Cohomological Obstruction Research Articles

Uniqueness of the extremal Schwarzschild de Sitter spacetime

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.

The modular class of a singular foliation

The modular class of a regular foliation is a cohomological obstruction to the existence of a volume form transverse to the leaves which is invariant under the flow of the vector fields of the foliation. It has been used in dynamical systems and operator theory, from which the name modular comes from. By drawing on the relationship between Lie algebroids and regular foliations, this paper extends the notion of modular class to the realm of singular foliations. The singularities are dealt with by replacing the singular foliations by any of their universal Lie ∞-algebroids, and by picking up the modular class of the latter. The geometric meaning of the modular class of a singular foliation F is not as transparent as for regular foliations: it is an obstruction to the existence of a universal Lie ∞-algebroid of F whose Berezinian line bundle is a trivial F-module. This paper illustrates the relevance of using universal Lie ∞-algebroids to extend mathematical notions from regular foliations to singular ones, thus paving the road to defining other characteristic classes in this way.

A cohomological obstruction in higher-dimensional Chern–Simons gauge theories

We study a set of cohomology classes which emerge in the cohomological formulations of the calculus of variations as obstructions to the existence of (global) solutions of the Euler–Lagrange equations of Chern–Simons gauge theories in higher dimensions [Formula: see text].

The fibration method over real function fields

Let mathbb R(C) be the function field of a smooth, irreducible projective curve over mathbb R. Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over mathbb R(C). Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over mathbb R(C)-valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

Graded rings of modular forms of rational weights

In a previous paper, for any odd integer $$N\ge 5$$ we constructed $$(N-1)/2$$ modular forms of $$\varGamma (N)$$ of rational weight $$(N-3)/2N$$ and proved that the graded rings of modular forms of weight $$\ell (N-3)/2N$$ ($$\ell \in {\mathbb {Z}}_{\ge 0}$$) are generated by our forms for $$N=5$$, 7, 9. The proof was given by a direct calculation of the structure of the ring. In this paper, we generalize the result to cases when $$N=11$$ and 13 by using Castelnuovo–Mumford criterion on normal generation and Fujita criterion on relations of sections of invertible sheaves. For this purpose, it is needed to handle modular forms of small weight where Riemann Roch theorem has cohomological obstruction. Both rings for $$N=11$$ and 13 are generated by 5 and 6 generators with 15 and 35 concrete fundamental relations, respectively. These relations also give equations of the corresponding modular varieties. We will show that the similar claim does not hold for $$N=15$$ and $$N=23$$. We also give remarks on relations of some theta constants and further problems.

Proof of Kobayashi's rank conjecture on Clifford–Klein forms

T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford-Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan-Chevalley-Koszul-Weil.

Extensions of hom-Lie color algebras

Abstract In this paper, we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie color algebra 𝔤 {\mathfrak{g}} by another hom-Lie color algebra 𝔥 {\mathfrak{h}} and discuss the case where 𝔥 {\mathfrak{h}} has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i.e., we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology.

Asymptotic properties of the Hitchin–Witten connection

We explore extensions to $\operatorname{SL}(n,\mathbb{C})$-Chern-Simons theory of some results obtained for $\operatorname{SU}(n)$-Chern-Simons theory via the asymptotic properties of the Hitchin connection and its relation to Toeplitz operators developed previously by the first named author. We define a formal Hitchin-Witten connection for the imaginary part $s$ of the quantum parameter $t = k+is$ and investigate the existence of a formal trivialisation. After reducing the problem to a recursive system of differential equations, we identify a cohomological obstruction to the existence of a solution. We explicitly find one for the first step, in the specific case of an operator of order $0$, and show in general the vanishing of a weakened version of the obstruction. We also find a solution of the whole recursion in the case of a surface of genus $1$.

Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford–Klein forms problem

In this article, we develop the algorithm for finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford–Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford–Klein forms. By a numerical experiment, we show that there is a large class of homogeneous spaces satisfying Tholozan’s condition.

Outer automorphisms of classical algebraic groups

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely $^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide explicit examples where the Tits class obstruction vanishes, and yet the group does not have outer automorphism. As a crucial tool, we use generic sums of algebras with involution.

Braid groups and discrete diffeomorphisms of the punctured disk

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.

English

We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra g by another hom-Lie algebra h and discuss the case where h has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology.

Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory

We relate the existence of Noether global conserved currents associated with locally variational field equations to the existence of global solutions for a local variational problem generating global equations. Both can be characterized as the vanishing of certain cohomology classes. In the case of a 3-dimensional Chern–Simons gauge theory, the variationally featured cohomological obstruction to the existence of global solutions is sharp and equivalent to the usual obstruction in terms of the Chern characteristic class for the flatness of a principal connection. We suggest a parallelism between the geometric interpretation of characteristic classes as obstruction to the existence of flat principal connections and the interpretation of certain de Rham cohomology classes to be the obstruction to the existence of global extremals for a local variational principle.

On the Cohomology of Contextuality

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.

A cohomological obstruction to the existence of compact Clifford–Klein forms

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq 1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q \geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply H. Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $\epsilon$-families of semisimple symmetric pairs.

Control of fixed points and existence and uniqueness of centric linking systems

A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermak's proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the $p$-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by $p$-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for $p=2$ in the presence of offenders.

Dynamics and the Cohomology of Measured Laminations

In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy.

Computational topology of equipartitions by hyperplanes

We compute a primary cohomological obstruction to the existence of an equipartition for $j$ mass distributions in $\mathbb{R}^d$ by two hyperplanes in the case $2d-3j = 1$. The central new result is that such an equipartition always exists if $d=6\cdot 2^k +2$ and $j=4\cdot 2^k+1$ which for $k=0$ reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. The theorem follows from a Borsuk-Ulam type result claiming the non-existence of a $\mathbb{D}_8$-equivariant map $f \colon S^{d}\times S^d\rightarrow S(W^{\oplus j})$ for an associated real $\mathbb{D}_8$-module $W$. This is an example of a genuine combinatorial geometric result which involves $\mathbb{Z}/4$-torsion in an essential way and cannot be obtained by the application of either Stiefel-Whitney classes or cohomological index theories with $\mathbb{Z}/2$ or $\mathbb{Z}$ coefficients. The method opens a possibility of developing an ``effective primary obstruction theory'' based on $G$-manifold complexes, with applications in geometric combinatorics, discrete and computational geometry, and computational algebraic topology.

Hodge theory for twisted differentials

Abstract We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodgetheoretical proof of the characterization of solvmanifolds in class C of Fujiki, first stated by D. Arapura.

Cohomological obstructions in locally variational field theories

We study cohomological obstructions to the existence of global conserved quantities. In particular, we show that, if a given local variational problem is supposed to admit global solutions, certain cohomology classes cannot appear as obstructions. Vice versa, we obtain a new type of cohomological obstruction to the existence of global solutions for a variational problem.