Geometric topology
Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic.
[1] Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above.
Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?"
Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a homotopy of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery.
The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory.
In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality.
Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms.
A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds.
Handle decompositions of manifolds arise naturally via Morse theory.
The modification of handle structures is closely linked to Cerf theory.
Local flatness is a property of a submanifold in a topological manifold of larger dimension.
Brown and Mazur received the Veblen Prize for their independent proofs[2][3] of this theorem.
To gain further insight, mathematicians have generalized the knot concept in several ways.
Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.
The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not.
Surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by Milnor (1961).
It is a major tool in the study and classification of manifolds of dimension greater than 3.
The classification of exotic spheres by Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.
