constructions of right (resp. left) Leibniz H-pseudoalgebras and explore the characterization of symmetric Leibniz H-pseudoalgebras. Furthermore, we introduce the notion of relative Rota-Baxter H-operator and show that a right (resp. left) Leibniz H-pseudoalgebra together with a relative Rota-Baxter H-operator gives rise to a representation of related right (resp. left) Leibniz H-pseudoalgebra. Finally, we describe the central extension and the formal deformation in terms of the corresponding second cohomology group.

A refinement of Morse–Novikov cohomology on manifolds with boundary and the cohomology of the space of solutions of kerΔ𝜃

In this paper, we consider the Morse–Novikov coboundary operator [Formula: see text] where [Formula: see text], and which is given by [Formula: see text], and for which the associated Morse–Novikov cohomology groups are denoted by [Formula: see text]. The ultimate objective of this paper is to uncover more geometric and topological insights about [Formula: see text] when the boundary of [Formula: see text] is nonvanishing. To begin, we employ a different approach to prove that the concrete realizations of the Morse–Novikov cohomology groups of [Formula: see text] intersect at the origin, and then we use the long exact sequence and cohomological algebra as powerful tools to decompose the absolute and relative Morse–Novikov cohomologies into interior and boundary portions, respectively. On the other hand, we use a novel reasoning based on cohomological algebra to determine the cohomology of [Formula: see text], and as a result the absolute Morse–Novikov cohomology of [Formula: see text] can be characterized in terms of two successive degrees in just one degree of the cohomology of [Formula: see text]. Consequently, the cohomology of [Formula: see text] can be recovered from the interior and boundary sections.

On the Čech cohomology of Morse boundaries

We consider cusped hyperbolic n–manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most n−3 and does not vanish in dimension n−2. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank–1 simple Lie groups.