Some of this content comes from work done trying to understand what I have found. As such, it represents the current state of my understanding (and ignorance) of the topic. This content, too, will be cheerfully amended as my understanding increases.Some notation:
R^{n} denotes the open set of reals in n dimensions.
S^{n} denotes the n-dimensional sphere (S^{1} is a circle, S^{2} is the surface of a ball, etc.).
B^{n} denotes the n-dimensional ball (B^{1} is a line segment with end points, B^{2} is a disc, etc.).
I^{n} is the unit line segment, square, cube, etc.
Z_{n} is the group of integers modulo n.A group is a set of elements closed under an associative multiplication operator (a product is a member of the group), which possesses an identity element (I), such that every element has an inverse (G G^{-1} = I). The number of elements in the group is the order.
Q = G / H is the quotient group, such that for all elements g in G - H and h in H, g h g^{-1} is in H (h is conjugated by g and H is a normal subgroup), and H is the equivalency class of the identity element in Q.
O(n) is the group of rotations in n dimensions; SO(n) is the subgroup of O(n) which includes the identity. As with all special groups, its elements have unit determinant.
A manifold equipped with an atlas of coordinate charts (a set of local coordinate systems mapping open patches of the manifold to R^{n}) is a G-manifold when the transition functions (which map overlapping coordinate charts) are members of the group G. Atlases are not unique.A manifold is C^{n} if its derivatives up to and including order n are all continuous. A C^{∞} manifold is called smooth.
A space X is compact if every open cover (a possibly infinite collection of open subsets of X whose union is equal to X) has a finite sub-cover (a finite sub-collection which still covers X). S^{1} is compact, R^{1} is not.(Tao)
A group acts freely if only the identity element leaves fixed points.(Thurston, p. 243)
(Hu, p. 290, 291)
- if any element of Γ changes the orientation, n must be even;
- if any element other than the identity of Γ preserves the orientation, n must be odd;
- if n is even and Γ is nontrivial, it must be a cyclic group of order 2, and H_{p}(M) = 0 if p is even, H_{p}(M) = Γ if p is odd;
- if n is odd and Γ is abelian, Γ is cyclic, H_{n}(M) = Z and H_{p}(M) = 0 if p is even, H_{p}(M) = Γ if p is odd;
A group is abelian if its elements all commute (g_{1} g_{2} = g_{2} g_{1}). Cyclic groups are abelian. The number of linearly independent elements in the maximal Abelian subgroup of a group is called the rank of the group.In general the rank of a group is the minimum number of group generators such that every element in the group is equal to a finite product of the generators.The maximal rank of an isometry group for an (n > 2)-dimensional geometry is n (n - 1) / 2.- a discrete group Γ acting freely on a compact manifold implies that Γ is finite.
A discrete group has no infinite sequence of elements that converges to the identity. (Marden, section 2.2)A free group has no finite product of elements equal to the identity. (Mumford et. al., chapter 5)
An h-cobordism is a cobordism where the inclusions of M and N into W are homotopy equivalences.
Any h-cobordism between simply connected manifolds of dimension > 5 is diffeomorphic to the cylinder. (Averett, p. 8)
Consider a triangulated space. The link of a vertex is the complex consisting of the neighboring vertices. For an n-dimensional space, if the link of every vertex is homeomorphic to an n-1 sphere, the space is a manifold. (Thurston, p. 121)
The set of equivalence classes of such maps forms a group π_{k}(M), the k^{th} homotopy group of M.
π_{1} is called the fundamental group.
π_{1} are isomorphic for homeomorphic manifolds.
If M = M_{1} ⊗ M_{2}, π_{k}(M) = π_{k}(M_{1}) +
π_{k}(M_{2}).
For k > 1, π_{k} is abelian.
For k > 1, π_{k}(S^{1}) = 0.
For k < n, π_{k}(S^{n}) = 0.
π_{k}(S^{k}) = Z.
For n > 1, π_{1}(RP^{n}) = Z_{2}.
(Greenberg, p. 25, 34, 36)
The free product is formed by taking all products of the elements of each group (where any two adjacent factors are from different groups), and reducing the product by removing any adjacent factors which are equivalent to the identity.
Every closed manifold homotopy equivalent to S^{n} is homeomorphic to S^{n} (the Poincare Conjecture).
A homotopy sphere is an n-manifold homotopically equivalent to S^{n}. An exotic sphere is a homotopy sphere which is not diffeomorphic to S^{n}. An "ordinary sphere" has a smooth structure inherited from its embedding into R^{n+1}. The following table gives the number of diffeomorphism classes of spheres in dimensions 1 through 15:
(Averett, p. 7, 8, 10)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 1 1 ∞ ? 1 1 28 2 8 6 992 1 3 2 16256
The genus is the number of "handles". g = 1 for the surface of a donut.The following illustrate the nonintuitive result that higher-dimensional spheres can be mapped onto lower-dimensional spheres:
π_{k}(S^{n}) = π_{k}(S^{2n-1}) = π_{k-1}(S^{n-1}).(Hu, p. 58, 153, 325, 328, 329, 332)
π_{n+1}(S^{n}) = π_{n+2}(S^{n}) = Z_{2} for n > 2.
π_{k}(S^{2}) = π_{k}(S^{3}) for k > 2.
π_{3}(S^{2}) = Z.
π_{k}(S^{2}) = Z_{2} for k = 4, 5, 7 and 8.
π_{6}(S^{2}) = Z_{12}.
π_{9}(S^{2}) = Z_{3}.
π_{10}(S^{2}) = Z_{15}.
A fiber bundle is a total space E which locally is a Cartesian product of the fiber space F and the base space M, with a projection E → M.SO(3) is topologically RP^{3}, with π_{1} = Z_{2}, and universal cover S^{3}.A connection is a field transverse to the fibers which defines motion among fibers relative to motion in the base manifold; it is the same dimension as the base manifold. Except in the trivial (flat) case, it can only be defined within a local coordinate patch.
If M is a closed surface other than S^{2} or RP^{2}, π_{1}(M) is infinite and the universal cover is R^{2}. (Hu, p. 96)
The index of a subgroup H is the number of elements x h_{i} in G where h_{i} is in H and x is not in H.Of S^{n}, T^{2} (of arbitrary genus), RP^{n}, CP^{n} and HP^{n}, only RP^{n} for n even is not orientable. (Greenberg, p. 162, 168)
The tangent space of a Lie group at the identity has the structure of a Lie algebra: a vector space equipped with a commutator [x,y] = - [y,x] which statisfies the Jacobi identity [[x,y],z] + [[y,z],x] + [[z,x],y] = 0.Diffeomorphism classes of closed n-Euclidean manifolds are in 1:1 correspondence, via their fundamental groups, with torsion-free groups containing a subgroup of finite index isomorphic to Z^{ n}.The stabilizers of a group are those elements which leave fixed points.
A (G, X) structure on a space X is a maxmial coordinate chart such that the coordinate systems in every overlapping patch are related by elements of the group G; a diffeomorphism class of metrics on M.
The elements g_{i} form a conjugacy class if for all elements x in G, x^{-1} g_{i} x is one of the g_{i}.
A torsion element x satisfies x^{m} = I for some integer m.(Thurston, p. 157, 167, 222)
For M hyperbolic and k > 1, π_{k}(M) = 0. (Marden, section 5.2)
H_{p}(M;G) = 0 for p > n = dim(M).
If M is connected, H_{0}(M;G) = G.
If M is orientable, H_{n}(M;G) = G.
If M is orientable and G is R, C or Z_{2}, H_{p}(M;G) = H_{n-p}(M;G)
(Poincare Duality). (Eguchi, Gilkey and Hanson, p. 233)
H_{p}(S^{n}) is isomorphic to H_{p-1}(S^{n-1}) for p > 1.
H_{p}(RP^{n}) = G_{2} (submodule annihilated by multiplication by 2) for p even > 1;
G / 2 for p odd > 0, < n; G for p = 0 or n odd. (Greenberg, p. 83, 121)
H_{1} = π_{1} / [π_{1}, π_{1}]. (Bott and Tu, p. 225)
[G, G] is the commutator subgroup of G: the subgroup of G generated by all of the commutators of G. It is the smallest normal subgroup of G such that G / [G, G] is abelian.If E is the universal cover of M, π_{2}(M) is isomorphic to H_{2}(E).
b_{0} = the number of connected components.
b_{p} = 0 (0 < p < n) for a compact, orientable manifold of positive constant curvature (M is a homology sphere).
b_{p} <= (n p) for a compact, orientable, locally flat manifold. T^{n} saturates that bound.
b_{p} = 0 (0 < p < n) for a compact, orientable conformally flat manifold of positive definite Ricci curvature.
(Goldberg, p. 89, 118; see also pp. 91, 92, 131)
b_{1} = b_{2} = 0 for a compact semi-simple Lie group (metric is non-degenerate). b_{3} = 1 for a compact simple Lie group. (Goldberg, p. 140, 145; see also p. 144)
b_{p} = 0 for RP^{n} except for b_{0} = 1, and b_{n} = 1 if n is odd.
If M is a compact submanifold of R^{n} (n > 1) with k connected components, b_{n-1}(R^{n} - M) = k.
(Greenberg, p. 129, 167)
An n-manifold M is equivalent to the product of a 1-manifold and an (n-1)-manifold iff χ(M) = 0. (Thurston, p. 115)
Further, χ(M) = χ(M_{1}) χ(M_{2}). (Eguchi, Gilkey and Hanson, p. 238)
If M and N are connected n-manifolds, χ(M # N) = χ(M) + χ(N), - 2 if n is even. (Greenberg, p. 131)
H^{0}(M;R) is the space of constant functions.
H^{n}(M;R) is the space of volume forms.
(Eguchi, Gilkey and Hanson, p. 234)
Note that an n-manifold is orientable iff it has a nowhere-vanishing n-form.
Two manifolds with the same homotopy type have the same de Rham cohomology. (Bott and Tu, p. 29, 36)
M is a boundary of a smooth compact manifold iff the Stiefel-Whitney numbers of M are all zero.
Characteristic numbers are topologically invariant integrals of elements of H^{p}(M;G) over p-chains.Two smooth closed n-manifolds belong to the same cobordism class iff their Stiefel-Whitney numbers are all equal. (Milnor and Stasheff, p. 52, 53)
Det(I - Ω / 2π) = 1 + p_{4} + p_{8} + ...where Ω is the O(k) curvature of the bundle. (Eguchi, Gilkey and Hanson, p. 311)
If some Pontrjagin number of M^{4 k} is nonzero, M cannot possess any orientation-reversing diffeomorphism, and M cannot be the boundary of a smooth compact oriented manifold with boundary.
M is an oriented boundary iff all Pontrjagin numbers and Stiefel-Whitney numbers are zero. (Milnor and Stasheff, p. 186, 217)
tr A ^ dA + (2/3) A ^ A ^ Awhere A is the connection one-form. (Eguchi, Gilkey and Hanson, p. 313, 350)
χ(M) = 2 κ^{n/2} Vol(M) / Vol(S^{n})This is the Gauss-Bonnet theorem.
If M is also orientable and hyperbolic, χ(M) is even. If n is even and M is hyperbolic and of finite volume,
Vol(M) = (-1)^{n/2} χ(M) Area(S^{n-1}) / 2(Marden, section 4.10)
If f is not an isometry, the Euler characteristic is the signed sum of the number of zeroes, where the sign is determined by whether or not the diffeomorphism preserves the orientation. (Eguchi, Gilkey and Hanson, p. 335)
The sign here comes from the index, which is +1 for a source or a sink, and -1 for a saddle. (Thurston, p. 23)
A bundle section is a continuous map from the base space to the total space. It associates a single point on the fiber over each point in the base space.If such a section exists, the manifold is said to be parallelizable.
The Euler class of a parallelizable manifold is zero.
If G is a group of analytic diffeomorphisms on a simply connected space X, any complete (G,X) manifold can be reconstructed from its holonomy group Γ as X / Γ.
If the isotropy group of every point is compact, the manifold is complete. (The isotropy subgroup of a group leaves one or more points fixed. isotropy is uniformity in all directions.)If H is a subgroup of G, the G-structure can be stiffened to an H-structure iff its holonomy group is conjugate to a subgroup of H. (Thurston, p. 141, 143, 151, Thurston Notes, p. 35)Completeness is equivalent to geodesic completeness. (Ratcliffe, p. 367)
The holonomy group Hol(g) depends on the connection. Hol(g) ⊂ O(n). For loops isotopic to I, the subgroup of Hol(g) is Hol^{0}(g). Hol^{0}(g) ⊂ SO(n). If π_{1}(M) = 0, Hol^{0}(g) = Hol(g).
SO(n) U(n/2) for n ≥ 4 M is Kahler SU(n/2) for n ≥ 4 M is Kahler and Ricci-flat (Calabi-Yau) Sp(1) ⋅ Sp(n/4) for n ≥ 8 M is Einstein and quaternionic Kahler Sp(n/4) for n ≥ 8 M is Riici-flat Spin(9) for n = 16 M is Einstein and either flat or the Cayley plane Spin(7) for n = 8 M is Ricci-flat G2 for n = 7 M is Ricci-flat (Spin(n) is the double cover of SO(n))
A Kahler metric is a Hermitian metric whose complex operator is covariantly constant.
If M admits an exterior form α : Dα = 0, and if α is not a power of the Kahler form associated with the metric, M is Einstein.
(Besse, ch. 10)
A Riemannian space V_{ n} (n > 2) has constant curvature iff it locally admits an isotropy group of rank n (n - 1) / 2 at each point. (Kramer et al, Th. 8.13, 8.14)
(Besse, pp. 6, 17, 157-158)
N^{D}_{0} = ( D - 2 ) ( D - 1 ) D ( D + 3 ) / 12while for order k there are:
N^{D}_{k} = D ( k + 1 ) ( D + k + 1 ) ! / ( 2 ( D - 2 ) ! ( k + 3 ) ! )(Haskins)
In both cases, geodesics are the intersection with a (timelike) plane in the embedding. (Ratcliffe, chs. 2, 3)
positive (S^{n}) negative (H^{n}) embedding Σ x_{i}^{2} = r^{2} in R^{n+1} x_{0}^{2} - Σ x_{i}^{2} = - r^{2} in R^{1,n} (one sheet) embedding map x_{1} = r cos θ_{1} x_{0} = r cosh η x_{2} = r sin θ_{1} cos θ_{2} x_{1} = r sinh η cos θ_{1} ... ... x_{n} = r sin θ_{1} sin θ_{2} ... sin θ_{n-1} cos θ_{n} x_{n-1} = r sinh η sin θ_{1} ... sin θ_{n-2} cos θ_{n-1} x_{n+1} = r sin θ_{1} sin θ_{2} ... sin θ_{n-1} sin θ_{n} x_{n} = r sinh η sin θ_{1} ... sin θ_{n-2} sin θ_{n-1} isometry group O(n+1) PO(1,n) (one sheet) arc length (ds^{2}) r^{2} (dθ_{1}^{2} + sin^{2}θ_{1} dθ_{2}^{2} + ... + sin^{2}θ_{1} ... sin^{2}θ_{n-1} dθ_{n}^{2}) r^{2} (dη^{2} + sinh^{2} η dθ_{1}^{2} + ... + sinh^{2} η sin^{2}θ_{1} ... sin^{2}θ_{n-2} dθ_{n-1}^{2}) curvature n (n-1) / r^{2} -n (n-1) / r^{2} volume element r^{n} (sin^{n-1} θ_{1}) (sin^{n-2} θ_{2}) ... (sin θ_{n-1}) Π dθ_{i} r^{n} (sinh^{n-1} η) (sin^{n-2} θ_{1}) ... (sin θ_{n-2}) dη Π dθ_{i} volume 2 π^{(n+1)/2} r^{n} / Γ((n+1)/2) 2 π^{n/2} r^{n} / Γ(n/2) * (-1/i^{n}) Cosh(η) _{2}F_{1}(1/2, 1-n/2, 3/2, Cosh^{2}(η))
(η runs from 0 to ∞; all θ coordinates run from 0 to π except the last, which runs to 2π; _{2}F_{1} is a hypergeometric function)
Any manifold finitely covered by S^{2} ⊗ S^{1} does not admit a metric with constant sectional curvature. (Besse, p. 158)
The Poincare ball model of H^{n} is a unit ball in Euclidean space. r=0 at the center and the boundary is r=∞. Any arc of a circle orthogonal to the boundary is a hyperbolic geodesic. Inversions (see below) in (n-1)-spheres orthogonal to the boundary are hyperbolic isometries. A metric for the model is
ds^{2} = 4 dx^{2} / (1 - r^{2})^{2}where dx^{2} is the ordinary Euclidean metric.
SO(4) is isomorphic to (S^{3} ⊗ S^{3}) / C_{2}(Thurston, pp. 64,107)
An inversion smoothly exchanges the interior with the exterior, keeping the sphere fixed. Inversions are conformal: they preserve angles. They also preserve orientation.These transformations form the Moebius group. The group of Euclidean isometries is a sub-group of the Moebius group. (Ratcliffe, p. 111)
The subgroup of Moeb_{n-1} which takes an S^{k} to itself is isomorphic to Moeb_{k} ⊗ O(n-k).
Moeb_{n} is equivalent to the homeomorphisms of S^{n} which map S^{n-1}s to each other.
(Thurston, pp. 61, 62, 83)
If the n-gon is embedded in a surface of positive curvature, the sum is greater; if it is embedded in a surface of negative curvature, the sum is less.
An example of a compressible surface is a cross-sectional disc of a solid torus perpendicular to its longitudinal axis. Such a disc is also called a compressing disc. In general, a compressing disc either divides M into pieces, or M - S is connected (our example).(Marden, section 3.7)A trivial example of an incompressible boundary is a solid formed by a pair of annuli whose boundaries are joined by cylinders (which are topologically annuli themselves, making this a 3-dimensional manifold bounded by 4 incompressible annuli).
The latter rules out a disc which "slices" a piece off the "end" of a solid torus.A compressing disc is essential. (Marden, section 3.7)
A closed hyperbolic surface can be smoothly (C^{∞}) and isometrically embedded in R^{17}. A noncompact hyperbolic surface can be smoothly and isometrically embedded in R^{51}. (Marden, section 2.7)
2π/k (2 - 2g - n - Σ (1 - 1/r_{i}))(k = 1 for S^{2} / G, -1 for H^{2} / G) (Marden, section 3.14)
The two tori represent rectangular and hexagonal tilings of the Euclidean plane, respectively.
After identifying the single-hashed sides of the hexgonal torus, the resultant "penne" must be twisted 180 degrees before the other sides will match up. (Thurston, p. 5)Omit the singly-hashed identification on the Klein bottle to make a Moebius strip. (Henle, p. 106, 111, 112)
If you want to construct these, use a large piece of cloth and safety pins. Make each identified edge pair a noticeably different length, and pin them in the order of longest to shortest. It helps to use an additional set of pins for the identified vertices to keep them organized as you pin.Since the boundary of a Moebius strip is S^{1}, we can remove a B^{2} from an S^{2} and glue in a Moebius strip to obtain RP^{2}. Do it twice to obtain a Klein bottle. This operation is termed adding a cross-cap.A simple notation is often used in such constructions. Label each singly-hatched side "a", each doubly-hatched side "b" and each triply-hatched side "c" (etc.). Then, typically starting with an "a" side whose direction is clockwise, proceed clockwise; whenever the side's arrow is parallel to your clockwise direction, write the letter; when the side has an arrow which is in the opposite direction, write the letter as an inverse. Hence we have:
- rectangular torus: a b^{-1} a^{-1} b
- hexagonal torus: a b^{-1} c^{-1} a^{-1} b c
- Klein bottle: a b a^{-1} b
- RP^{2}: a b a b
A sphere with g cross-caps has χ = 2 - g. (Edelbrunner and Harer, p. 34, 36)
Consider a surface of genus 2. A two-holed torus can be constructed using identifications on an octagon:
a b a^{-1} b^{-1} c d c^{-1} d^{-1}(Another construction identifies parallel sides; this can also be done with a decagon to produce a genus-2 torus.)
An interior angle of a regular octagon is 3π/4. At each vertex, two octagons is not enough and three is too many. But as we see below, χ for the genus-2 torus is -2, which means that it can only be given a hyperbolic structure. When the octagon is embedded in H^{2}, the sides become curved; by an appropriate choice of size for the octagon, the interior angles will be π/4, and the octagon can then be used to tile H^{2}.
(Thurston, section 1.2)
a_{1} b_{1} a_{1}^{-1} b_{1}^{-1} ... a_{g} b_{g} a_{g}^{-1} b_{g}^{-1}will produce a genus-g torus. There are many identifications which will produce the same handlebody. In most cases twists (as in the hexagonal torus) will be required
An identification of the sides of a 2g-gon of the form
a_{1} a_{1} ... a_{g} a_{g}will produce a non-orientable genus-g surface. All non-orientable surfaces can be obtained this way. Any two non-orientable closed surfaces with the same χ are homeomorphic.
(Thurston, p. 26)
(Gray, p. 46 ^{(1)}) (Henle, p. 167 ^{(2)}, 193 ^{(3)}, 295 ^{(4)}) (Hu, p. 28 ^{(5)})
surface b_{1} b_{2} χ π_{1} E^{2} 0 1 2 0 H^{2} 0 1 2 0 annulus 1 1 0 Z cylinder 1 1 0 Z disc 0 1 1 0 S^{2} 0 1 2 ^{(4)} 0 T^{2} 2 1 0 ^{(2)} Z ⊗ Z Riemann surface genus g 2 g 1 2 - 2 g ^{(4)} Z^{ 2g} | aba^{-1}b^{-1}cdc^{-1}d^{-1}... = 1 (Bott and Tu, p. 240) Moebius strip 1 0 0 Z Klein bottle 2 ^{(3)} 1 ^{(3)} 0 ^{(2)} Z ⊗ Z | abab^{-1} = 1 RP^{2} 0 0 1 ^{(2)} Z_{2} connected sum of n RP^{2} n - 1 0 2 - n ^{(4)} Z^{n} | aabbcc... = 1 ^{(1)} non-orientable surface genus g g - 1 ^{(5)} 0 ^{(5)} 2 - g
The Uniformization Theorem states that a simply connected Riemann surface can be conformally mapped into exactly one of S^{2} (if χ > 0), C (if χ = 0) or B^{2} (if χ < 0).
The group of conformal automorphisms of a Riemann surface is discrete IFF π_{1} is nonabelian. For g ≥ 2, it has at most 84(g-1) elements.A Riemann surface admits a positive-definite metric of the form ds^{2} = E du^{2} + 2 F du dv + G dv^{2}. Such a metric has area element √ (E G - F^{2}) du ∧ dv.Every Riemann surface whose π_{1} is nonabelian admits a hyperbolic metric compatible with its complex structure. It has finite hyperbolic area IFF it is a closed surface with n punctures (open discs removed) and 2g+n≥3.
(Marden, sections 2.7, 2.9)
Every closed 2-manifold is homeomorphic to S^{2}, a connected sum of tori, a connected sum of tori and one projective plane, or a connected sum of tori and one Klein bottle. (Averett, p. 4)
Every compact, connected 2-manifold is topologically equivalent to a sphere (2-cell notation aa^{-1}), a connected sum of tori (aba^{-1}b^{-1}cdc^{-1}d^{-1}...), or a connected sum of projective planes (aabbccdd...). Every compact, connected 2-manifold with boundary is equivalent as just stated, with some finite number of discs removed. (Henle, p. 122, 129) (Christenson, p. 439)
Every compact connected surface is diffeomorphic to S^{2} - some number of B^{2}, with either I^{2} or Moebius bands connecting the holes.
Two surfaces are diffeomorphic iff they have the same genus, Euler characteristic and number of boundary components.
Two surfaces are diffeomorphic iff they have the same Euler characteristic and number of boundary components, and are both either orientable or non-orientable.
Every connected compact orientable 2-manifold is diffeomorphic to the surface obtained from an orientable surface of genus 1 - (χ + δ) / 2 by the removal of δ disjoint B^{2}. (Hirsch, p. 189, 205, 207)
Every open surface admits a hyperbolic structure. (Scott, p. 421)
χ = #vertices - #edges + #facesA closed surface has hyperbolic, Euclidean or spherical (elliptical) structure iff its Euler characteristic is negative, zero or positive, respectively.
The Euler characteristic of a surface formed by gluing two surfaces with boundary along a boundary component is the sum of the original Euler characteristics.
Note that RP^{2} is equal to a Moebius strip glued to a disc.The Euler characteristic of the union of two surfaces is equal to the sum of their Euler characteristics minus the Euler characteristic of their intersection.
A sphere with g handles, c cross-caps and d open discs removed has Euler characteristic
χ = 2 - 2g - c - d
f(z) = (a z +b) / (c z + d),where a, b, c and d are complex numbers subject to a d ≠ b c (the usual normalization is a d - b c = 1, which makes the group isomorphic to PSL(2,C)). These are the most general functions on E^{2} which map circles to circles.
f(z) is often represented as a matrix:Conjugation can be used to effect a coordinate transformation on an iterated Moebius map {z, f(z), f^{2}(z),...}.(also denoted ((a,b),(c,d))). If the trace of f(z) (Tr(f) = a+d) is
a b c d
- <-2 or >2, f is loxodromic: a screw motion (conjugate to z → λ^{2}z, |λ|>1);
- <-2 or >2 and real, f is hyperbolic: a loxodromic transformation with λ>1 (this term sometimes is used to include loxodromic);
- >-2 and <2 and real, f is elliptic: a rotation (conjugate to z → e^{2iθ}z, θ ≠ π; NB: z → -z is conjugate to z → 1/z);
- equal to ±2, f is parabolic: a Euclidean translation on the sphere at ∞ (conjugate to z → z+1).
A parabolic map has a single fixed point; a loxodromic, hyperbolic or elliptic map has two fixed points.Some convenient trace identities are:
- Tr(f) = Tr(f^{-1})
- Tr(gfg^{-1}) = Tr(f) (Tr is unchanged by conjugation)
- Tr(product) is unchanged under cyclic permutation of the product
- Tr(fg) = Tr(f^{-1}g^{-1})
- Tr(fg) + Tr(f^{-1}g) = Tr(f) Tr(g)
If Tr(f) = 0, f^{2} = I (so f is elliptic).
Suppose groups G and H of Moebius transformations are each generated by two parabolic elements. G and H are conjugate (therefore isomorphic) if their invariants Tr(ab)-2 are equal up to sign. (Lyndon and Ullman, p. 1388)
The set of all points in E^{2} that can be reached by iteration of the maps in a Moebius sub-group G is called the regular set (Ω(G)) of the group. It has either one, two, or infinitely many connected components, each of which is either simply connected or contractible (to a point).Considering the quotient space obtained by Ω(G) / G.Λ(G) ≡ E^{2} - Ω(G) is called the limit set. These are the fixed points of G. If G has only elliptic elements, it is finite and Λ(G) is empty. If Λ(G) is not empty, it has one, two, or uncountably infinite points.
In the latter case it is a closed set, whose components are either topologically circles, fractal sets, or totally disconnected (every component is a point). (Marden, sections 2.4.1, 2.8)(Series, p. 8)
It turns out that if G has no elliptic elements, the quotient space is the boundary of a hyperbolic 3-manifold M = H^{3} / G, where π_{1}(M) = G and ∂M ≡ Ω(G) / G. If G contains elliptic elements, M is an orbifold. Unless specified below, we will assume G contains no elliptic elements.Restricting ourselves to 2-generator groups, generators a and b with inverses A and B, respectively:The fundamental group of ∂M is isomorphic to G, which means the Kleinian groups constitute representations of the fundamental group for a given quotient space topology.
The limit set passes through the points of tangency, which are fixed by cyclic permutations of abAB, and the group is called a quasifuchsian group. If the curve is a circle (as above) the group is Fuchsian and the traces of all the group elements are real.The region exterior to the paired curves is made up of two disjoint tiles (labeled "1" and "2" above), which become the two sides of the torus separated by the puncture. All quasifuchsian groups yield the same type of quotient surface.
rank 2 with 1 parabolic → genus 2 with 1 element of π_{1} pinched → 2 once-punctured tori
Λ(G) of such a group (with Tr(abAB) = -2) is a closed curve: a quasicircle, which separates Ω(G) into two disjoint regions; one is tiled by the ideal quadralateral "1", the other is tiled by the ideal quadralateral "2":
In general, if tr(abAB) = 2, a and b share a fixed point; a point fixed by a parabolic operator induces a puncture on ∂M, and each puncture increases the dimension of π_{1} by 1;The images here and below are the results of randomly iterating nontrivial products of a, b, A and B, up to 200 factors each, on the fixed points of a, b, A, B, abAB, AbaB, BabA and ABab, for a reasonably large number of iterations on each fixed point.
The green point is the fixed point of a, and the magenta point is the fixed point of b. The region external to the circles corresponds to the surface of the 4-punctured sphere.
rank 2 with 2 parabolic → genus 2 with 2 elements of π_{1} pinched → 4-punctured sphere
Using the Riley parameter λ, we have a = ((1,2),(0,1)) and b = ((1,0),(λ,1)). Tr(ab) is 2(1+λ) and Tr(abAB) is 2(1+2λ^{2}). The Riley slice is the corresponding 1-complex-dimensional slice of Teichmuller space for groups of two generators. The only point of contact with the Maskit slice (below) is the point λ=i, corresponding to the (0,1) gasket (μ=-2i).The Riley slice is symmetric about both the real and imaginary axes. Here is a portion of the limit set for λ=0.05+0.93i:
To make this image, a and b were conjugated by ((i,-1),(1,-i))/√2 so that the points at real ∞ have been translated to (0,i) and the origin has been translated to (0,-i).
As drawn, the radius of circle "a" is infinite; the green point is the fixed point of a, and the magenta point is the fixed point of b.
rank 2 with 3 parabolic → genus 2 with 3 elements of π_{1} pinched → 2 thrice-punctured spheres
For each additional independent parabolic element (besides abAB), one half of the regular set closes up into a set of tangent discs:
The sets of tiles (on each side) are disjoint from each other.
When constructing the quotient space ∂M, all of the gray discs are identified, and the white discs will similarly be identified
(the gray region outside the limit set is a disc after adding in the point at ∞).
NB: The map maps circles, not discs; here we have colored them in to facilitate identification.
The circles are also iteratively mapped, so not all circles which should be gray are necessarily colored in.We will call such groups gasket groups (also called double-cusp groups).
An acylindrical manifold is boundary incompressible and contains no essential cylinders. Such a manifold M of genus g with n punctures has 3g+n-3 simple loops which divide M into 2g+n-2 "pairs of pants", and whose corresponding group elements can become parabolic in the algebraic limit. Hence each of the loops can be pinched to become a puncture, and each pair of pants becomes a component of ∂M.
(Marden, section 5.3)
(Here, the blue circle represents the actions of ab and ba.)
This modular group is unique up to conjugation, and the space is rigid (Marden, section 2.10);
Here, only one side of Ω(G) has closed up into a set of tangent discs; the part of ∂M corresponding to them is a triply-punctured sphere. The other side is simply connected, and its portion of ∂M is the once-punctured torus:It is instructive to look at the discrete groups associated with the first few values of p and q. There are only 9 independent ones for p or q ≤ 5 because translations of μ by ±2k (k ∈ Z) are equivalent; also, the negative of the complex conjugate of μ yields the same limit set, but reflected in the imaginary axis. For each limit set below, the associated parabolic word is in parentheses:
rank 2 with 2 parabolic → genus 2 with 2 elements of π_{1} pinched → 1 once-punctured torus plus 1 thrice-punctured sphereG is a single-cusp group.An a^{p}B^{q} group element can be found by constructing the set starting with {1}, adding q or subtracting p so that the elements are always positive and never greater than p+q, until you return to 1. The group element is the product of a's and B's created from writing an a for each subtraction and a B for each addition (in reverse order). For instance, the 2,5 set is
{1, 6, 4, 2, 7, 5, 3, 1}and the corresponding group word is aaaBaaB. This is simply an arithmetic prescription for a consistent winding of q a's and p B's around the torus.If we want G to be a gasket group, we will force Tr(b) to be 2 and look for a p,q element to be the other parabolic element (abAB will be parabolic with our parameterization). ∂M will then be a pair of triply-punctured spheres.
Parameterizing a by the Maskit parameter μ we have a = ((-iμ, -i), (-i, 0)) and b = ((1, 2), (0, 1)). The trace of the 2,5 group element is
-i (μ^{5} - 4μ^{4} + 9μ^{3} - 12μ^{2} + 9μ - 4),which must be ±2, from which we can obtain μ.
- Since -1 < im(μ) < 1 cannot yield a discrete group,
- and μ and its complex conjugate yield conjugate groups,
- we need only consider μ with im(μ) ≥ 1
- (im(μ) > 2 always yields a discrete group).
We therefore have three possible solutions: 0.375189 + 1.30024i, 1.06548 + 1.2824i and 0.766588 + 1.64214i. When generating Λ(G)s of the first two, one obtains chaotic, overlapping patterns characteristic of non-discrete groups; but the last one yields a discrete group (see below).
For p or q < 0, begin with -p,1, for which the parabolic word is ab^{p}, and 0,1 (parabolic word a). For any p,q find relatively prime integers r,s for which ps - rq = ±1. A new parabolic word will be
w_{(p+q),(r+s)} = w_{p,q}w_{r,s}.So, for example, w_{-1,2} = w_{-1,1}w_{0,1} = aba.(This construction works for positive p and q as well.)p,q and -p,-q will produce the same word.
0,1 (a) | 0,3 (aaa) | 0,4 (aaaa) |
0,5 (aaaaa) | 1,2 (aaB) | 1,3 (aaaB) |
1,4 (aaaaB) | 1,5 (aaaaaB) | 2,5 (aaaBaaB) |
±1,1 and ±2,1 and ±3,1 and ±4,1 and ±5,1 are translations of 0,1 (0,2 is the same as 0,1);
-5,2 and -3,2 and -1,2 and 1,2 and 3,2 and 5,2 are equivalent under translation;
-5,3 and -2,3 and 1,3 and 4,3 are equivalent under translation;
-4,3 and -1,3 and 2,3 and 5,3 are equivalent to 1,3 under reflection (and translation);
-3,4 and 1,4 and 5,4 are equivalent under translation;
-5,4 and -1,4 and 3,4 are equivalent to 1,4 under reflection (and translation);
-4,5 and 1,5 are equivalent under translation;
-1,5 and 4,5 are equivalent to 1,5 under reflection (and translation);
-3,5 and 2,5 are equivalent under translation; and
-2,5 and 3,5 are equivalent to 2,5 under reflection (and translation).
Note that 0,3 and 0,4 and 0,5 are non-free; in each case the associated parabolic word is -I (where I is the identity matrix). The limit sets do not bound discs; they are fractal dust. The same holds true (up to sign) for all 0,q for q > 2.
ι = | ps = qr |is the number of times α crosses β on the quotient surface. (Marden, problem 2.6)
In both cases, Λ(G) is incomplete; on the left, given infinite time the program would completely fill in the white region outside the gray circles (which constitute, with the exterior region, the non-degenerate part of Ω(G)); on the right, given infinite time the program would fill the entire complex plane.
Degenerate groups are discrete and geometrically infinite (but M(G) may have finite volume).
The limit set of a doubly degenerate discrete group is the entire sphere at ∞. This is also true for closed hyperbolic manifolds. (Thurston, p. 172)The irrational slope of the winding is the ending lamination for the associated cusp. (Marden, problem 2.6)
For example, let
a = ((1 + 2i, 4), (1, 1 - 2i)),a and b have trace 2, and c, aB and cbCB have trace -2. This group corresponds to Mumford et. al., figure 11.6; the limit set is conjugate to their figure 11.7:
b = ((1 + 1.5i, 1), (2.25, 1 - 1.5i)) and
c = ((-13 - 4i, 8i), (-12 + 16i, 11 + 4i)).
The magenta and green points correspond to the fixed points of a and b, respectively; the red circle corresponds to the action of aB, and the blue circle corresponds to the action of acBC. The quotient surface is four triply-punctured spheres. The limit set has four mutually disjoint sets of components, corresponding to the four numbered tiles.
rank 3 with 6 parabolic → genus 3 with 6 elements of π_{1} pinched → 4 thrice-punctured spheres
For any two sets of three points, there is exactly one element of PSL(2,C) which maps the first set to the second. (Thurston, p. 87)
If a surface has empty boundary, two structures of S are equivalent in moduli space iff they have the same holonomy group (up to conjugacy). They are equivalent in Teichmuller space iff they have the same holonomy map.
The mapping class group of T^{2} is PSL(2,Z) (the projective special linear group acting on pairs of integers).
The Teichmuller space of a compact surface that admits a hyperbolic structure is homeomorphic to R^{ 3 |χ|}.
The maximum number of disjoint, non-parallel simple closed curves on a hyperbolic surface is 3g - 3. Cutting the surface along the corresponding geodesics divides the surface into 2g - 2 surfaces homeomorphic to S^{2} - 3 B^{2} (pairs of pants). The Teichmuller space of the original surface corresponds to the degrees of freedom defining the lengths of the boundary components of the pants, along with the number of twists with which the pants are glued back together.The mapping class group of a closed surface is isomorphic to the outer automorphism group of its fundamental group.If there are n punctures, the dimension is 3 |χ| - n (or 6g + 2n - 6). (Marden, sect. 2.10)
For groups of two generators, the quotient space has χ = -2, so the Teichmuller space is 6-dimensional. We can parameterize it by the complex traces Tr(a), Tr(b) and Tr(ab).The Maskit slice is a 1-complex-dimensional slice of Teichmuller space corresponding to Tr(b) = 2 and Tr(abAB) = -2. The slice has translational symmetry because Tr(ab) = Tr(a) ± 2k, k ∈ Z. From our considerations above, we need only examine the region 0 ≤ Re(μ) < 2, 1 ≤ Im(μ).
On the left, Im(μ) = 2 and Re(μ) varies from 0 to 1.9; on the right, Re(μ) = 1 and Im(μ) varies from 4 to 1.5:
The region outside the unit circle in these plots should be gray, but we have left it white to better show how the limit set "bleeds" out of the unit circle.As Im(μ) decreases (right), the first seven frames correspond to single-cusp groups: their topology is a triply-punctured sphere (gray discs) plus a (simply-connected) once-punctured torus (white region). As the torus "closes up" (the last remaining loop of π_{1} becomes "tighter"), the group approaches a double-cusp group. The eighth frame is the (1,2) gasket, and the last three frames are non-discrete. The latter show an increasing "overlap" in the limit sets, but with sufficient time, the limit set will fill all three frames.The "essential" portion of the slice is shown here (with some groups identified by (p,q):
The red points indicate the gasket groups on the Maskit boundary. Groups with values of μ above the boundary are single-cusp groups. Groups below the boundary are almost all non-discrete, with one class of exceptions: the magenta points are non-free groups. Note that a single value of (p,q) (i.e., 9,20) can have solutions to its trace equation which are a gasket and a non-free group (9,20 also has eight non-discrete groups). The non-free groups plotted above are (5,12), (7,16), (9,20), (11,24), (13,28) and (15,32).
Note that the region 0 ≤ Re(μ) < 2 appears to be symmetric about the line Re(μ) = 1. This is due to a combination of symmetries g(μ) ≈ g(μ^{*}) ≈ g(-μ^{*}) ≈ g(μ + 2), or equivalently, g(μ) ≈ g(2 - μ), where here "≈" denotes equivalence up to conjugacy (the "mirror image" groups labeled above are all conjugate).
The slice also shows a pattern in the ratio p/q:
Here red corresponds to p/q = 0 and blue corresponds to p/q = 21/22 (the highest ratio computed for this data set).
(Mumford et. al., ch. 10)
The outer automorphism group is the conjugacy class under diffeomorphisms of maps of π_{1} into itself.(Thurston, p. 260, 262, 264, 266, 271, 276, Thurston Notes, pp. 89-91)
For a given 3-manifold M, the minimum genus for all possible splittings is the Heegaard genus. The rank of π_{1}(M) cannot be larger than the Heegaard genus. (Marden, section 2.8.1)
χ = #vertices - (2 #edges - 3 #faces + 4 #tetrahedra) / 2is zero. So χ = 0 for any closed 3-manifold.
If X is a 3-space and it has k vertices v_{k},When identifying faces of polyhedra, the total number of faces must be even, and every identification must be orientation-reversing between pairs of faces. This produces an oriented manifold except near the vertices, which must be removed in order for the identification to produce a manifold. (Thurston Notes, p. 3)χ(X) = k - ½ Σ χ(link(v_{k}))(Thurston, p. 122)
If M is orientable and irreducible, then π_{2}(M) = 0. (Scott, p. 483)
Sphere / Prime Decomposition - every orientable closed 3-manifold has a finite connected sum decomposition into prime manifolds. (Borisenko, p. 8)
- S^{3} / Γ, where Γ is a finite subgroup of SO(4) acting freely by rotations; Γ = π_{1}(M). If Γ is cyclic, M is a lens space.
A (p,q)-lens space is generated by Γ = ((e^{2πi/p},0),(0,e^{2πiq/p})). Two lens spaces with indices (p,q) and (p',q') are homeomorphic iff p=p' and either q=±q'(mod p) or qq' = ±1(mod p). (Ratcliffe, p. 342)- S^{1} ⊗ S^{2}, with infinite cyclic fundamental group. It is the only orientable 3-manifold that is prime but not irreducible, and the only prime orientable 3-manifold with nontrivial π_{2}.
- aspherical manifolds, irreducible with infinite noncyclic π_{1}.
A Seifert fibration is a 3-manifold fibered by circles which are the orbits of a circle action which is free except on at most finitely many fibers.(Borisenko, p. 6, 10)
The vertices of the original tetrahedra are numbered 1 through 4 and 5 through 8, all identified, and the faces (identified) are labeled A through D. All of the red edges are identified, as are all of the blue edges. The edge labels (z_{i} and w_{i}) are vertex invariants, each corresponding to the the opposite dihedral angle. The link of each vertex is denoted by the directed circles; since the tetrahdrea are ideal, the links are Euclidean triangles.The identifications produce an object with 1 vertex, 2 edges, 4 sides and 2 tetrahedra; removal of the vertex (so that χ = 0) makes the object a 3-manifold.The vertex invariant of a vertex of a Euclidean triangle in the complex plane is the ratio of the adjacent sides. Labeling the vertices clockwise,
- z_{1} = z / 1
- z_{2} = (z - 1) / z
- z_{3} = 1 / (1 - z)
Note that z_{1} z_{2} z_{3} = -1.
By considering the neighborhood of the removed vertex, we see that the boundary is a torus (the grayed area is a fundamental domain for the torus, but the green area is the fundamental domain with respect to the knot):
Here, the numbers indicate the vertices, the vertex invariants are labeled, and each triangle represents the link of its associated vertex. The arrows correspond to the directed circles; rotation across a vertex invariant z_{i} in a counterclockwise direction introduces a factor of z_{i} in the numerator, while rotating across z_{i} in a clockwise direction introduces a factor of z_{i} in the denominator. The red line corresponds to the holonomy generator in the longitudinal direction of the torus (dH(l)/dl = z_{1}^{2}/w_{1}^{2}, after dividing by z_{1}^{2}z_{2}^{2}z_{3}^{2}w_{1}^{2}w_{2}^{2}w_{3}^{2} ≡ 1), and the blue line correspond to the generator of holonomy in the meridianal direction ( dH(m)/dm = w_{1}/z_{3}).
This calculation used the "gray" domain. If we use the "green" domain, the meridian stays the same, but the longitude becomesFor the manifold to be complete, the products of the vertex invariants around any intersection of the link diagram must be 1:1/√z_{1} * w_{3} * z_{2} * w_{3} * 1/z_{2} * w_{2} * w_{3} * z_{2} * w_{3} * z_{3} * z_{1} * w_{2} * √z_{1} * -1= - z_{1} z_{2} z_{3} w_{2}^{2} w_{3}^{4}(Neumann and Zagier, p. 329)= w_{3}^{2} / w_{1}^{2}
= 1 / (w_{1}^{2} (1- w_{1}^{2}))
(These are often called gluing consistency conditions.)In addition, completeness imposes two other conditions: dH(l)/dl and dH(m)/dm must both equal one (the holonomy must be trivial). The first condition tells us that z=w; the second determines z:
z (1 - z) ≡ 1Since we require solutions such that Im(z) > 0 (Im(w) > 0) (all dihedral angles positive and ≤ π), z = (-1)^{1/3}. Notes that this solution also satisfies the first completeness condition.
a(z) = z / (z + 1),and G is doubly degenerate. (Marden, section 3.14)(a = ((1,0),(1,1)) and is therefore parabolic.)b(z) = z + e^{i π/3},(b = ((1,e^{i π/3}),(0,1)) and is also parabolic.)
M = X / Γ. (Borisenko, p. 11, 16, 20, 22)So X is the universal cover of M, and π_{1}(M) = Γ.Thurston conjectured that every oriented closed prime 3-manifold has a torus decomposition such that its components have one of the following geometric structures. It was proved by Perelman in 2003.
X | S^{3} | H^{3} | E^{3} | H^{2} ⊗ R | S^{2} ⊗ R | SL(2,R)~ | Nil | Sol |
---|---|---|---|---|---|---|---|---|
G | SO(4) | PSL(2,C) | R^{3} ⊗ SO(3) | OPS Isom H^{2} ⊗ Isom E^{1} | OPS SO(3) ⊗ Isom E^{1} | SL(2,R)~ ⊗ R | (1) | (2) |
H_{0} | SO(3) | SO(3) | SO(3) | SO(2) | SO(2) | SO(2) | SO(2) | {e} |
ds^{2} | dθ_{1}^{2} + sin(θ_{1})^{2} dθ_{2}^{2} + sin(θ_{1})^{2} sin(θ_{2})^{2} dφ^{2} | (dx^{2} + dy^{2} + dz^{2}) / z^{2} | dx^{2} + dy^{2} + dz^{2} | dx^{2} + cosh^{2}x dy^{2} + dz^{2} | dθ^{2} + sin(θ)^{2} dφ^{2} + dz^{2} | dx^{2} + dy^{2} - dz^{2} | dx^{2} + dy^{2} + (dz - x dy)^{2} | e^{2z} dx^{2} + e^{-2z} dy^{2} + dz^{2} |
isometry group | O(4) | Moeb(E^{2}) | O(3) ⊗ R^{3} | Isom(H^{2}) ⊗ R | O(3) ⊗ R | SL(2,R)~ ⊗ R | Nil ⊗ S^{1} | Sol |
topologically = | H^{2} ⊗ R | R^{3} | R^{3} | |||||
bundle structure | nontrivial S^{1} over S^{2} | trivial S^{1} over T^{2} | trivial S^{1} over surface g > 1 | S^{1} over S^{2} | nontrivial S^{1} over surface g > 1 | nontrivial S^{1} over T^{2} | nontrivial T^{2} over S^{1} | |
Bianchi class | IX | V, VII | I | III | (none) | III, VIII | II | VI |
topological invariants | e ≠ 0 | e = 0 | e = 0 | e = 0 | e ≠ 0 | e ≠ 0 | ||
R | 6 | -6 | 0 | -2 | 2 | 0 | -1/2 | -2 |
R_{a b}R^{a b} | 12 | 12 | 0 | 2 | 2 | 0 | 3/4 | 4 |
X | "round" T^{3} | H^{3} | H^{3} |
---|---|---|---|
ds^{2} | (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2} dc_{1}^{2} + (r_{b} - c_{3} r_{c})^{2} / (1 - c_{2}^{2}) dc_{2}^{2} + r_{c}^{2} / (1 - c_{3}^{2}) dc_{3}^{2} | dx^{2} + dy^{2} + dz^{2} - Σ x_{i} x_{j} dx_{i} dx_{j} / (1 + r^{2}) | dr^{2} + cosh^{2} r dx^{2} + sinh^{2} r dθ^{2} |
R | 2 c_{3} (r_{a} - 2 c_{2} r_{b} + 3 c_{2} c_{3} r_{c}) / (r_{c} (-r_{b} + c_{3} r_{c}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})), < 0 | -6 | -6 |
R_{a b}R^{a b} | (2 c_{3}^{2} (r_{a}^{2} - 3 c_{2} r_{a} r_{b} + 3 c_{2}^{2} r_{b}^{2} + 4 c_{2} c_{3} r_{a} r_{c} - 8 c_{2}^{2} c_{3} r_{b} r_{c} + 6 c_{2}^{2} c_{3}^{2} r_{c}^{2})) / (r_{c}^{2} (-r_{b} + c_{3} r_{c})^{2} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2}), > 0 | 12 | 12 |
(bundle structure Anderson, p. 185)
(curvature invariants from Koehler)
In the above coordinates, geodesics beginning in the x-y plane lift into the z direction; similarly with the x-z and y-z planes. In the x and z directions, straight lines are geodesics, but not in the y direction.
In the above coordinates, geodesics in the x-z or y-z planes stay in those planes; geodesics in the x-y plane do not. Straight lines in the z direction are geodesics, but not in the x or y directions.
The center of a group is the set of all members which commute with every member of the group.
d g(t) / dt = -2 Ric(g(t)) + λ(t) g(t)to approach the singularities and then excise them. To get a feel for how it works, consider Einstein metrics, which are "fixed points" of the flow. For metrics of positive curvature, the flow contracts the metric (to a point in finite time, unless the flow is volume-normalized); for metrics of negative curvature, the flow expands the metric forever. The Ricci Flow preserves isometries since it is diffeomorphism invariant, so the geometric structure is valid arbitrarily close to each singular region. Perelman showed that for an arbitrary 3-manifold, only a finite number of surgeries are required to decompose the manifold into sub-manifolds, each of which has one of the eight geometric structures above. (Anderson, p. 185-9, 191-2)
S^{3}, E^{3}, SL(2,R)~, Nil and Sol are unimodular Lie groups. (Scott, p. 464, 468, 470, 478)
Every closed Euclidean 3-manifold is the quotient of T^{2} ⊗ R by the action of a discrete group Γ:
The remaining manifolds are not orientable:
Three of these are Klein bottle bundles over circles. (Weeks (Shape), p. 248)
(Thurston, pp. 159, 233-238)
All 10 closed Euclidean 3-manifolds are finitely covered by the 3-torus. (Scott, p. 448)
Every elliptic 3-manifold is either a lens space, or the quotient of RP^{3} by one of the following groups:
(Thurston, pp. 243, 250)
3-manifolds with geometries E^{3}, H^{2} ⊗ R or S^{2} ⊗ R are, up to finite covers, trivial circle bundles over oriented surfaces of genus g, with g = 1, g > 1 or g = 0, respectively.
3-manifolds with geometries S^{3}, SL(2,R)~ or Nil are, up to finite covers, nontrivial circle bundles over oriented surfaces of genus g, with g = 0, g > 1 or g = 1, respectively.
These are also twisted line bundles over S^{2}, H^{2} and E^{2}, respectively. (Weeks (Shape), p. 253, 254)A closed 3-manifold possessing Sol geometry is finitely covered by a torus bundle over S^{1} with holonomy given by a hyperbolic automorphism of T^{2} (an element of SL(2,Z) with distinct real eigenvalues). (Borisenko, p. 29)
(Thurston, pp. 281, 283)
A knot is a simple closed curve.A knot complement has a cusp along each link component. (Gukov, p. 17)
A link is a union of disjoint knots.
Every noncompact hyperbolic 3-manifold of finite volume can be decomposed into a finite number of ideal hyperbolic tetrahedra. (Callahan et al., p. 321)
The toral cross-section of a cusp can be seen as the open tubular neighborhood of the link components strung along the edges of an ideal tetrahedron. (Weeks)
The link of a cusp (topologically a torus or a Klein bottle) is triangulated by the links of the vertices with appropriate identification.
If the first homotopy group is generated by {x^{i}} and the product of the vertex invariants along each x is Z_{i}, the holonomy along a path a_{i}x^{i} is Z_{i}^{ai} (implicit summation). If the manifold is complete, the holonomy is trivial.
(Ratcliffe, section 10.5)
(p,q)-Dehn surgery is a Dehn filling where the meridian wraps p times around the meridian and q times around the longitude (p and q relatively prime, q ≠ 0).
The meridian is the homotopy generator of "shorter" radius (if the torus is filled, it bounds an embedded disc). The longitude is the homotopy generator of "larger" radius (orthogonal to the meridian, intersecting it once).(p,q) and (-p,-q) surgeries produce the same manifold.
(p,q) Dehn surgery on a hyperbolic manifold will produce another hyperbolic manifold except in finitely many (at most 12) cases. (Marden, section 4.10)The result of Dehn surgery always has a cyclic fundamental group. (Thurston Notes, p. 2)
Every 3-manifold can be obtained from Dehn filling of a finite-degree covering space of the Borromean Rings: they are a universal link.
The Borromean Rings are 3 linked S^{1} such that cutting any one unlinks the other two:(Thurston Notes, p. 57, 61) (Thurston, p. 131) (Thurston How..., p. 2569) (Hilden)
The Whitehead link:
and the 9_{46} knot:
are also universal.
π_{1}(knot complement) = Z iff the knot is trivial. (Thurston Notes, p. 2)
For the figure 8 knot complement, the generator of translations in the meridianal direction is (from the link diagram above)
w_{1} / z_{3}and the generator of translations in the longitudinal direction is the product of the longitudinal holonomy generator with the square of the meridianal holonomy generator:= w_{1} (1 - z_{1})
w_{2}^{2} w_{3}^{2} z_{1}^{4} z_{2}^{2} z_{3}^{2} * (w_{1} (1 - z_{1}))^{2}The gluing (completeness) conditions for (p,q) Dehn surgery can then be written as= z_{1}^{2} (1 - z_{1})^{2}.(which by the gluing consistency conditions is equal to the result we obtained above.)
where z = z_{1} is the vertex invariant of the first tetrahedron and w = w_{1} of the second.
(Ratcliffe, p. 497)
Consider the (1,2) Dehn twist of the figure 8 knot complement. We find z and w by requiring both that the original gluing condition be satisfied and that the product of one meridian traversal and two longitude traversals gives the identity:
There are 16 solutions to these equations; only one satisfies the Dehn surgery gluing conditions and produces z_{i} and w_{i} in the interval (0,π):
Adding all of the interior angles in the torus boundary of the removed vertex, we obtain 1440 degrees. Therefore the torus is described by a 10-sided polygon:
|z_{1}| arg(z_{1}) |z_{2}| arg(z_{2}) |z_{3}| arg(z_{3}) |w_{1}| arg(w_{1}) |w_{2}| arg(w_{2}) |w_{3}| arg(w_{3}) 0.61071 113.458 2.23268 42.2824 0.733397 24.2599 1.5253 27.4688 0.516168 35.8646 1.27014 116.667 (arg values in degrees)
Traversal of a meridian results in the length of a vector increasing by a factor of |w_{1} (1 - z_{1})| = 2.07978 and a rotation of arg(w_{1} (1 - z_{1})) = 3.20887 degrees counterclockwise; traversal of a longitude results in the length of a vector decreasing by a factor of |z_{1}^{2} (1 - z_{1})^{2}| = 0.693412 and a rotation of arg(z_{1}^{2} (1 - z_{1})^{2}) = 178.396 degrees:
Each "twisted quadrilateral" is a slice of the Dehn-filled cusp, with meridian and longitude edge labeling indicating traversal along two longitudes (red to green and green to blue) and one meridian (blue back to red). Since the manifold is now complete and compact, the red and blue quadrilaterals share a meridian edge (M_{0}): traversal along one meridian and two longitudes brings you back to where you started. (See Weeks.)
The volume of an ideal tetrahedron with dihedral angles α, β and γ is Π(α)+Π(β)+Π(γ). (Thurston Notes, pp. 160, 167-169)
is 6 Π(π/3) = 2.02988321282... (since an equilateral tetrahedron has dihedral angles of π/3).
The volume of the (1,2) Dehn-filled figure 8 complement is
Π(z_{1}) + Π(z_{2}) + Π(z_{3}) + Π(w_{1}) + Π(w_{2}) + Π(w_{3})= 1.3985088842...
v(S^{3} - C_{k}) = 2 k (2 Π(β/2) + Π(α + π/k) + Π(α - π/k))(Thurston Notes, p. 147, 167)
Here are the volumes produced by (p,q) surgery on the figure 8 knot complement for p,q ≤ 20. Note that for (p,q) = (1,1), (2,1), (3,1) and (4,1), the resulting manifolds are not hyperbolic. The minimum volume (0.981369) is for the (5,1) surgery; the maximum volume shown (2.02326) is for (19,20) surgery; the limiting volume is 2.02988.
Every closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic.
The interior of every compact irreducible atoroidal 3-manifold with boundary is hyperbolic.
Drilling out a geodesic increases the volume of a finite-volume hyperbolic manifold. (Marden, section 4.10)The volumes of complete hyperbolic manifolds are indexed by countable ordinals in subsets (numbers in parentheses are SnapPy identifiers for orientable manifolds):
Geometrically, a cusp is a region isometric to the psuedosphere: the t → ∞ end of the surface of revolution of (x,y) = (t - tanh t, sech t) (parameterized in two dimensions as ds^{2} = tanh^{2}(u) du^{2} + sech^{2}(u) dv^{2}, u ∈ [0,∞), v ∈ [0, 2π]):(Thurston Notes, p. 38, 116, 129, 139) (Thurston How..., p. 2559, 2563)
It is topologically T^{2} ⊗ [0,∞), has finite surface area (2π) and (interior) volume (π/3), and constant negative curvature; the hyperbolic radian is taken to be unit distance. Note that the surface is not extensible at the t=0 end; the tangent with respect to t is infinite there. The limit surface of the bounding torus is a horosphere (isometric to R^{2}).
For a nonorientable cusp, substitute a Klein bottle for the T^{2}. (Ratcliffe, p. 445)
The first integral betti number of a hyperbolic 3-manifold with cusps must be ≥ the number of cusps. (Callahan et al., p. 329)
Topological invariants of a cusped manifold include:
The Chern-Simons invariant is defined modulo 1/2 for orientable manifolds. It changes sign with a change in orientation, so if there is an orientation reversing homomorphism of the manifold, the Chern-Simons invariant must be 0 or 1/4 mod 1/2.
(Callahan et al., p. 329)
A Kleinian group is a discrete torsion-free group of isometries of H^{3} (and is therefore a discrete subgroup of Moeb(E^{2})), which acts properly discontinuously (the inverse image of any compact set is also compact). The latter implies that a discrete group has a countable number of elements. Kleinian groups are usually taken to be nonelementary - they leave an infinite number of fixed points on the sphere at ∞ (the dimension of the limit set dim(Λ) > 0).∂M(G) = Ω(G) / G, and is the union of a finite number of hyperbolic Riemann surfaces with at most a finite number of punctures (arising from parabolic fixed points) and conical points (arising from elliptic fixed points). If g is the genus of the surface, n is the number of punctures and r_{i} is the order of the i^{th} cone point (r_{i} finite and ≥ 2), each boundary component is constrained byΛ(G) of a Kleinian group in H^{3} is on the sphere at ∞, and the regular set is tiled by solids whose vertices meet Λ(G) at ∞ (the surface tiles we discussed above are the faces of the solid tiles where they meet the sphere at ∞). If the solid tiles of a Kleinian group only touch the sphere at ∞ at ideal vertices, Λ(G) is the entire sphere. The figure 8 knot complement is an example of such a group; the solid tiles have infinitely many sides. (Mumford et. al., ch. 12)
2 g + n - 2 + Σ (1 - 1 / r_{i}) > 0 (see area formula above).If G has no elliptic elements (as we will usually assume), 3g + n -3 ≥ 1, and the deformation (Teichmuller) space has complex dimension 3g + n - 3. The boundary surfaces have finite area 2π (2g + n - 2) > 0, so the boundary components can be:
If G has N generators, Σ g_{i} ≤ N. If G is purely loxodromic, ∂M(G) has ≤ N/2 components.
Every closed geodesic in M(G) is the projection of a loxodromic axis (connecting the fixed points of the loxodromic element) which is independent of any elliptic elements of G.
The fixed point of a parabolic map is on the sphere at ∞. There are an infinite number of horospheres tangent to the fixed point which are left invariant by the map. A horoball is the union of a horosphere with its interior.
The projections of the universal horoballs of the group G into M(G) are mutually disjoint; there is one for each parabolic fixed point.If the fixed point is not also fixed by an elliptic map, its stablizer is either cyclic or abelian with rank 2. If cyclic, the quotient of H^{3} by the stablizer is homeomorphic to a solid cusp cylinder (a solid infinitely long cylinder with its axis removed). If rank 2, the quotient is homeomorphic to a solid cusp torus (a solid torus with its core curve removed, having finite volume and surface area). The boundary of either surface is incompressible.
π_{1} of the cusp torus injects into π_{1}(M).The (mutually disjoint) ε-thin parts of M are disjoint from the universal horoballs and are eitherThe process of Dehn-filling a cusp of a finite-volume hyperbolic manifold described above follows the removal of the interior of a solid cusp torus. The removal gives M a torus boundary component but does not change π_{1}(M).
Think of the construction of the figure 8 knot complement; the removal of the vertex (necessary to make the object a manifold) removes the core curve of the torus, making it a solid cusp torus. The knot (in general, link) is the removed axis of the solid cusp torus (in general, tori).
A solid pairing tube is a finite cylinder with its axis removed, joining small discs around two punctures associated with a rank one cusp. (Think of a pairing tube as a "pinch" joining two ε-thick parts of M, with the pinch stretched out slightly so that each puncture is on one thick part.)
M is geometrically finite iff it is compact except for a finite number of cusps, and any rank 1 cusps correspond to pairs of punctures, each determining a solid pairing tube.
If the volume of M is finite, Λ(G) is S^{2}, ∂M is empty, there are no rank 1 cusps and at most a finite number of rank 2 cusps.
(Series, p. 1, 2, 5, 7) (Marden, sections 2.2, 2.3, 3.1 - 3.4, 3.6, problem 3.20, 4.9, 4.10)
The projection of a measured lamination Λ to R = H^{2} / G is a set of simple closed geodesics. The space formed by projecting out scalar multiplication is homeomorphic to S^{6g+2n-7} (for n punctures). That projective space is the boundary of the Teichmuller space of R; the resulting closure is homeomorphic to B^{6g+2n-6}.
(Marden, sections 3.9, 5.11)
The quotient of the convex hull by G is called the convex core of M(G). It is a subset of the interior of M. Every closed geodesic in M(G) lies in the convex core (the convex core is the smallest convex set containing all closed geodesics of M(G)).
The boundary components of the convex core are incompressible iff Λ(G) is connected.The convex core has finite volume iff G is geometrically finite. If G is geometrically finite without rank 1 cusps, all rank two cusps have solid cusp tori contained in the convex core.
(Marden, section 3.11)
M(G) can be separated into a "non-cuspidal part" M_{nc}(G) and a finite set of solid cusp cylinders and solid cusp tori. The ∂M_{nc}(G) is either a component of ∂M(G) or a doubly infinite cusp cylinder or a cusp torus.(Marden, section 3.12)The relative compact core C_{rel} is defined similarly to the compact core, but has additional boundary components: a closed incompressible annulus which intersects each solid cusp cylinder, and the (also incompressible) cusp tori.
π_{1}(C_{rel}) is isomorphic to π_{1}(M).
And end of M(G) (or M_{nc}) is geometrically finite if it has a neighborhood which does not intersect the convex core of M(G). Otherwise, it is geometrically infinite, and the convex core extends into the end.
(Marden, section 5.5)
Δ_{h}u = ((1 - r^{2})^{2} / 4) (Δu + (2r/(1 - r^{2})) ∂u/∂r,where Δu is the Euclidean laplacian.
If G is nonelementary and M(G) has infinite volume,
where λ_{0} is the lowest eigenvalue of the hyperbolic Laplace-Beltrami operator on M(G) and d is the Hausdorff dimension of Λ(G) of G. (Marden, section 5.15)
The crossing number of a knot is the minimum number of crossings for all planar projections of the knot. It is an isotopy invariant.
An isotopy is a homotopy between embeddings (making it a homeomorphism).For an oriented link, we can assign a value to each crossing of different components:
The linking number is half of the sum of those crossing numbers, and is also an isotopy invariant.
- +1 if rotating the overstrand in a counterclockwise direction will align it with the understrand;
- -1 if rotating the overstrand in a clockwise direction will align it with the understrand.
For a projection of an oriented link, the writhe is the sum of the crossing numbers for all crossings. It is not an isotopy invariant.
A knot which can be separated into nontrivial sub-knots, each of which can be enclosed in a sphere intersected by only two arcs, is composite (denoted as K_{1} # K_{2}). If such a decomposition is not possible, the knot is prime.
The number of prime knots with each crossing number up to 15 have been computed:A knot which is isotopically equivalent to its mirror image is amphicheiral.
3 4 5 6 7 8 9 10 11 12 13 14 15 1 1 2 3 7 21 49 165 552 2176 9988 46972 253293
Of the knots up to 9 crossings, the only achiral knots are 3_{1}, 6_{3}, 8_{3}, 8_{9}, 8_{12}, 8_{17} and 8_{18}.A sub-knot whose planar projection intersects a surrounding circle four times is a tangle.
A knot formed by rotation or inversion of a tangle is a mutant.
An alternating knot is a knot whose crossing signs alternate as the knot is traversed in a given direction.
A knot which can be invertibly mapped to the surface of a torus is a torus knot.
A (p,q)-torus knot wraps around the meridian p times and around the longitude q times. A (9,2) torus knot:
If a knot is embedded in an unknotted solid torus, and that solid torus is then knotted, the resultant embedded knot is a satellite knot. (Adams, pp. 2-3, 8-9, 15, 19, 41, 49, 108, 115, 152) (Lickorish, p. 6)
The crossing number for a (p,q)-torus knot is min (p (q - 1), q (p - 1)). (Adams, pp. 110-111)
There are similar expansions for Lissajous knots and the knots 3_{1}, 4_{1}, 5_{1} and 8_{19}. (Kauffman (Fourier), p. 366) (Trautwein, pp. 355-356, 359, 361)
A hyperbolic knot is a knot whose complement can be given a metric of constant curvature -1.(Adams, pp. 119-120)
P_{L} for an arbitrary link is constructed recursively from polynomials for simpler links.
P_{L1 # L2} = P_{L1} P_{L2}
P_{L} is unchanged by mutation of L.
P_{L} is invariant under reversal of orientation of all components. (Lickorish, p. 179, 180)
The Alexander polynomial Δ_{L} (t) = P_{L} (i, i (t^{1/2} - t^{-1/2}). (Lickorish, p. 180)
The Kauffman polynomial F(a,z) = a^{- writhe} Λ (a,z) is defined by the following skein relations:
F(a,z) is an isotopy invariant independent of P(l,m). (Lickorish, p. 174)
F_{L1 # L2} = F_{L1} F_{L2}
F_{L} is unchanged by mutation of L.
F_{L} is invariant under reversal of orientation of all components. (Lickorish, p. 179, 180)
©2018, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.