相关论文: Principal fibrations from noncommutative spheres
Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within…
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional…
We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential…
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We…
The irreducible *-representations of the polynomial algebra O(S^3_{pq}) of the quantum 3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C*-algebra are shown to coincide with their classical…
We obtain explicit formulas for the trivialization functions of the ${\rm SU}(3)$ principal bundle $G_2 \to S^6$ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the…
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres in Connes and Landi [8] and in Connes and Dubois Violette [7], by using the differential and integral calculus on these spaces that is…
M. Kontsevich constructed universal characteristic classes of smooth bundles with fiber a framed odd-dimensional integral homology sphere. In dimension 3, they are known to give a universal finite type invariants of homology 3-spheres.…
In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert…
We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum…
Singular fibrations generalize achiral Lefschetz fibrations of 4-manifolds over surfaces while sharing some of their properties. For instance, relatively minimal singular fibrations are determined by their monodromy. We explain how to…
Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection…
In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to which extend the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by 8) which can be defined…
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature.…
In this short article we review how the classical theory of principal fibre bundles (PFB) transcribes in an algebraic formalism. In this dual formulation, a PFB is given by a right co-module algebra ${\cal P}$ over a Hopf algebra ${\cal H}$…
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible…
We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that…
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An…
We provide a modular construction of the Laza--Sacc\`a--Voisin compactification of the intermediate Jacobian fibration of a cubic fourfold. Additionally, we construct infinitely many $20$-dimensional families of polarized hyper-K\"ahler…
The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal…