Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents.

Some geometric obstructions to warped product immersions in locally conformal Kaehler space forms

Being motivated by a well-known Nash’s embedding theorem, Chen introduced a method to discover the relationship for the extrinsic invariants controlled by the intrinsic one. In this paper, we extend Chen’s inequality for the intrinsic and extrinsic invariants for pointwise bi-slant warped products in locally conformal Kaehler space forms with quarter-symmetric and semi-symmetric connections. The equality case of the inequality is also investigated. Several applications of the inequality are given. Furthermore, we provide two non-trivial examples of such immersions.

The Disc Embedding Theorem

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

The Collar Adding Lemma

The collar adding lemma is a key ingredient in the proof of the disc embedding theorem. Specifically, it proves that a skyscraper with an added collar is homeomorphic to the standard 4-dimensional 2-handle. The proof is similar to the proof in a previous chapter that the Alexander gored ball with an added collar is homeomorphic to the standard 3-ball. Roughly speaking, a skyscraper is seen as the quotient space of the 4-ball corresponding to a certain decomposition. The added collar allows the decomposition to be modified so that the resulting decomposition shrinks; that is, the corresponding quotient space, which is identified with the skyscraper with an added collar, is homeomorphic to the original 4-ball.

Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds

Surgery theory and the classification of simply connected 4-manifolds comprise two key consequences of the disc embedding theorem. The chapter begins with an introduction to surgery theory from the perspective of 4-manifolds. In particular, the terms and maps in the surgery sequence are defined, and an explanation is given as to how the sphere embedding theorem, with the added ingredient of topological transversality, can be used to define the maps in the surgery sequence and show that it is exact. The surgery sequence is applied to classify simply connected closed 4-manifolds up to homeomorphism. The chapter closes with a survey of related classification results.

Shrinking Starlike Sets

‘Shrinking Starlike Sets’ presents a proof that null decompositions with recursively starlike-equivalent elements shrink. Unlike in previous chapters, the shrink is produced abstractly rather than explicitly. The chapter proceeds in steps, first establishing the shrinking of null decompositions with starlike elements, then of those with starlike-equivalent elements, and then finally of those with recursively starlike-equivalent elements. In the eventual proof of the disc embedding theorem, a decomposition of the 4-ball consisting of recursively starlike-equivalent elements will be produced. These consist of the ‘holes’, along with certain ‘red blood cell discs’. The result from this chapter will be used to show that the decomposition shrinks; this is called the α-shrink.

The Ball to Ball Theorem

The ball to ball theorem is presented, which states that a map from the 4-ball to itself, restricting to a homeomorphism on the 3-sphere, whose inverse sets are null and have nowhere dense image, is approximable by homeomorphisms relative to the boundary. The approximating homeomorphisms are produced abstractly, as in the previous chapter, with no need to investigate the decomposition elements further. In the proof of the disc embedding theorem, a decomposition of the 4-ball will be constructed, called the gaps+ decomposition. The ball to ball theorem will be used to prove that this decomposition shrinks; this is called the β-shrink.

The Schoenflies Theorem after Mazur, Morse, and Brown

‘The Schoenflies Theorem after Mazur, Morse, and Brown’ provides two proofs of the Schoenflies theorem. The Schoenflies theorem states that every bicollared embedding of an (n – 1)-sphere in the n-sphere splits the n-sphere into two balls. This chapter provides two proofs. The first is due to Mazur and Morse; it utilizes an infinite ‘swindle’ and a classical technique called push-pull. The second proof, due to Brown, serves as an introduction to shrinking, or decomposition space theory. The latter is a beautiful, but outmoded, branch of topology that can be used to produce non-differentiable homeomorphisms between manifolds, especially from a manifold to a quotient space. Techniques from decomposition space theory are essential in the proof of the disc embedding theorem.