Pontryagin Classes Research Articles

Almost complex torus manifolds - a problem of Petrie type

The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n T^n -action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.

Motivic Pontryagin classes and hyperbolic orientations

AbstractWe introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups , , , ). We show that hyperbolic orientations of ‐periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that ‐orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that ‐periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space . Finally, we construct the universal hyperbolically oriented ‐periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum .

Algebraic independence of topological Pontryagin classes

Abstract We show that the topological Pontryagin classes are algebraically independent in the rationalised cohomology of B ⁢ Top ⁢ ( d ) {B\mathrm{Top}(d)} for all d ≥ 4 {d\geq 4} .

Constructing the Virasoro Groups Using Differential Cohomology

Abstract The Virasoro groups are a family of central extensions of $\textrm{Diff}^+(S^1)$, the group of orientation-preserving diffeomorphisms of $S^1$, by the circle group $\mathbb T$. We give a novel, geometric construction of these central extensions using “off-diagonal” differential lifts of the first Pontryagin class, thus affirmatively answering a question of Freed–Hopkins.

The positive scalar curvature cobordism category

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside R+(Sd) if d≥6. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen’s Theorem B and instances of the Gromov–Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal’s theory of Γ-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen–Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes. A delooped version of the Atiyah–Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real K-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

The weak form of Hirzebruch’s prize question via rational surgery

We present a relatively elementary construction of a spin manifold with vanishing first rational Pontryagin class satisfying the conditions of Hirzebruch’s prize question, using a modification of Sullivan’s theorem for the realization of rational homotopy types by closed smooth manifolds. As such this is an alternative to the solutions of the problem given by Hopkins–Mahowald, though without the guarantee of the constructed manifold admitting a string structure. We present a particular solution which is rationally 7 connected with eighth Betti number equal to one; our approach yields many other solutions with complete knowledge of their rational homotopy type.

Rational Pontryagin classes of Euclidean fiber bundles

The rational Pontryagin classes, evaluated on fiber bundles where the fiber is a 2n-dimensional euclidean space, can be nonzero in cohomology dimensions much greater than 4n. This makes a striking contrast with the Pontryagin classes of vector bundles.

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.

The Hamilton–Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes

By using the Hamilton–Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental HJ differential where the characteristic equations and the symmetries of the theory are identified. Moreover, we work in both theories with the same phase space variables and we show that in spite of Pontryagin and Euler classes share the same equations of motion their symmetries are different. In addition, we report all HJ Hamiltonians and we compare our results with other formulations reported in the literature.

Vertical genera

The classical construction of a genus on a closed manifold is extended to the case of families of manifolds parametrised by a space X. The genera considered here are built from certain generalised Pontryagin and Chern classes as maps from bordism cohomology to the singular cohomology of X.

AST419 - Chiral Differential Operators via Batalin-Vilkovisky Quantization

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book Factorization algebras in quantum field theory, by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold.

A 15-Vertex Triangulation of the Quaternionic Projective Plane

In 1992, Brehm and Kühnel constructed an 8-dimensional simplicial complex \(M^8_{15}\) with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold “like a projective plane” in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to \({\mathbb {H}}P^2\). This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of \(M^8_{15}\). As a result, we obtain that it is indeed a minimal triangulation of \({\mathbb {H}}P^2\).

On the composition and exterior products of double forms and p -pure manifolds

We translate into double forms formalism the basic Greub and Greub-Vanstone identities that were previously obtained in mixed exterior algebras. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; we show that the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We define and study a refinement of the notion of pure curvature of Maillot, namely $p$-pure curvature, and we use one of the basic identities to prove that if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its Pontryagin class of degree $4k$ vanishes.

H4(Co0; Z) = Z/24

AbstractWe show that the 4th integral cohomology of Conway’s group $\mathrm{Co}_0$ is a cyclic group of order $24$, generated by the 1st fractional Pontryagin class of the $24$-dimensional representation.

Shifted Symplectic Lie Algebroids

AbstractShifted symplectic Lie and $L_{\infty }$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological field theory defined by Alexandrov--Kontsevich--Schwarz--Zaboronsky. In this paper, we classify zero-, one-, and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric “higher structures”, such as Courant algebroids twisted by $\Omega ^{2}$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well-known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the $C^{\infty }$, holomorphic, and algebraic settings and are based on a number of technical results on the homotopy theory of $L_{\infty }$ algebroids and their differential forms, which may be of independent interest.

Spherical T-duality and the spherical Fourier–Mukai transform

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel-Whitney class. Unless $\mathrm{dim}(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincar\'e virtual line bundle $\mathcal{P}$ on $S^3 \times S^3$ (actually also for $S^n\times S^n$) and the spherical Fourier-Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial $S^3$-bundle. This is then used to prove that all spherical T-dualities induce natural degree-shifting isomorphisms between the 7-twisted K-theories of the pairs $(P,H)$ and $(\hat P, \hat H)$ when $\mathrm{dim}(M)\leq 4$, improving our earlier results.

The First Pontryagin Class of a Quadratic Lie 2-Algebroid

In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H^5(M). Associated to a CLWX 2-algebroid, there is a quadratic Lie 2-algebroid naturally. Conversely, we show that the first Pontryagin class of a quadratic Lie 2-algebroid is the obstruction class of the existence of a CLWX-extension. Finally we construct a quadratic Lie 2-algebroid from a trivial principle 2-bundle with a Gamma-connection and show that its first Pontryagin class is trivial.

Metric Inequalities with Scalar Curvature

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities appear as generalisations of the classical bounds on the distances between conjugates points in surfaces with positive sectional curvatures. The techniques of our proofs is based on the Schoen–Yau descent method via minimal hypersurfaces, while the overall logic of our arguments is inspired by and closely related to the torus splitting argument in Novikov’s proof of the topological invariance of the rational Pontryagin classes.

Faddeev–Jackiw quantization of topological invariants: Euler and Pontryagin classes

The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev–Jackiw context. The Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev–Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.

Comments on MacDowell–Mansouri gravity and torsion

Starting with the MacDowell–Mansouri formulation of gravity with a [Formula: see text] gauge group, we introduce new parameters into the action to include the nondynamical Holst term, and the topological Nieh–Yan and Pontryagin classes. Then, we consider the new parameters as fields and analyze the solutions coming from their equations of motion. The new fields introduce torsional contributions to the theory that modify Einstein’s equations.