Natural operations on differential forms

Navarro, José; Sancho, J. B.

doi:10.1016/j.difgeo.2014.12.003

Cited by 10 publications

(11 citation statements)

References 7 publications

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“…The proof closely follows the one given in [10] for the case of differentiable manifolds, which, in turn, applies techniques already used by the Czech school ( [7], [8]).…”

Section: Introductionmentioning

confidence: 78%

“…Quite recently, there have been a couple of reformulations of this result ( [6], [10]) that, essentially, confirm that the only natural operation between differential forms is the exterior differential. These theorems have been used for several different applications ( [4], [6], [9], [10], [15]). As an example, these ideas have been succesfully extended to study differential operations on forms on contact manifolds, allowing a deeper understanding of the Rumin operator ( [2]).…”

Section: Introductionmentioning

confidence: 88%

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Natural operations on holomorphic forms

Navarro¹,

Navarro²,

Prieto³

2018

Arch. Math. (Brno)

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We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalenceà la Galois.MSC : 58A32, 32L05.

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“…The proof closely follows the one given in [10] for the case of differentiable manifolds, which, in turn, applies techniques already used by the Czech school ( [7], [8]).…”

Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 88%

Natural operations on holomorphic forms

Navarro¹,

Navarro²,

Prieto³

2018

Arch. Math. (Brno)

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“…More general operations in a much broader context are studied in [KMS93], but the operations relevant to us are still linear (and we are interested in nonlinear ones as well); there it is shown that all operations that raise the form degree by one are multiples of the exterior derivative, and linearity follows from naturality. More recently, operations (both linear and nonlinear) acting on 1-forms (connections) were considered in [FH13], and generalized to differential forms of all degrees in [NS15]. We will make use of this for our construction of cohomology operations on closed differential forms Ω * cl in stacks.…”

Section: Introductionmentioning

confidence: 99%

Primary operations in differential cohomology

Grady

Sati

2018

Advances in Mathematics

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We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U (1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a Künneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics.where B n U (1) ∇ represents the moduli stack of n-bundles equipped with connection, studied in [FSSt12][FSS13] [FSS15a]. The homotopy classes of such morphisms will in turn be describe the differential cohomology groupOne of the main goals of this paper is to characterize this group for various values of k and n. This in turn will lead to to a full characterization of primary cohomology operations in differential cohomology.Since differential cohomology operations, as we will see, involve various coefficients, we find it useful to point out the interrelations that already exist between these (we found the discussion in [FFG86] particularly useful). This should also help us develop some intuition for the full differential case. Note that for coefficients being one of Z, Z/p or Q i.e. an abelian group, then the set of all cohomology operations H m (K(G, n); G), where G and G ′ are from the above set, will also be abelian.Operations from Z/p to Q. We know that H q (K(G, n); Q) = 0 for all q > 0 when G is a finite abelian group, i.e. for us Z/p. This shows that there are nontrivial cohomology operations from Z/p-coefficients to Q-coefficients.Operations from Z to Q. We will distinguish the odd and even cases. For the first, H q (K(Z, 2n+ 1); Q) is nonzero only for q = 2n + 1, where it is equal to Q, with generator the image of the fundamental class ι under the homomorphism r : H 2n+1 (K(Z, 2n+1); Z) → H 2n+1 (K(Z, 2n+1); Q) induced by the natural embedding Z ֒→ Q. Thus, every operation from an odd-dimensional integral class to rational cohomology preserves the dimension, i.e. takes α ∈ H 2n+1 (X; Z) to λr(α) ∈ H 2n+1 (X; Q) for some fixed rational number λ corresponding to the operation. In the even case, H q (K(Z, 2n); Q) = Q[r(ι)], so that every operation from even integral cohomology to rational cohomology is given as the power α → λα k , where k ∈ Z, λ ∈ Q are determined by the operation.Operations from Q to Q. The group H n (K(Q, m); Q) can be straightforwardly calculated via e.g. the Serre spectral sequence. In this case, one has that any cohomology operation assigns to an element α ∈ H n (X; Q) the element λα k ∈ H nk (X; Q), where k ∈ Z, λ ∈ Q both fixed by the operation.Operations from H * (−) dR to H * (−) dR . On the other hand, operations in de Rham cohomology can be deduced from those on rational cohomology via the de Rham theorem. Hence, de Rham operations should be systematically characterized. The de Rham cohomology groups and the rational cohomology groups hav...

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“…Again, more operators appear, for instance ω → ω ∧ ω or ω → ω ∧ P a,i ω ∧ dω. Using the Peetre-Slovák theorem as in [19], one can show that a natural operator (satisfying an additional regularity assumption) on ContMan 2n+1 must be a differential operator. We do not have a complete description of these operators.…”

Section: Introductionmentioning

confidence: 99%

“…We do not have a complete description of these operators. v) One could also, as in [19], consider natural operations on k-tuples of forms. There are many of them, for instance (ω 1 , ω 2 ) → P a 1 +a 2 ,i (ω 1 ∧ ω 2 ) or (ω 1 , ω 2 ) → d(ω 1 ∧ P a 2 ,i ω 2 ).…”

Section: Introductionmentioning

confidence: 99%