In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
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 on objects; 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.
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 connected 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 (beyond cobordism theory), such as Extraordinary (co)homology, is little-used in the classification of manifolds, because these invariants 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.
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:
For instance, is a compact manifold, is a closed manifold, and is an open manifold, while is none of these.
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 (they are not a complete set of invariants): for instance, orientable 3-manifolds are parallelizable (Steenrod's theorem in low-dimensional topology), 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 (homeomorphic, diffeomorphic). This is due to the unsolvability of the word problem for groups, or more precisely, the triviality problem (given a finite presentation for a group, is it the trivial group?). 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 (or higher). 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 .
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 (and a single non-orientable) closed surface with positive curvature (the sphere and projective plane), and likewise for zero curvature (the torus and the Klein bottle), 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.
Thus dimension 4 differentiable manifolds are the most complicated: they are neither geometrizable (as in lower dimension), nor are they classified by surgery (as in higher dimension or topologically), 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 (cut into geometrizable pieces as in 3 dimensions, into symplectic manifolds, and so forth). 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 (but for different reasons, or via different proofs) and in what ways are they unusual?
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 compact 1-dimensional manifold without boundary is homeomorphic (or diffeomorphic if it is smooth) to the circle. A second countable, non-compact 1-dimensional manifold is homeomorphic or diffeomorphic to the real line. Dropping the assumption of second countability one gets two additional manifolds: the long line, and a space formed from a ray of the real line and a ray of the long line meeting at a point. [1]
The study of maps of 1-dimensional manifolds are a non-trivial area. For example:
Every connected closed 2-dimensional manifold (surface) admits a constant curvature metric, by the uniformization theorem. [3] There are 3 such curvatures (positive, zero, and negative). This is a classical result, and as stated, easy (the full uniformization theorem is subtler). 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. While the classification of surfaces is classical, maps of surfaces is an active area; see below.
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 (the Solution of the Poincaré conjecture) is analytic, not topological.
Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), 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 (topological) 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.
In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory.
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.
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".
Analogously to the classification of manifolds, in high codimension (meaning more than 2), embeddings are classified by surgery, while in low codimension or in relative dimension, they are rigid and geometric, and in the middle (codimension 2), one has a difficult exotic theory (knot 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 Matthias Kreck [4]
In 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, 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.
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 solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Topology is the part of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
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 up to homotopy equivalence.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
In mathematics, the Chern theorem states that the Euler–Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogous to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
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 John Milnor. Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is
In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after Michael Spivak.
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 Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X, a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.