Whitney disk

Last updated April 29, 2019

In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.^[1]

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

Manifold topological space that at each point resembles Euclidean space

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

Their existence and embeddedness is crucial in proving the cobordism theorem, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc. Casson handles are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological four-manifolds.

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and Michael Freedman (1982) introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifolds.

Pseudoholomorphic Whitney discs are counted by the differential in Lagrangian intersection Floer homology.

In mathematics, a Lagrangian system is a pair $(Y, L)$ , consisting of a smooth fiber bundle $Y \to X$ and a Lagrangian density $L$ , which yields the Euler–Lagrange differential operator acting on sections of $Y \to X$ .

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analog of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Hamiltonian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

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In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X|_p, Y|_p) is a smooth function. The family g_p of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

Cobordism (n+1)-dimensional manifold-with-boundary W linking two n-dimensional manifolds M and N, in the sense that the boundary of W consists of M and N

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R³ passing through the origin.

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

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In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold $into projective space. An ample line bundle is one such that some positive power is very ample. Globally generated sheaves are those with enough sections to define a morphism to projective space.$

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Originally developed for differentiable manifolds, surgery techniques also apply to PL and topological manifolds.

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

References

↑ Scorpan, Alexandru (2005), The Wild World of 4-manifolds, American Mathematical Society, p. 560, ISBN 9780821837498 .

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[1] Scorpan, Alexandru (2005), The Wild World of 4-manifolds, American Mathematical Society, p. 560, ISBN 9780821837498 .