4-manifold
Encyclopedia
In mathematics
, 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 (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
4-manifolds are of importance in physics because, in General Relativity
, spacetime
is modeled as a pseudo-Riemannian 4-manifold.
on the middle dimensional homology. A famous theorem of implies that the homeomorphism
type of the manifold only depends on this intersection form, and on a Z/2Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form
and Kirby–Siebenmann invariant can arise, except that if the form is even the Kirby–Siebenmann invariant is the signature/8 (mod 2).
Examples:
Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is Z there is a classification similar to the one above using Hermitian forms over the group ring of Z. If the fundamental group is too large (for example, a free group on 2 generators) then Freedman's techniques seem to fail and very little is known about such manifolds.
For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.
A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures. First, the Kirby Siebenmann invariant must vanish.
In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surface
s, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R4; see exotic R4
.
Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds,
using Seiberg-Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifold
s, possibly with orientations reversed, have been disproved.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 4-manifold is a 4-dimensional topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
. A smooth 4-manifold is a 4-manifold with a smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
. 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 (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
4-manifolds are of importance in physics because, in General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
is modeled as a pseudo-Riemannian 4-manifold.
Topological 4-manifolds
The homotopy type of a simply connected compact 4-manifold only depends on the intersection formIntersection form (4-manifold)
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd cohomology group of the 4-manifold...
on the middle dimensional homology. A famous theorem of implies that the homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
type of the manifold only depends on this intersection form, and on a Z/2Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form
Unimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
and Kirby–Siebenmann invariant can arise, except that if the form is even the Kirby–Siebenmann invariant is the signature/8 (mod 2).
Examples:
- In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecturePoincaré conjectureIn mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
. - If the form is E8, this gives a manifold called the E8 manifoldE8 manifoldIn mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...
, a manifold not homeomorphic to any simplicial complex. - If the form is Z, there are two manifolds depending on the Kirby–Siebenmann invariant: one is 2 dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic (and with no smooth structure).
- When the rank of the form is greater than about 28, the number of positive definite unimodular forms starts to increase extremely rapidly with the rank, so there are huge numbers of corresponding simply connected topological 4-manifolds (most of which seem to be of almost no interest).
Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is Z there is a classification similar to the one above using Hermitian forms over the group ring of Z. If the fundamental group is too large (for example, a free group on 2 generators) then Freedman's techniques seem to fail and very little is known about such manifolds.
For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.
Smooth 4-manifolds
For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
- Which topological manifolds are smoothable?
- Classify the different smooth structures on a smoothable manifold.
There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures. First, the Kirby Siebenmann invariant must vanish.
- If the intersection form is definite Donaldson's theoremDonaldson's theoremIn mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
gives a complete answer: there is a smooth structure if and only if the form is diagonalizable. - If the form is indefinite and odd there is a smooth structure.
- If the form is indefinite and even we may as well assume that it is of nonpositive signature by changing orientations if necessary, in which case it is isomorphic to a sum of m copies of II1,1 and 2n copies of E8(−1) for some m and n. If m ≥ 3n (so that the dimension is at least 11/8 times the |signature|) then there is a smooth structure, given by taking a connected sum of n K3 surfaceK3 surfaceIn mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
s and m − 3n copies of S2×S2. If m ≤ 2n (so the dimension is at most 10/8 times the |signature|) then Furuta proved that no smooth structure exists . This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown. (The smallest case not covered above has n=2 and m=5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II7,55 of rank 62 with n=3 and m=7.) The "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times the |signature|.
In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surface
Dolgachev surface
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.-Properties:...
s, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R4; see exotic R4
Exotic R4
In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...
.
Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds,
using Seiberg-Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
s, possibly with orientations reversed, have been disproved.)
Special phenomena in 4-dimensions
There are several fundamental theorems about manifolds that can be proved by low dimensional methods in dimensions at most 3, and by completely different high dimensional methods in dimension at least 5, but which are false in dimension 4. Here are some examples:- In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H4(M,Z/2Z) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
- In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures.
- 4 is the only dimension n for which Rn can have an exotic smooth structure. R4 has an uncountable number of exotic smooth structures; see exotic R4Exotic R4In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...
.
- The solution to the smooth Poincaré conjecturePoincaré conjectureIn mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphereExotic sphereIn differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
- The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
- A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
- There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex is an open problem (as of 2007).
See also
- Kirby calculusKirby calculusIn mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves...
- Algebraic surfaceAlgebraic surfaceIn mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
- 3-manifold3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
- 5-manifold5-manifoldIn mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first...
- Enriques–Kodaira classification
- Casson handleCasson handleIn 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced...