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 ...
, specifically
geometry and topology
In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Ri ...
, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
Main themes
Overview
* Low-dimensional manifolds are classified by geometric structure; high-dimensional manifolds are classified algebraically, by
surgery theory
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 . Milnor called this technique ''surgery'', while An ...
.
: "Low dimensions" means dimensions up to 4; "high dimensions" means 5 or more dimensions. The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly (but not topologically); see
discussion of "low" versus "high" dimension.
* Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories.
* Positive curvature is constrained, negative curvature is generic.
* The abstract classification of high-dimensional manifolds is
ineffective: given two manifolds (presented as
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
es, for instance), there is no algorithm to determine if they are isomorphic.
Different categories and additional structure
Formally, classifying
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 ...
s is classifying objects up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
.
There are many different notions of "manifold", and corresponding notions of
"map between manifolds", each of which yields a different
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
and a different classification question.
These categories are related by
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
s: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor
.
These functors are in general neither one-to-one nor onto; these failures are generally referred to in terms of "structure", as follows. A topological manifold that is in the image of
is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold".
Thus given two categories, the two natural questions are:
* Which manifolds of a given type admit an additional structure?
* If it admits an additional structure, how many does it admit?
:More precisely, what is the structure of the set of additional structures?
In more general categories, this ''structure set'' has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.
Many of these structures are
G-structure
In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes var ...
s, and the question is
reduction of the structure group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes vario ...
. The most familiar example is orientability: some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
Enumeration versus invariants
There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants.
For instance, for orientable surfaces,
the
classification of surfaces
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
enumerates them as the connect sum of
tori, and an invariant that classifies them is the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
or
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
.
Manifolds have a rich set of invariants, including:
*
Point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
**
Compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
**
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...
* Classic
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
**
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
**
Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
**
Cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
*
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
** normal invariants (
orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
,
characteristic classes, and characteristic numbers)
**
Simple homotopy (
Reidemeister torsion)
**
Surgery theory
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 . Milnor called this technique ''surgery'', while An ...
Modern algebraic topology (beyond
cobordism
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 (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
theory), such as
Extraordinary (co)homology, is little-used
in the classification of manifolds, because these invariant are homotopy-invariant, and hence don't help with the finer classifications above homotopy type.
Cobordism groups (the bordism groups of a point) are computed, but the bordism groups of a space (such as
) are generally not.
Point-set
The point-set classification is basic—one generally fixes point-set assumptions and then studies that class of manifold.
The most frequently classified class of manifolds is closed, connected manifolds.
Being homogeneous (away from any boundary), manifolds have no local point-set invariants, other than their dimension and boundary versus interior, and the most used global point-set properties are compactness and connectedness. Conventional names for combinations of these are:
* A compact manifold is a compact manifold, possibly with boundary, and not necessarily connected (but necessarily with finitely many components).
* A closed manifold is a compact manifold without boundary, not necessarily connected.
* An open manifold is a manifold without boundary (not necessarily connected), with no compact component.
For instance,
_invariant,_and_thus_can_be_Effectively_computable">effectively_computed_given_a_CW_complex.html" ;"title="Homology (mathematics)">homological