Extended Regge Complex for Linearized Riemann-Cartan Geometry and Cohomology

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech–de Rham spectral sequence by making use of the factor map which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus T θ T_\theta is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over T θ T_\theta .

The title refers to the nilcommutative or NC -schemes introduced by M. Kapranov in ‘Noncommutative Geometry Based on Commutator Expansions’, J. Reine Angew. Math 505 (1998) 73–118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson or NP -schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC - and NP -schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas , Masson, Paris (1968), pp. 306–358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.

In this chapter, we study the topology of C ∞-manifolds. We define the de Rham cohomology of a manifold, which is the vector space of closed differential forms modulo exact forms. After sheafifying the construction, we see that the de Rham complex forms a so-called acyclic resolution of the constant sheaf ℝ. We prove a general result that sheaf cohomology can be computed using such resolutions, and deduce a version of de Rham’s theorem that de Rham cohomology is sheaf cohomology with coefficients in ℝ. It follows that de Rham cohomology depends only on the underlying topology. Using a different acyclic resolution that is dual to the de Rham complex, we prove Poincaré duality. This duality makes cohomology, which is normally contravariant, into a covariant theory. We devote a section to explaining these somewhat mysterious covariant maps, called Gysin maps. We end this chapter with the remarkable Lefschetz trace formula, which in principle, calculates the number of fixed points for a map of a manifold to itself.

The principal objective in this chapter is a proof of the de Rham theorem, one version of which we have stated in 4.17. In its most complete form it asserts that the homomorphism from the de Rham cohomology ring to the differentiable singular cohomology ring given by integration of closed forms over differentiable singular cycles is a ring isomorphism. The approach will be to exhibit both the de Rham cohomology and the differentiable singular cohomology as special cases of sheaf cohomology and to use a basic unique­ness theorem for homomorphisms of sheaf cohomology theories to prove that the natural homomorphism between the de Rham and differentiable singular theories is an isomorphism. As an added dividend of this approach we shall also obtain the existence of canonical isomorphisms of the de Rham and differentiable singular cohomology theories with the continuous singular theory, the Alexander-Spanier theory, and the Čech cohomology theory for differentiable manifolds. From these isomorphisms we shall conclude that the de Rham cohomology theory is a topological invariant of a differentiable manifold.

Geometry of the nonabelian 2-index potential and twisted de rham cohomology

Simplicial de rham cohomology and characteristic classes of flat bundles

Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds

We study ‐gauges for de Rham cohomology of smooth algebraic varieties in characteristic . Applying to good reductions of Shimura varieties of Hodge type, we recover the Ekedahl–Oort stratifications by constructing universal de Rham ‐gauges with ‐structure. We also study the cohomology of de Rham ‐gauges on these varieties. In particular, in the PEL‐type case and when the weights of the flat automorphic vector bundles are ‐small, we determine the ‐gauge structure on their de Rham cohomology by the associated dual Bernstein‐Gelfand‐Gelfand (BGG) complexes.

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between K K -theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.

General perversities and [formula omitted] de Rham and Hodge theorems for stratified pseudomanifolds

Simplicial cochain algebras for diffeological spaces

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R\rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to ${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

The space Γ_X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ_X , equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.

D functions used to construct the stiffness matrix of a finite element should possess the following properties. 1) Infinitesimal rigid body motions should be accurately represented. If this requirement is not met the conditions of equilibrium of the element are not satisfied.*• 2) The displacement functions should contain all the lower terms of a complete set of functions. This requirement insures monotonic convergence by mesh size reduction. 3) A minimum degree of interelement continuity must be maintained between adjacent elements. This minimum degree of compatibility must insure a perfect match for the inplane and the out of plane components of displacement. Also for the out of plane component, slopes tangent and normal to all common edges of two adjacent elements must match. This requirement then insures convergence to an exact result by mesh size reduction. The importance of the last two requirements is firmly established; however, the first requirement has been shown to be problem dependent. If the structure to be analyzed is so constrained that no element of the structure is ever going to undergo any rigid body motion, then obviously this requirement can be violated. For example, axisymmetric elements acted upon by axisymmetric loads need to have only one rigid body mode: a rigid translation parallel to the axis of symmetry. For this particular type of element a truncated cone as used by Grafton and Strome always includes a rigid body motion parallel to the longitudinal axis. However, if the axisymmetric element is to have curvature in the longitudinal direction, then all the rigid body modes are absent. Jones and Strome recognized such a deficiency and reintroduced a longitudinal translation in their element. Later, Stricklin et al. reported on a similar improved element but omitted the longitudinal rigid body motion altogether. This last element is capable of handling asymmetric loading, therefore it is not difficult to imagine a loading in which many elements would have to undergo considerable transverse motion; a cantilevered structure would lead to such a situation. Haisler and Stricklin studied the influence of longitudinal translation and observed that such a rigid motion is recuperated by mesh size reduction. For elements of rectangular aspects, Bogner, Fox, and Schmit developed a systematic method for constructing acceptable displacement fields. However, for curved cylindrical elements, only two rigid body modes are accounted for. The same authors reported on a (48 X 48) stiffness matrix and mentioned that an eigenvalue analysis of such a matrix indicated that rigid body motions were adequately represented. However, as pointed out in our study of curved cylindrical elements, rigid body motions cannot be represented by independent displacement components. In the same reference, the importance of these rigid body motions is clearly illustrated in several examples. However, the inclusion of rigid body motions was done at the expense of rigorous interelement compatibility. This compromise resulted in a significant improvement in the behavior of the element. In this paper we develop a method to include rigid body motions without comprising deformational compatibility. The method is general and can be applied without difficulties to any element, curved or flat. The improvements of a curved cylindrical element are illustrated with one example.

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge-theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.