Classification Of Manifolds

In mathematics, specifically geometry and topology, 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.
: "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 complexes, for instance), there is no algorithm to determine if they are isomorphic.

Different categories and additional structure

Formally, classifying manifolds is classifying objects up to isomorphism.
There are many different notions of "manifold", and corresponding notions of
"map between manifolds", each of which yields a different category and a different classification question.

These categories are related by forgetful functors: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor $\mbox \to \mbox$.

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 $\mbox \to \mbox$ 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-structures, and the question is reduction of the structure group. 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 enumerates them as the connect sum of $n \geq 0$ tori, and an invariant that classifies them is the genus or Euler characteristic.

Manifolds have a rich set of invariants, including:
* Point-set topology
** Compactness
** Connectedness
* Classic algebraic topology
** Euler characteristic
** Fundamental group
** Cohomology ring
* Geometric topology
** normal invariants (orientability, characteristic classes, and characteristic numbers)
** Simple homotopy ( Reidemeister torsion)
** Surgery theory

Modern algebraic topology (beyond cobordism 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 $MO_*(M)$) 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.

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Computability

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