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 (i.e. there are
smooth 4-manifolds which are homeomorphic but not diffeomorphic).
The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of M. Freedman (1982) 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 then the Kirby–Siebenmann invariant must be the signature/8 (mod
2).
Examples:
In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture.
If the form is E8, this gives a manifold called the E8 manifold, 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 class must vanish.
If the intersection form is definite Donaldson's theorem (Donaldson 1983) 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 nK3 surfaces 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 (Furuta 2001). 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 surfaces,
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 manifolds, 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.
Four 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 R4.
The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic 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 was an open problem. Ciprian Manolescu
showed that there are manifolds in each dimension greater than or equal
to 5, that are not homeomorphic to a simplicial complex.
Failure of the Whitney trick in dimension 4
According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points. The "Whitney trick" uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of n-dimensional
embeddings to embeddings of 2-disks. But this not a reduction when the
embedding is 4: the 2 disks themselves are middle-dimensional, so trying
to embed them encounters exactly the same problems they are supposed to
solve. This is the phenomenon that separates dimension 4 from others."
An image from inside a 3-torus, generated by Jeff Weeks'
CurvedSpaces software. All of the cubes in the image are the same cube,
since light in the manifold wraps around into closed loops, the effect
is that the cube is tiling all of space. This space has finite volume
and no boundary.
In mathematics, a 3-manifold is a space
that locally looks like Euclidean 3-dimensional 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 enough observer, all 3-manifolds look like our universe does
to a small enough observer. This is made more precise in the definition
below.
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.
Thurston's
contributions to the theory allow one to also consider, in many cases,
the additional structure given by a particular Thurston model geometry
(of which there are eight). The most prevalent geometry is hyperbolic
geometry. Using a geometry in addition to special surfaces is often
fruitful.
The fundamental groups
of 3-manifolds strongly reflect the geometric and topological
information belonging to a 3-manifold. Thus, there is an interplay
between group theory and topological methods.
Important examples of 3-manifolds
Euclidean 3-space
Euclidean 3-space is the most important example of a 3-manifold, as
all others are defined in relation to it. This is just the standard
3-dimensional vector space over the real numbers.
3-sphere
Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal,
the curves intersect each other orthogonally (in the yellow points) as
in 4D. All curves are circles: the curves that intersect
<0> have infinite radius (= straight line).0>
A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions.
Real projective 3-space
Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4. It is a compact, smooth manifold of dimension 3, and is a special case Gr(1, R4) of a Grassmannian space.
The 3-dimensional torus is the product of 3 circles. That is:
The 3-torus, T3 can be described as a quotient of R3 under integral shifts in any coordinate. That is, the 3-torus is R3 modulo the action of the integer latticeZ3 (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.
A 3-torus in this sense is an example of a 3-dimensional compactmanifold. It is also an example of a compact abelianLie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone.
In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere.
In 2008, astronomers found the best orientation on the sky for the
model and confirmed some of the predictions of the model, using three
years of observations by the WMAP spacecraft.
However, there is no strong support for the correctness of the model, as yet.
Seifert-Weber space
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closedhyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron
to its opposite in a way that produces a closed 3-manifold. There are
three ways to do this gluing consistently. Opposite faces are
misaligned by 1/10 of a turn, so to match them they must be rotated by
1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber
space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.
With the 3/10-turn gluing pattern, the edges of the original
dodecahedron are glued to each other in groups of five. Thus, in the
Seifert–Weber space, each edge is surrounded by five pentagonal faces,
and the dihedral angle
between these pentagons is 72°. This does not match the 117° dihedral
angle of a regular dodecahedron in Euclidean space, but in hyperbolic space
there exist regular dodecahedra with any dihedral angle between 60° and
117°, and the hyperbolic dodecahedron with dihedral angle 72° may be
used to give the Seifert–Weber space a geometric structure as a
hyperbolic manifold.
It is a quotient space of the order-5 dodecahedral honeycomb, a regulartessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
Gieseking manifold
In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable
and has the smallest volume among non-compact hyperbolic manifolds,
having volume approximately 1.01494161. It was discovered by Gieseking (1912).
The Gieseking manifold can be constructed by removing the vertices from a tetrahedron,
then gluing the faces together in pairs using affine-linear maps.
Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the
face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the
face 3,2,1 in that order. In the hyperbolic structure of the Gieseking
manifold, this ideal tetrahedron is the canonical polyhedral
decomposition of Epstein-Penner. Moreover, the angle made by the faces
is .
The triangulation has one tetrahedron, two faces, one edge and no
vertices, so all the edges of the original tetrahedron are glued
together.
The classes are not necessarily mutually exclusive.
Some important structures on 3-manifolds
Contact geometry
Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold ('complete integrability').
Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry,
which belongs to the even-dimensional world. Both contact and
symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional extended phase space that includes the time variable.
Haken manifold
A Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sidedincompressible surface.
Sometimes one considers only orientable Haken manifolds, in which case a
Haken manifold is a compact, orientable, irreducible 3-manifold that
contains an orientable, incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
Haken manifolds were introduced by Wolfgang Haken. Haken proved that Haken manifolds have a hierarchy,
where they can be split up into 3-balls along incompressible surfaces.
Haken also showed that there was a finite procedure to find an
incompressible surface if the 3-manifold had one. Jaco and Oertel gave
an algorithm to determine if a 3-manifold was Haken.
Essential lamination
An essential lamination is a lamination
where every leaf is incompressible and end incompressible, if the
complementary regions of the lamination are irreducible, and if there
are no spherical leaves.
Essential laminations generalize the incompressible surfaces found in Haken manifolds.
Heegaard splitting
A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise.
This contrasts strongly with higher-dimensional manifolds which need
not admit smooth or piecewise linear structures. Assuming smoothness the
existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.
Taut foliation
A taut foliation is a codimension 1 foliation
of a 3-manifold with the property that there is a single transverse
circle intersecting every leaf. By transverse circle, is meant a closed
loop that is always transverse to the tangent field of the foliation.
Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface.
A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.
Kneser–Haken finiteness
Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements.
Loop and Sphere theorems
The loop theorem is a generalization of Dehn's lemma and should more properly be called the "disk theorem". It was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.
A simple and useful version of the loop theorem states that if there is a map
with not nullhomotopic in , then there is an embedding with the same property.
The sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of
having a representative that is an embedding.
Annulus and Torus theorems
The annulus theorem
states that if a pair of disjoint simple closed curves on the boundary
of a three manifold are freely homotopic then they cobound a properly
embedded annulus. This should not be confused with the high dimensional
theorem of the same name.
The torus theorem is as follows: Let M be a compact,
irreducible 3-manifold with nonempty boundary. If M admits an essential
map of a torus, then M admits an essential embedding of either a torus
or an annulus
JSJ decomposition
The JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
The Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott. The precise statement is as follows:
Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.
A simplified proof is given in, and a stronger uniqueness statement is proven in.
Lickorish-Wallace theorem
The Lickorish–Wallace theorem states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
Waldhausen's
theorems on topological rigidity say that certain 3-manifolds (such as
those with an incompressible surface) are homeomorphic if there is an
isomorphism of fundamental groups which respects the boundary.
Waldhausen conjecture on Heegaard splittings
Waldhausen
conjectured that every closed orientable 3-manifold has only finitely
many Heegaard splittings (up to homeomorphism) of any given genus.
The cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torusT, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times that two simple closed curves in T representing r and s must intersect) is at most 1. Consequently, there are at most three Dehn fillings of M with cyclic fundamental group.
Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem
Thurston's hyperbolic Dehn surgery theorem states: is hyperbolic as long as a finite set of exceptional slopes is avoided for the i-th cusp for each i. In addition, converges to M in H as all for all corresponding to non-empty Dehn fillings .
This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H.
Troels Jorgensen's study of the geometric topology further shows that
all nontrivial limits arise by Dehn filling as in the theorem.
Another important result by Thurston is that volume decreases
under hyperbolic Dehn filling. In fact, the theorem states that volume
decreases under topological Dehn filling, assuming of course that the
Dehn-filled manifold is hyperbolic. The proof relies on basic
properties of the Gromov norm.
Jørgensen also showed that the volume function on this space is a continuous, proper function. Thus by the previous results, nontrivial limits in H
are taken to nontrivial limits in the set of volumes. In fact, one can
further conclude, as did Thurston, that the set of volumes of finite
volume hyperbolic 3-manifolds has ordinal type. This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov.
Also, Gabai, Meyerhoff & Milley showed that the Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.
Thurston's hyperbolization theorem for Haken manifolds
One form of Thurston's geometrization theorem states:
If M is an compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.
The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
The conditions that the manifold M should be irreducible
and atoroidal are necessary, as hyperbolic manifolds have these
properties. However the condition that the manifold be Haken is
unnecessarily strong. Thurston's hyperbolization conjecture states that a
closed irreducible atoroidal 3-manifold with infinite fundamental
group is hyperbolic, and this follows from Perelman's proof of the
Thurston geometrization conjecture.
Tameness conjecture, also called the Marden conjecture or tame ends conjecture
The ending lamination theorem, originally conjectured by William Thurston and later proven by Minsky, Brock and Canary, states that hyperbolic 3-manifolds with finitely generatedfundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
Poincaré conjecture
The 3-sphere is an especially important 3-manifold because of the now-proven Poincaré conjecture. Originally conjectured by Henri Poincaré,
the theorem concerns a space that locally looks like ordinary
three-dimensional space but is connected, finite in size, and lacks any
boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.
After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attack the problem. Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery
to systematically excise singular regions as they develop, in a
controlled way. Several teams of mathematicians have verified that
Perelman's proof is correct.
Thurston's geometrization conjecture
Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connectedRiemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).
In three dimensions, it is not always possible to assign a single
geometry to a whole topological space. Instead, the geometrization
conjecture states that every closed 3-manifold can be decomposed in a
canonical way into pieces that each have one of eight types of geometric
structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds
satisfy the geometrization conjecture. Thurston announced a proof in
the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.
There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture
are corollaries of the geometrization conjecture, although there are
shorter proofs of the former that do not lead to the geometrization
conjecture.
Virtually fibered conjecture and Virtually Haken conjecture
In a posting on the ArXiv on 25 Aug 2009, Daniel Wise
implicitly implied (by referring to a then unpublished longer
manuscript) that he had proven the Virtually fibered conjecture for the
case where the 3-manifold is closed, hyperbolic, and Haken. This was
followed by a survey article in Electronic Research Announcements in
Mathematical Sciences.
Several more preprints have followed, including the aforementioned longer manuscript by Wise. In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds
. The proof built on results of Kahn and Markovic in their proof of the Surface subgroup conjecture and results of Dani Wise in proving the Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups. Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
Simple loop conjecture
If is a map of closed connected surfaces such that is not injectiυe, then there exists a non contractible simple closed
curve such that is homotopically trivial. It was proven by Gabai.
Surface subgroup conjecture
The surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group
has a surface subgroup. By "surface subgroup" we mean the fundamental
group of a closed surface not the 2-sphere. This problem is listed as
Problem 3.75 in Robion Kirby's problem list.
Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds.
A proof of this case was announced in the Summer of 2009 by Jeremy
Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the
FRG (Focused Research Group) Conference hosted by the University of
Utah. A preprint appeared in the arxiv.org server in October 2009. Their paper was published in the Annals of Mathematics in 2012
. In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.
Important conjectures
Cabling conjecture
The
cabling conjecture states that if Dehn surgery on a knot in the
3-sphere yields a reducible 3-manifold, then that knot is a (p,q)-cable
on some other knot, and the surgery must have been performed using the
slope pq.
Lubotzy-Sarnak conjecture
The fundamental group of any finite volume hyperbolic n-manifold does
not have Property τ .
The first four spatial dimensions, represented in a two-dimensional picture.
Two points can be connected to create a line segment.
Two parallel line segments can be connected to form a square.
Two parallel squares can be connected to form a cube.
Two parallel cubes can be connected to form a tesseract.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line
has a dimension of one because only one coordinate is needed to specify
a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.
In mathematics, the dimension of an object is, roughly speaking, the number of degrees of freedom of a point that moves on this object. In other words, the dimension is the number of independent parameters or coordinates
that are needed for defining the position of a point that is
constrained to be on the object. For example, the dimension of a point
is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two, etc.
The dimension is an intrinsic property of an object, in the sense
that it is independent of the dimension of the space in which the
object is or can be embedded. For example, a curve, such as a circle
is of dimension one, because the position of a point on a curve is
determined by its signed distance along the curve to a fixed point on
the curve. This is independent from the fact that a curve cannot be
embedded in a Euclidean space of dimension lower than two, unless if it is a line.
The dimension of Euclidean n-spaceEnis n. When trying to generalize to other types of spaces, one is faced with the question "what makes Enn-dimensional?" One answer is that to cover a fixed ball in En by small balls of radius ε, one needs on the order of ε−n such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces.
A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4".
The rest of this section examines some of the more important mathematical definitions of dimension.
Vector spaces
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.
The uniquely defined dimension of every connected topological manifold can be calculated. A connected topological manifold is locallyhomeomorphic to Euclidean n-space, in which the number n is the manifold's dimension.
In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to "work"; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.
Complex dimension
The
dimension of a manifold depends on the base field with respect to which
Euclidean space is defined. While analysis usually assumes a manifold
to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number (x + iy) has a real partx and an imaginary party, where x and y are both real numbers; hence, the complex dimension is half the real dimension.
Conversely, in algebraically unconstrained contexts, a single
complex coordinate system may be applied to an object having two real
dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension.
Varieties
The dimension of an algebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of the tangent space at any Regular point of an algebraic variety. Another intuitive way is to define the dimension as the number of hyperplanes
that are needed in order to have an intersection with the variety that
is reduced to a finite number of points (dimension zero). This
definition is based on the fact that the intersection of a variety with a
hyperplane reduces the dimension by one unless if the hyperplane
contains the variety.
An algebraic set
being a finite union of algebraic varieties, its dimension is the
maximum of the dimensions of its components. It is equal to the maximal
length of the chains of sub-varieties of the given algebraic set (the length of such a chain is the number of "").
Each variety can be considered as an algebraic stack,
and its dimension as variety agrees with its dimension as stack. There
are however many stacks which do not correspond to varieties, and some
of these have negative dimension. Specifically, if V is a variety of dimension m and G is an algebraic group of dimension nacting on V, then the quotient stack [V/G] has dimension m−n.
Krull dimension
The Krull dimension of a commutative ring is the maximal length of chains of prime ideals in it, a chain of length n being a sequence
of prime ideals related by inclusion. It is strongly related to the
dimension of an algebraic variety, because of the natural correspondence
between sub-varieties and prime ideals of the ring of the polynomials
on the variety.
For any normal topological spaceX, the Lebesgue covering dimension of X is defined to be n if n is the smallest integer for which the following holds: any open cover
has an open refinement (a second open cover where each element is a
subset of an element in the first cover) such that no point is included
in more than n + 1 elements. In this case dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞. Moreover, X has dimension −1, i.e. dim X = −1 if and only if X is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".
An inductive dimension may be defined inductively as follows. Consider a discrete set
of points (such as a finite collection of points) to be 0-dimensional.
By dragging a 0-dimensional object in some direction, one obtains a
1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an (n + 1)-dimensional object by dragging an n-dimensional object in a new direction. The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
Similarly, for the class of CW complexes, the dimension of an object is the largest n for which the n-skeleton is nontrivial. Intuitively, this can be described as follows: if the original space can be continuously deformed into a collection of higher-dimensional triangles joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.
Hausdorff dimension
The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals. The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also have non-integer real values. The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions
that work for highly irregular sets and attain non-integer positive
real values. Fractals have been found useful to describe many natural
objects and phenomena.
Hilbert spaces
Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.
In physics
Spatial dimensions
Classical physics theories describe three physical dimensions: from a particular point in space,
the basic directions in which we can move are up/down, left/right, and
forward/backward. Movement in any other direction can be expressed in
terms of just these three. Moving down is the same as moving up a
negative distance. Moving diagonally upward and forward is just as the
name of the direction implies; i.e., moving in a linear combination
of up and forward. In its simplest form: a line describes one
dimension, a plane describes two dimensions, and a cube describes three
dimensions.
Time
A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension"
for this reason, but that is not to imply that it is a spatial
dimension. A temporal dimension is one way to measure physical change.
It is perceived differently from the three spatial dimensions in that
there is only one of it, and that we cannot move freely in time but
subjectively move in one direction.
The equations used in physics to model reality do not treat time
in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).
In
physics, three dimensions of space and one of time is the accepted
norm. However, there are theories that attempt to unify the four fundamental forces by introducing extra dimensions. Most notably, superstring theory requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called M-theory
which subsumes five previously distinct superstring theories. To date,
no experimental or observational evidence is available to confirm the
existence of these extra dimensions. If extra dimensions exist, they
must be hidden from us by some physical mechanism. One well-studied
possibility is that the extra dimensions may be "curled up" at such tiny
scales as to be effectively invisible to current experiments. Limits on
the size and other properties of extra dimensions are set by particle
experiments such as those at the Large Hadron Collider.
At the level of quantum field theory, Kaluza–Klein theory unifies gravity with gauge
interactions, based on the realization that gravity propagating in
small, compact extra dimensions is equivalent to gauge interactions at
long distances. In particular when the geometry of the extra dimensions
is trivial, it reproduces electromagnetism.
However at sufficiently high energies or short distances, this setup
still suffers from the same pathologies that famously obstruct direct
attempts to describe quantum gravity. Therefore, these models still require a UV completion,
of the kind that string theory is intended to provide. In particular,
superstring theory requires six compact dimensions forming a Calabi–Yau manifold.
Thus Kaluza-Klein theory may be considered either as an incomplete
description on its own, or as a subset of string theory model building.
In addition to small and curled up extra dimensions, there may be
extra dimensions that instead aren't apparent because the matter
associated with our visible universe is localized on a (3 + 1)-dimensional subspace. Thus the extra dimensions need not be small and compact but may be large extra dimensions. D-branes
are dynamical extended objects of various dimensionalities predicted by
string theory that could play this role. They have the property that
open string excitations, which are associated with gauge interactions,
are confined to the brane
by their endpoints, whereas the closed strings that mediate the
gravitational interaction are free to propagate into the whole
spacetime, or "the bulk". This could be related to why gravity is
exponentially weaker than the other forces, as it effectively dilutes
itself as it propagates into a higher-dimensional volume.
Some aspects of brane physics have been applied to cosmology. For example, brane gas cosmology
attempts to explain why there are three dimensions of space using
topological and thermodynamic considerations. According to this idea it
would be because three is the largest number of spatial dimensions where
strings can generically intersect. If initially there are lots of
windings of strings around compact dimensions, space could only expand
to macroscopic sizes once these windings are eliminated, which requires
oppositely wound strings to find each other and annihilate. But strings
can only find each other to annihilate at a meaningful rate in three
dimensions, so it follows that only three dimensions of space are
allowed to grow large given this kind of initial configuration.
Extra dimensions are said to be universal if all fields are equally free to propagate within them.
Networks and dimension
Some complex networks are characterized by fractal dimensions. The concept of dimension can be generalized to include networks embedded in space. The dimension characterize their spatial constraints.
In literature
Science fiction texts often mention the concept of "dimension" when referring to parallel or alternate universes or other imagined planes of existence.
This usage is derived from the idea that to travel to
parallel/alternate universes/planes of existence one must travel in a
direction/dimension besides the standard ones. In effect, the other
universes/planes are just a small distance away from our own, but the
distance is in a fourth (or higher) spatial (or non-spatial) dimension,
not the standard ones.
One of the most heralded science fiction stories regarding true
geometric dimensionality, and often recommended as a starting point for
those just starting to investigate such matters, is the 1884 novella Flatland by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions."
The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include Robert A. Heinlein's —And He Built a Crooked House (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract; and Alan E. Nourse's Tiger by the Tail and The Universe Between (both 1951). Another reference is Madeleine L'Engle's novel A Wrinkle In Time
(1962), which uses the fifth dimension as a way for "tesseracting the
universe" or "folding" space in order to move across it quickly. The
fourth and fifth dimensions were also a key component of the book The Boy Who Reversed Himself by William Sleator.
In philosophy
Immanuel Kant,
in 1783, wrote: "That everywhere space (which is not itself the
boundary of another space) has three dimensions and that space in
general cannot have more dimensions is based on the proposition that not
more than three lines can intersect at right angles in one point. This
proposition cannot at all be shown from concepts, but rests immediately
on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain."
"Space has Four Dimensions" is a short story published in 1846 by German philosopher and experimental psychologistGustav Fechner under the pseudonym
"Dr. Mises". The protagonist in the tale is a shadow who is aware of
and able to communicate with other shadows, but who is trapped on a
two-dimensional surface. According to Fechner, this "shadow-man" would
conceive of the third dimension as being one of time. The story bears a strong similarity to the "Allegory of the Cave" presented in Plato's The Republic (c. 380 BC).
Simon Newcomb wrote an article for the Bulletin of the American Mathematical Society in 1898 entitled "The Philosophy of Hyperspace".
Linda Dalrymple Henderson coined the term "hyperspace philosophy", used
to describe writing that uses higher dimensions to explore metaphysical themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art. Examples of "hyperspace philosophers" include Charles Howard Hinton, the first writer, in 1888, to use the word "tesseract"; and the Russian esotericistP. D. Ouspensky.