In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, differential topology is the field dealing with the
topological properties and
smooth properties of
smooth manifolds. In this sense differential topology is distinct from the closely related field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its
homotopy type, or the structure of its
diffeomorphism group. Because many of these coarser properties may be captured
algebraically, differential topology has strong links to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
The central goal of the field of differential topology is the
classification of all smooth manifolds up to
diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (
connected) manifolds in each dimension separately:
* In dimension 1, the only smooth manifolds up to diffeomorphism are the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, the
real number line, and allowing a
boundary, the half-closed
interval