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

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.
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:
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 tori, and an invariant that classifies them is the genus or Euler characteristic.
Manifolds have a rich set of invariants, including:
Modern algebraic topology, such as
Extraordinary 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 are computed, but the bordism groups of a space 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, 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:
For instance, is a compact manifold, is a closed manifold, and is an open manifold, while is none of these.

Computability

The Euler characteristic is a homological invariant, and thus can be effectively computed given a CW structure, so 2-manifolds are classified homologically.
Characteristic classes and characteristic numbers are the corresponding generalized homological invariants, but they do not classify manifolds in higher dimension : for instance, orientable 3-manifolds are parallelizable, so all characteristic classes vanish. In higher dimensions, characteristic classes do not in general vanish, and provide useful but not complete data.
Manifolds in dimension 4 and above cannot be effectively classified: given two n-manifolds presented as CW complexes or handlebodies, there is no algorithm for determining if they are isomorphic. This is due to the unsolvability of the word problem for groups, or more precisely, the triviality problem. Any finite presentation of a group can be realized as a 2-complex, and can be realized as the 2-skeleton of a 4-manifold. Thus one cannot even compute the fundamental group of a given high-dimensional manifold, much less a classification.
This ineffectiveness is a fundamental reason why surgery theory does not classify manifolds up to homeomorphism. Instead, for any fixed manifold M it classifies pairs with N a manifold and a homotopy equivalence, two such pairs, and, being regarded as equivalent if there exist a homeomorphism and a homotopy.

Positive curvature is constrained, negative curvature is generic

Many classical theorems in Riemannian geometry show that manifolds with positive curvature are constrained, most dramatically the 1/4-pinched sphere theorem. Conversely, negative curvature is generic: for instance, any manifold of dimension admits a metric with negative Ricci curvature.
This phenomenon is evident already for surfaces: there is a single orientable closed surface with positive curvature,
and likewise for zero curvature, and all surfaces of higher genus admit negative curvature metrics only.
Similarly for 3-manifolds: of the 8 geometries,
all but hyperbolic are quite constrained.

Overview by dimension

Thus dimension 4 differentiable manifolds are the most complicated:
they are neither geometrizable,
nor are they classified by surgery,
and they exhibit unusual phenomena, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Notably, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.
One can take a low-dimensional point of view on high-dimensional manifolds
and ask "Which high-dimensional manifolds are geometrizable?",
for various notions of geometrizable. In dimension 4 and above not all manifolds
are geometrizable, but they are an interesting class.
Conversely, one can take a high-dimensional point of view on low-dimensional manifolds
and ask "What does surgery predict for low-dimensional manifolds?",
meaning "If surgery worked in low dimensions, what would low-dimensional manifolds look like?"
One can then compare the actual theory of low-dimensional manifolds
to the low-dimensional analog of high-dimensional manifolds,
and see if low-dimensional manifolds behave "as you would expect":
in what ways do they behave like high-dimensional manifolds
and in what ways are they unusual?

Dimensions 0 and 1: trivial

There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics.
A connected 1-dimensional manifold without boundary is either the circle or the real line.
However, maps of 1-dimensional manifolds are a non-trivial area; see below.

Dimensions 2 and 3: geometrizable

Every connected closed 2-dimensional manifold admits a constant curvature metric, by the uniformization theorem. There are 3 such curvatures.
This is a classical result, and as stated, easy. The study of surfaces is deeply connected with complex analysis and algebraic geometry, as every orientable surface can be considered a Riemann surface or complex algebraic curve.
Every closed 3-dimensional manifold can be cut into pieces that are geometrizable, by the geometrization conjecture, and there are 8 such geometries.
This is a recent result, and quite difficult. The proof is analytic, not topological.
While the classification of surfaces is classical, maps of surfaces is an active area; see below.

Dimension 4: exotic

Four-dimensional manifolds are the most unusual: they are not geometrizable, and surgery works topologically, but not differentiably.
Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?"
Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4.
Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

Dimension 5 and more: surgery

In dimension 5 and above, manifolds are classified by surgery theory.
requires 2+1 dimensions, hence the two Whitney disks of surgery theory require 2+2+1=5 dimensions.
The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above, by general position.
In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension.
More subtly, dimension 5 is the cut-off because the middle dimension has codimension more than 2: when the codimension is 2, one encounters knot theory, but when the codimension is more than 2, embedding theory is tractable, via the calculus of functors. This is discussed further below.

Maps between manifolds

From the point of view of category theory, the classification of manifolds is one piece of understanding the category: it's classifying the objects. The other question is classifying maps of manifolds up to various equivalences, and there are many results and open questions in this area.
For maps, the appropriate notion of "low dimension" is for some purposes "self maps of low-dimensional manifolds", and for other purposes "low codimension".

Low-dimensional self-maps

Analogously to the classification of manifolds, in high codimension, embeddings are classified by surgery, while in low codimension or in relative dimension, they are rigid and geometric, and in the middle, one has a difficult exotic theory.
Particularly topologically interesting classes of maps include embeddings, immersions, and submersions.
Geometrically interesting are isometries and isometric immersions.
Fundamental results in embeddings and immersions include:
Key tools in studying these maps are:
One may classify maps up to various equivalences:
Diffeomorphisms up to cobordism have been classified by :