mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, given two

submanifold 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 \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

s ''A'' and ''B'' of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

''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''.. Their existence and

embeddedness In economics and economic sociology, embeddedness refers to the degree to which economic activity is constrained by non-economic institutions. The term was created by economic historian Karl Polanyi as part of his substantivist approach. Polanyi ...

is crucial in proving the

h-cobordism theorem In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M ...

, 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 handle 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 "fle ...

s are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological

four-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. T ...

s. Pseudoholomorphic Whitney discs are counted by the differential in

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

intersection

Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...

References

{{topology-stub Geometric topology