Computable Constants for Korn’s Inequalities on Riemannian Manifolds

A method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.

Quantum Cosmology with Third Quantisation

We reviewed the canonical quantisation of the geometry of the spacetime in the cases of a simply and a non-simply connected manifold. In the former, we analysed the information contained in the solutions of the Wheeler–DeWitt equation and showed their interpretation in terms of the customary boundary conditions that are typically imposed on the semiclassical wave functions. In particular, we reviewed three different paradigms for the quantum creation of a homogeneous and isotropic universe. For the quantisation of a non-simply connected manifold, the best framework is the third quantisation formalism, in which the wave function of the universe is seen as a field that propagates in the space of Riemannian 3-geometries, which turns out to be isomorphic to a (part of a) 1+5 Minkowski spacetime. Thus, the quantisation of the wave function follows the customary formalism of a quantum field theory. A general review of the formalism is given, and the creation of the universes is analysed, including their initial expansion and the appearance of matter after inflation. These features are presented in more detail in the case of a homogeneous and isotropic universe. The main conclusion in both cases is that the most natural way in which the universes should be created is in entangled universe–antiuniverse pairs.

ON THE TRANSITIVITY OF THE GROUP OF ORBIFOLD DIFFEOMORPHISMS

Consider a connected manifold of dimension at least two and the group of compactly supported diffeomorphisms that are isotopic to the identity through a compactly supported isotopy. This group acts n-transitively: any n-tuple of points can be moved to any other n-tuple by an element of this group. The group of diffeomorphisms of an orbifold is typically not n-transitive: simple obstructions are given by isomorphism classes of isotropy groups of points. In this paper we investigate the transitivity properties of the group of compactly supported diffeomorphisms of orbifolds that are isotopic to the identity through a compactly supported isotopy. We also study an example in the category of area preserving mappings.

In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T*Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

Genus Integration, Abelianization, and Extended Monodromy

Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in [9]. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of [6], but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

The covariant derivative and the curvature

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the chapter summarizes the elementary introduction to the exterior calculus of Chapter 5 of Book 2. Next, the chapter studies the Lie bracket and Lie derivative, before moving on to the covariant derivative and a connected manifold. The covariant derivative in particular is introduced to ensure the effectiveness of the Lie brackets and the Lie derivative. From here, this chapter considers the torsion of a covariant derivative and finally to the curvature of a covariant derivative.

Representation stability for homotopy groups of configuration spaces

We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of [6], and that their derived dual integral homotopy groups are finitely generated as{{\mathsf{FI}}}-modules in the sense of [4]. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-{{\mathsf{FI}}}-space to properties of its dual homotopy groups. We also discuss several other applications, including free Lie and Gerstenhaber algebras.

On Stacked Triangulated Manifolds

We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $\Delta$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $\Delta$ is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.