Notes on Topology and Geometry

Some notation:

Rⁿ denotes the open set of reals in n dimensions.
Sⁿ denotes the n-dimensional sphere (S¹ is a circle, S² is the surface of a ball, etc.).
Bⁿ denotes the n-dimensional ball (B¹ is a line segment with end points, B² is a disc, etc.).
Iⁿ 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).

Q = G / H is the quotient group, such that

• for all elements g in G and h in H, g h g^-1 is in H (H is a normal subgroup), and
• for all elements q₁ and q₂ in Q, q₁ q₂^-1 is in H.

Therefore the elements of H all act as the identity element for Q.

O(n) is the group of rotations in n dimensions; SO(n) is the subgroup of O(n) which includes the identity.

Contents

• On n-Manifolds
  • Homotopy
  • Homology
  • Cohomology
  • Characteristic Classes
  • Holonomy
  • Curvature
• On Surfaces
• On 3-manifolds
• On Knots
• References

• A homeomorphism is a continuous invertible map.
• Quantities which are unchanged under homeomorphisms are called topological invariants.
• A diffeomorphism is an invertible differentiable map.
• A manifold is a topological space locally homeomorphic to Rⁿ.
• A manifold with boundary is a topological space locally homeomorphic to Bⁿ.
• If M and N are manifolds with boundary, ∂(M x N) = the union of (M x ∂N) and (∂M x N). (Christenson, p. 465)
• The doubling of a manifold M with boundary is the operation of gluing two copies of M along the boundaries. (Scott, p. 425)
• A closed manifold is compact without boundary.
• The torus (Tⁿ) is obtained by pairwise identifying diametrically opposite sides of an n-cube. It can also be obtained by identifying corresponding points on a pair of coaxial T^n-1 (one inside the other). It is topologically equivalent to T^n-1 x S¹. (Alexandroff, p. 10, 11)
• S^n-1 x I is an n-annulus; S^n-1 x (0,1) is an open n-annulus; S^n-1 x (0,1] is a half-open n-annulus. (Christenson, p. 451)
• Real projective space RPⁿ is Sⁿ / identification of antipodal points. This is the only order-two element of O(n+1) that acts freely on Sⁿ.

  A group acts freely if only the identify element leaves fixed points.

  (Thurston, p. 243)
• If M is an S^n-1 in Sⁿ, then Sⁿ / M has 2 components with boundaries M. (Christenson, p. 456)
• S^n-1 = SO(n) / SO(n-1). (Bott and Tu, p. 195)
• SO(n) is the component of the identity of O(n). (Hirsch, p. 14)

  O(n) is the isometry group of S^n-1. (Thurston, p. 64)

• If G is finite and operates freely on Sⁿ and M = Sⁿ / G:

  • if any element of G changes the orientation, n must be even;
  • if any element other than the identity of G preserves the orientation, n must be odd;
  • if n is even and G is nontrivial, it must be a cyclic group of order 2, and H_p(M) = 0 if p is even, H_p(M) = G if p is odd;
  • if n is odd and G is abelian, G is cyclic, H_n(M) = Z and H_p(M) = 0 if p is even, H_p(M) = G if p is odd;

    A group is abelian if its elements all commute (g₁ g₂ = g₂ g₁).

  • a discrete group G acting freely on a compact manifold implies that G is finite.

  (Hu, p. 290, 291)

• M compact can be considered the quotient space resulting from identifications on the boundary of a Bⁿ. (Christenson, p. 461)
• Every differentiable n-manifold can be imbedded in R²ⁿ. (Greenberg, p. 228)
• A closed n-manifold M is the connected sum N₁#N₂ if there is an embedded S^n-1 in M which separates M into M₁ and M₂ with boundaries S^n-1. N_i are the closed manifolds formed by filling the M_i boundary spheres with Bⁿs. (Borisenko, p. 7)
• Two n-manifolds belong to the same (co)bordism class iff their disjoint union is the boundary of a smooth compact (n+1)-manifold.
• Replacement of a point by the set of tangent lines at that point is called blowing up the point. (Thurston, p. 30)

• Two maps of S^k into M are said to be homotopically equivalent if they are continuously deformable into one another.

  The set of equivalence classes of such maps forms a group p_k(M), the k^th homotopy group of M.
  p₁ is called the fundamental group.

  p₁ are isomorphic for homeomorphic manifolds.

  If M = M₁ x M₂, p_k(M) = p_k(M₁) + p_k(M₂).
  For k > 1, p_k is abelian.
  For k > 1, p_k(S¹) = 0.
  For k < n, p_k(Sⁿ) = 0.
  p_k(S^k) = Z.
  For n > 1, p₁(RPⁿ) = Z₂. (Greenberg, p. 25, 34, 36)

  p₁(M) is non-abelian if M is a torus of genus > 1, or a connected sum of RP².

  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:

  p_k(Sⁿ) = p_k(S^2n-1) = p_k-1(S^n-1).
  p_n+1(Sⁿ) = p_n+2(Sⁿ) = Z₂ for n > 2.
  p_k(S²) = p_k(S³) for k > 2.
  p₃(S²) = Z.
  p_k(S²) = Z₂ for k = 4, 5, 7 and 8.
  p₆(S²) = Z₁₂.
  p₉(S²) = Z₃.
  p₁₀(S²) = Z₁₅.

  (Hu, p. 58, 153, 325, 328, 329, 332)
• A manifold is simply connected if p₁ = 0.
• The universal cover of a manifold M is locally homeomorphic to M but is simply connected, with discrete fibers over M. If the cover is locally pathwise connected, the fiber group is isomorphic to p₁(M).

  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 RP³, with p₁ = Z₂, and universal cover S³.
  p₁(SO(n)) = Z₂.
  The proper Lorentz Group is topologically RP³ x R³, with universal cover SL(2,C).
  For k > 1, p_k(M) = p_k(E) if E is the universal cover of M. (Greenberg, p. 24, 30, 35)

  If M is a closed surface other than S² or RP², p₁(M) is infinite and the universal cover is R². (Hu, p. 96)

• If G is a simply connected group and H is a discrete normal subgroup, then in the neighborhood of the identity, p₁(G/H) is isomorphic to H (and is abelian). (Greenberg, p. 18)
• Every connected non-orientable manifold has a 2-fold connected orientable cover.
  Every connected manifold whose p₁ does not contain a subgroup of index 2 is orientable.

  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 Sⁿ, T² (of arbitrary genus), RPⁿ, CPⁿ and HPⁿ, only RPⁿ for n even is not orientable. (Greenberg, p. 162, 168)
• If G is a Lie group acting transitively, analytically and with compact stabilizers on a simply connected manifold X and M is a closed differentiable manifold, (G, X) structures on M (up to diffeomorphism) are in 1:1 correspondence with conjugacy classes of discrete subgroups of G that are isomorphic to p₁(M) and act freely on X with quotient M.

  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.

  The stabilizers of a group are those elements which leave fixed points.

  A (G, X) structure is a manifold X such that the coordinate systems in every overlapping patch are related by elements of the group G.

  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.

  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.

  A torsion element x satisfies x^m = 0 for some integer m.

  (Thurston, p. 157, 167, 222)

• Let M be a smooth connected manifold. A p-chain is S_i c_i N_i, where the N_i are smooth, oriented p-submanifolds of M. (Eguchi, Gilkey and Hanson, p. 233)
• A cycle is a chain with no boundary.
• The p^th homology group is H_p(M;G) = Q_p / B_p, where G is one of R, C, Z or Z₂, Q_p is the set of p-cycles and B_p is the set of p-boundaries in M (a subset of Q_p). It is the set of equivalence classes of cycles which differ only by boundaries; equivalently, the set of equivalence classes of cycles which are not boundaries of a sub-manifold N^p+1.

  H_p(M;G) = 0 for p > n = dim(M).
  If M is connected, H₀(M;G) = G.
  If M is orientable, H_n(M;G) = G.
  If M is orientable and G is R, C or Z₂, H_p(M;G) = H_n-p(M;G) (Poincare Duality). (Eguchi, Gilkey and Hanson, p. 233)

  H_p(Sⁿ) is isomorphic to H_p-1(S^n-1) for p > 1.
  H_p(RPⁿ) = G₂ (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)

  If [G, G] is the smallest normal subgroup of G such that G / [G, G] is abelian, H₁ = p₁ / [p₁, p₁]. (Bott and Tu, p. 225)

If E is the universal cover of M, p₂(M) is isomorphic to H₂(E).
If p_k(M) = 0 for k < p, p_p(M) is isomorphic to H_p(M). (Hurewicz Theorem) (Hu, p. 154, 167)

• b_p is the p^th Betti number, equal to dim(H_p(M;R)). It is a topological invariant.

  b₀ = 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. Tⁿ 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)

  b₁ = b₂ = 0 for a compact semi-simple Lie group (metric is non-degenerate). b₃ = 1 for a compact simple Lie group. (Goldberg, p. 140, 145; see also p. 144)

  b_p = 0 for RPⁿ except for b₀ = 1, and b_n = 1 if n is odd.
  If M is a compact submanifold of Rⁿ (n > 1) with k connected components, b_n-1(Rⁿ - M) = k. (Greenberg, p. 129, 167)

• The Euler Characteristic of a manifold without boundary M is c(M) = S_p (-1)^p b_p.

  An n-manifold M is equivalent to the product of a 1-manifold and an (n-1)-manifold iff c(M) = 0. (Thurston, p. 115)

• If M = M₁ x M₂, H_k(M;R) = S_{p + q = k} H_p(M;R) x H_q(M;R) (Kunneth formula).

  Further, c(M) = c(M₁) c(M₂). (Eguchi, Gilkey and Hanson, p. 238)

  If M and N are connected n-manifolds, c(M # N) = c(M) + c(N), - 2 if n is even. (Greenberg, p. 131)

• If K is an n-fold cover of M, c(K) = n c(M). (Scott, p. 426)
• H_p(M;Z) is isomorphic to the cobordism group of M for p < 4. (Kauffman, p. 319)

• For G one of R, C or Z₂, the vector space dual to H_p(M;G) is the cohomology group H^p(M;G). It is the set of equivalence classes of closed forms (dw = 0) which differ only by exact forms; equivalently, the set of closed forms which are not exact differential forms (dw = 0 but w != df). When G = R, it is called de Rham cohomology.

  H⁰(M;R) is the space of constant functions.
  Hⁿ(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)

• For a compact manifold without boundary, H_p(M;R) and H^p(M;R) are isomorphic. Note that when G = Z, the cohomology group does not include torsion and the two vector spaces are not isomorphic. (Eguchi, Gilkey and Hanson, p. 236)
• For a compact manifold without boundary, H^p(M;R) is isomorphic to the set of harmonic p-forms on M (whose Laplacian vanishes). (Eguchi, Gilkey and Hanson, p. 238)

• The Stiefel-Whitney classes of a real vector bundle are the Z₂ cohomology classes.
  H¹(M;Z₂) is zero iff M is orientable.
  If H²(M;Z₂) is nonzero, it is not possible to consistently define spinors on M. (Eguchi, Gilkey and Hanson, p. 316)

  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)
• The Pontrjagin class of a real vector bundle of dimension > 3 is

  Det(I - W / 2p) = 1 + p₄ + p₈ + ...

  where W 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)

• The Euler class of a real even-dimensional orientable vector bundle is the sum of the traces of the wedge products of the SO(k) curvature 2-form.

  In 2 dimensions, we have the Gauss-Bonnet theorem: ∫ W dA = 2p c.

  If the manifold has boundary, the integral over M which defines the Euler characteristic must be corrected by the Chern-Simons term; ie., in four dimensions:

  tr A ^ dA + (2/3) A ^ A ^ A

  where A is the connection one-form. (Eguchi, Gilkey and Hanson, p. 313, 350)
• The Lefschetz fixed point theorem states that the Euler characteristic is the number of fixed points of an isometry on M. It is also equal to the number of zeroes of the Killing vector field defined by f.

  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)

• The characteristic classes are all topological invariants. They must vanish if a global section is to exist on the bundle.

  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 a smooth closed orientable n-manifold can be embedded in R^{n + k}, the Euler class of the normal bundle (normal to the tangent bundle in M x R^{n + k}) must be zero. (Milnor and Stasheff, p. 101, 120)

• If M has a (group) G structure, the map from p₁(M) -> G is the holonomy group of M.

  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 G as X / G.

  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.)

  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 )

• The Riemann curvature tensor has n² (n² - 1) / 12 independent components. (Wald, p. 54)
• A Riemannian space V_n has constant curvature iff it locally admits an isometry group of rank n (n + 1) / 2.

  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)

• The number of independent curvature invariants of order 0 (involving no covariant derivatives) in D > 2 dimensions is

  N^D₀ = ( D - 2 ) ( D - 1 ) D ( D + 3 ) / 12

  while for order k there are:

  N^D_k = D ( k + 1 ) ( D + k + 1 ) ! / ( 2 ( D - 2 ) ! ( k + 3 ) ! )

  (Haskins)
• An n-dimensional Riemannian space is completely characterized by the curvature tensor and its n (n + 1) / 2 first covariant derivatives. This number is an upper limit; fewer derivatives may be needed. (Karlhede)

• Some planar constructions:
  

  The two tori represent rectangular and hexagonal tilings of the Euclidean plane, respectively. Omit the singly-hashed identification on the Klein bottle to make a Moebius strip. (Henle, p. 106, 111, 112) (Thurston, p. 5)

  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 S¹, we can remove a B² from an S² and glue in a Moebius strip to obtain RP². Do it twice to obtain a Klein bottle. This operation is termed adding a cross-cap.
  A sphere with g cross-caps has c = 2 - g. (Edelbrunner and Harer, p. 34, 36)

  A manifold obtained by identification is orientable iff all face identifications relative to a consistent orientation are orientation-reversing. (Thurston, p. 121)

• A compact oriented surface S embedded in a manifold N is incompressible if for every closed disc D embedded in N such that the intersection of D with S is a subset of ∂D, ∂D is contractible in S. (Borisenko, p. 9)
• Every oriented, closed 2-manifold is the quotient of either S², E² or H², by a discrete subgroup of isometries acting freely on it. (Borisenko, p. 2)

  Every compact, connected 2-manifold is topologically equivalent to a sphere (2-cell notation aa^-1), a connected sum of tori (aba^-1b^-1cdc^-1d^-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 S² - some number of B², with either I² 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 manifold is diffeomorphic to the surface obtained from an orientable surface of genus 1 - (c + d) / 2 by the removal of d disjoint B².

  Every orientable surface bounds a compact 3-manifold.
  Every non-orientable surface of even genus bounds a compact 3-manifold. (Hirsch, p. 189, 193, 194, 205, 207)

  Every open surface admits a hyperbolic structure. (Scott, p. 421)

• The Euler characteristic of a surface can be computed from its triangulation as

  c = #vertices - #edges + #faces

  A closed surface has hyperbolic, Euclidean or spherical (elliptical) structure iff its Euler characteristic is negative, zero or positive, respectively.
  This holds for surfaces in general, except that surfaces homeomorphic to R², S¹ x R or the Moebius strip can be given both Euclidean and hyperbolic structures. (Thurston, p. 28, 118)
• Any isometry of a surface can be generated by reflections. (Scott, p. 406, 411, 417)
• The moduli space of a surface is the quotient of the space of structures admissible to the surface, by the action of the diffeomorphisms of the surface (Diff).
  The Teichmuller space restricts the diffeomorphisms to those homotopic to the identity by a homotopy which takes the boundary into itself at all times (Diff₀).
  The mapping class group is Diff S / Diff₀ S. Therefore the moduli space is the quotient of the Teichmuller space by the mapping class group.

  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 T² 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 |c|}.

  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 S² - 3 B² (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.

  The outer automorphism group is the conjugacy class under diffeomorphisms of maps of p₁ into itself.

  (Thurston, p. 260, 262, 264, 266, 271, 276, Thurston Notes, pp. 89-91)

• surface	b1	b2	c	p1
  E2	0	1	2	0
  H2	0	1	2	0
  annulus	1	1	0	Z
  cylinder	1	1	0	Z
  disc	0	1	1	0
  S2	0	1	2 (4)	0
  T2	2	1	0 (2)	Z x Z
  Riemann surface genus g	2 g	1	2 - 2 g (4)	Z 2g | aba-1b-1cdc-1d-1... = 1 (Bott and Tu, p. 240)
  Moebius strip	1	0	0	Z
  Klein bottle	2 (3)	1 (3)	0 (2)	Z x Z | abab-1 = 1
  RP2	0	0	1 (2)	Z2
  connected sum of n RP2	n - 1	0	2 - n (4)	Zn | aabbcc... = 1 (1)
  non-orientable surface genus g	g - 1 (5)	0 (5)	2 - g

  (Gray, p. 46 ⁽¹⁾) (Henle, p. 167 ⁽²⁾, 193 ⁽³⁾, 295 ⁽⁴⁾) (Hu, p. 28 ⁽⁵⁾)
• Removal of an open disc reduces the Euler characteristic by 1.

  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 RP² 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

  c = 2 - 2g - c - d

• S² x S¹ is obtained by identifying corresponding points on a pair of S², one inside the other; it can also be obtained by identifying corresponding points on two congruent solid tori. (Alexandroff, p. 11)
• All compact orientable 3-manifolds are boundaries. (Greenberg, p. 240)
• Every orientable 3-manifold can be obtained from Dehn surgery, cutting a tubular neighborhood of a knot or a link and gluing it back together with a different identification.

  A knot is a simple closed curve.
  A link is a union of disjoint knots.

  Every 3-manifold can be obtained by gluing together the boundaries of two solid tori of some genus. (Thurston Notes, pp. 1-3)
• The Dehn filling of a 3-manifold M with torus boundary is the gluing of a solid torus to the boundary of M. Dehn fillings are parametrized by H₁(∂M, Z) (without regard to orientation). The class parametrizing the filling is called the slope. (Dunfield, p. 419)

  (p,q)-Dehn surgery is a Dehn filling where the meridian wraps p times around the meridian and q times around the longitude.

  The meridian is the homotopy generator of shorter radius. The longitude is the homotopy generator of larger radius.

  (Thurston Notes, p. 57)
• A 3-dimensional gluing of tetrahedra is a manifold iff

  c = #vertices - (2 #edges - 3 #faces + 4 #tetrahedra) / 2

  is zero. So c = 0 for any closed 3-manifold. (Thurston, p. 122)
• A closed 3-manifold is irreducible if every embedded S² is the boundary of a B³.

  If M is orientable and irreducible, then p₂(M) = 0. (Scott, p. 483)

• A closed 3-manifold is prime if every separating embedded S² is the boundary of a B³. The three main classes are

  • S³ / G, where G is a finite subgroup of SO(4) acting freely by rotations; G = p₁(M). If G is cyclic, M is a lens space.
  • S¹ x S², 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 p₂.
  • aspherical manifolds, irreducible with infinite noncyclic p₁.

  Sphere / Prime Decomposition - every orientable closed 3-manifold has a finite connected sum decomposition into prime manifolds. (Borisenko, p. 8)
• Torus Decomposition - every closed orientable irreducible 3-manifold has a finite collection of disjoint incompressible tori which decompose it into a finite collection of compact 3-manifolds with toral boundary, each of which is either torus-irreducible or Seifert fibered.

  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)
• Mostow Rigidity - If N and N' are 3-manifolds carrying complete hyperbolic metrics of finite volume, and their fundamental groups are isomorphic, then N and N' are diffeomorphic. This implies volume is a topological invariant.

  Any metric of curvature +1 on a spherical 3-manifold is unique up to isometry. (Borisenko, p. 13)

• A complete oriented hyperbolic 3-manifold with finite volume is the union of a compact submanifold bounded by tori and a finite number of horoballs modulo Z + Z actions.

  A horoball, which is centered at a point x on the sphere at infinity, is the union of the surface orthogonal to all lines through x, with its interior.

  The volumes of two homotopy equivalent, closed, oriented hyperbolic manifolds are equal.

  The volumes of complete hyperbolic manifolds are indexed by countable ordinals in subsets, first with no cusps, then with one cusp, etc., with the volume increasing with the number of cusps.

  A cusp is a region isometric to the t -> ∞ end of the surface of revolution of (x,y) = (t - tanh t, sech t):

  

  It has constant negative curvature.

  (Thurston Notes, p. 38, 116, 129, 139)

  A knot complement has a cusp along the knot. (Gukov, p. 17)

  The first integral betti number of a hyperbolic 3-manifold with cusps must be ≥ the number of cusps. (Callahan et al., p. 329)

• Every compact 3-manifold contains a simple closed curve whose complement admits a noncompact hyperbolic structure of finite volume.

  Every noncompact hyperbolic 3-manifold of finite volume can be decomposed into a finite number of ideal hyperbolic tetrahedra.

  An ideal tetrahedron has its vertices on the sphere at infinity.

  (Callahan et al., p. 321)
• For a square-free integer d > 0:
  1. PSL(2, O_d) is a discrete subgroup of PSL(2, C). Let G be a torsion-free subgroup of finite index in PSL(2, O_d). Then H³ / G is an oriented hyperbolic 3-manifold of finite volume.
  2. Let D = d if d is 3 mod 4, otherwise D = 4 d.
  3. Let (-D n) be defined by
     • (-D n) = P_{prime factors p} (-D p_i)
     • (-D 1) = 1
     • (-D 2) = 1 if -D is 1 mod 8, -1 if -D is 5 mod 8
     • (-D p) = 0 if p is a factor of D
     • (-D p) = 1 if -D is some square mod p, otherwise -1
  4. The Lobachevsky function p(q) = - ∫₀^q ln |2 sin u| du
  5. the volume of H³ / G = (D / 12) S_{k mod D} (-D k) p(p k / D) times the index of G in PSL(2, O_d)

  (Thurston Notes, pp. 167-169)

• For the simple k-linked "closed" chain C_k, let a = cos^-1 (cos (p/k) / √2) and b = p - 2a.
  Then the volume of the complement for k > 1 is

  v(S³ - C_k) = 2 k (2 p(b/2) + p(a + p/k) + p(a - p/k))

  Another family of chain complements is D_2k (k > 2), with volumes

  v(S³ - D_2k) = 8 k (p(p/4 + p/(2k)) + p(p/4 - p/(2k)))

  (Thurston Notes, p. 147, 151, 167)
• The Weeks manifold, of volume 0.9427, appears to be the unique closed orientable hyperbolic 3-manifold of smallest volume. (Gabai, Meyerhoff and Milley)
• Let M be a hyperbolic manifold formed by (p,q)-Dehn surgery on a knot complement. The volume of M is less than the volume of the original (cusped) complement, approaching it in the limit p² + q² -> ∞. (Gukov, p. 17)
• The minimum dimension of the Teichmuller space of a 3-manifold that admits a hyperbolic structure is -6 c. (Thurston Notes, p. 88)
• If M is a smooth manifold, the mapping torus of a diffeomorphism f: M -> M is the manifold obtained from M x I by identifying the ends of the cylinder via f.

  Every closed Euclidean 3-manifold is the quotient of T² x R by the action of a discrete group G:

  • T³, the mapping torus of T² by a translation
  • the mapping torus of T² by the cyclic group of order 2 (C₂)
  • the mapping torus of T² by C₃, C₄ or C₆, each of which have two forms distinguished by handedness
  • G = C₂ * C₂, acting as rotations around perpendicular axes

    The remaining manifolds are not orientable:

  • G = C₂ * C₂, acting as glide-reflections in a plane, of index 2 or 4
  • G = C₂ * C₂, acting as one screw motion and one reflection, in orthogonal planes, with two possible spacings

  (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 orientable.

  Every elliptic 3-manifold is either a lens space, or the quotient of RP³ by one of the following groups:

  • the product of the dihedral group (of a regular polygon), the tetrahedral group (order 12), the octahedral group (order 24) or the icosahedral group (order 60), with a cyclic group of order relatively prime to the order of the first factor
  • a subgroup of index 3 in the product of the tetrahedral group and the cyclic group C_3m, m odd
  • a subgroup of index 2 in the product C_2m x D_2n, n even and m and n relatively prime (D is the dihedral group)

  (Thurston, p. 243, 250)

• The group of isometries of Hⁿ is generated by reflections and is isomorphic to the Moebius group Moeb_n-1: the group of homeomorphisms of the Sⁿ at infinity which takes (n-1)-spheres to (n-1)-spheres. The group of Euclidean similarities, O(n-1) and O(n+1) are all subgroups of Moeb_n-1.

  Euclidean similarities are generated by O(n), scalar dilations and translations. There are no hyperbolic or spherical similarities.

  The subgroup of Moeb_n-1 which takes an S^k to itself is isomorphic to Moeb_k x O(n-k).
  The group of orientation-preserving isometries of H³ is PSL(2,C). (Thurston, p. 61, 62, 83, 99)
• There are seven manifolds with S² x R structures: S² x R and two line bundles over P² (noncompact); two S² bundles over S¹, P² x S¹ and P³ # P³ (compact). The latter is the only closed 3-manifold with a geometric structure that is a non-trivial connected sum. (Scott, p. 457)
• M has a Sol structure iff M is either a bundle over S¹ with fiber the torus or Klein bottle, or is the union of two twisted 1-bundles over the torus or Klein bottle. (Scott, p. 477)
• For the following, X is a connected and simply connected manifold, G is the maximal Lie Group of diffeomorphisms acting transitively on X, H₀ is the compact identity component of the isotropy group of G, and there exists a compact manifold M and a discrete subgroup G of G such that

  M = X / G. (Borisenko, p. 11, 16, 20, 22)

  So X is the universal cover of M, and p₁(M) = G.

  X	S3	H3	E3	H2 x R	S2 x R	SL(2,R)~	Nil	Sol
  G	SO(4)	PSL(2,C)	R3 x SO(3)	OPS Isom H2 x Isom E1	OPS SO(3) x Isom E1	SL(2,R)~ x R	(1)	(2)
  H0	SO(3)	SO(3)	SO(3)	SO(2)	SO(2)	SO(2)	SO(2)	{e}
  ds2	dq12 + sin(q1)2 dq22 + sin(q1)2 sin(q2)2 df2	(dx2 + dy2 + dz2) / z2	dx2 + dy2 + dz2	dx2 + cosh2x dy2 + dz2	dq2 + sin(q)2 df2 + dz2	dx2 + dy2 - dz2	dx2 + dy2 + (dz - x dy)2	e2z dx2 + e-2z dy2 + dz2
  isometry group	O(4)		O(3) x R3	Isom(H2) x R	O(3) x R	SL(2,R)~ x R	Nil x S1	Sol
  topologically =						H2 x R	R3	R3
  topological invariants	e ≠ 0, c > 0		e = c = 0	e = 0, c < 0	e = 0, c > 0	e ≠ 0, c < 0	e ≠ 0, c = 0
  R	6	-6	0	-2	2	0	-1/2	-2
  Ra bRa b	12	12	0	2	2	0	3/4	4

  X	"round" T3	H3	H3
  ds2	(ra - c2 rb + c2 c3 rc)2 dc12 + (rb - c3 rc)2 / (1 - c22) dc22 + rc2 / (1 - c32) dc32	dx2 + dy2 + dz2 - S xi xj dxi dxj / (1 + r2)	dr2 + cosh2 r dx2 + sinh2 r dq2
  R	2 c3 (ra - 2 c2 rb + 3 c2 c3 rc) / (rc (-rb + c3 rc) (ra - c2 rb + c2 c3 rc)), < 0	-6	-6
  Ra bRa b	(2 c32 (ra2 - 3 c2 ra rb + 3 c22 rb2 + 4 c2 c3 ra rc - 8 c22 c3 rb rc + 6 c22 c32 rc2)) / (rc2 (-rb + c3 rc)2 (ra - c2 rb + c2 c3 rc)2), > 0	12	12

  • "~" denotes the Universal Cover.
  • Nil = 3-dimensional Heisenberg Group (matrices of the form {{1,x,y},{0,1,z},{0,0,1}}).
  • Sol = Lie Group with topology R x R² such that (t,(x,y))->(e^tx,e^-ty); isotropy group is trivial.
  • "OPS" denotes Orientation Preserving Subgroup.
  • (1) semidirect product of X with S¹, acting by rotations on the quotient of X by its center.
  • (2) extension of X by an automorphism group of order 8.
  • e denotes the Euler number of the associated Seifert bundle; c is the Euler number of the base manifold.
  • For T³, the r_i parametrize the radii of curvature (with r_a > r_b + r_c and r_b > r_c) and the c_i the cosines of the angles for each of the S¹ factors.

  S³, E³, SL(2,R)~, Nil and Sol are unimodular Lie groups. (Scott, p. 464, 468, 470, 478)

  (Curvature invariants from Koehler)

• A closed 3-manifold possesses a geometric structure other than H³ or Sol iff it is a Seifert fiber space.

  3-manifolds with geometries E³, H² x R or S² x 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 S³, 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.

  A closed 3-manifold possessing Sol geometry is finitely covered by a torus bundle over S¹ with holonomy given by a hyperbolic automorphism of T² (an element of SL(2,Z) with distinct real eigenvalues). (Borisenko, p. 29)

On Knots

If you're going to play with knots, I recommend building a jig; tying knots without something to hold the crossovers in place can be very difficult. I simply cut a square piece of laminate and hammered 25 nails into it. Knot 6₃ (from the standard tables) is shown here:

• S¹ is the unknot.

  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.

  For an oriented link, we can assign a value to each crossing of different components:

  • +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.

  The linking number is half of the sum of those crossing numbers, and is also an isotopy invariant.

  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 K₁ # K₂). 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:

  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

  A knot which is isotopically equivalent to its mirror image is amphicheiral.

  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.

  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)

• If K₁ # K₂ is an alternating knot, their crossing numbers are simply additive. (Adams, p. 69)
• A (p,q)-torus knot is also a (q,p)-torus knot.

  The crossing number for a (p,q)-torus knot is min (p (q - 1), q (p - 1)). (Adams, pp. 110-111)

• A (p,q)-torus knot can be parametrized on a torus of radius 1/4 by
  • x(t) = cos (p t) + cos (p t) cos (q t ) / 4
  • y(t) = sin (p t) + sin ( p t) cos ( q t) / 4
  • z(t) = sin ( q t) / 4

  There are similar expansions for Lissajous knots and the knots 3₁, 4₁, 5₁ and 8₁₉. (Kauffman (Fourier), p. 366) (Trautwein, pp. 355-356, 359, 361)

• Torus knots and satellite knots are not hyperbolic. All other knots are.

  A hyperbolic knot is a knot whose complement can be given a metric of constant curvature -1.

  (Adams, pp. 119-120)
• The HOMFLY polynomial P(l,m) is defined by the following skein relations between three links L₊, L_- and L₀, which differ by only one crossing:
  • P_unknot = 1
  • l P_L+ + l^-1 P_L- + m P_L0 = 0

  P_L for an arbitrary link is constructed recursively from polynomials for simpler links.

  P_L is an isotopy invariant. (Lickorish, p. 168)

• P_{disjoint union of L1 and L2} = -(l + l^-1) m^-1 P_L1 P_L2

  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 Jones polynomial V_L (t) = P_L (i t^-1, i (t^-1/2 - t^1/2).

  The Alexander polynomial D_L (t) = P_L (i, i (t^1/2 - t^-1/2). (Lickorish, p. 180)

• If U is the 0-crossing planar projection of the unknot, and defining the following diagrams:

  

  The Kauffman polynomial F(a,z) = a^{- writhe} L (a,z) is defined by the following skein relations:
  • L_U = 1
  • L_L = a L₀
  • L_D+ + L_D- = z (L_D0 + L_D&infin)

  F(a,z) is an isotopy invariant independent of P(l,m). (Lickorish, p. 174)

• F_{disjoint union of L1 and L2} = ((a + a^-1) z^-1 - 1) F_L1 F_L2

  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)

• p_{k > 1} = 0 for any knot complement in S³. (Lickorish, p. 114)

• Adams, C. C., The Knot Book, AMS (2001)
• Alexandroff, P., Elementary Concepts of Topology, (1932), Dover (1961)
• Borisenko, A., "An Introduction to Hamilton and Perelman's work on the conjectures of Poincare and Thurston"
• Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Springer (1982)
• Callahan, P. J., Hildebrand, M. V. and Weeks, J. R., "A Census of Cusped Hyperbolic 3-manifolds", Mathematics of Computation 68 pp. 321-332 (1999)
• Christenson, C. O. and Voxman, W. L., Aspects of Topology, Marcel Dekker (1977)
• Dunfield, N. M. and Thurston, W. P., "The virtual Haken conjecture: Experiments and examples", Geometry and Topology 7 pp. 399-441 (2003)
• Edelsbrunner, H. and Harer, J., Computational Topology: An Introduction (2008)
• Eguchi, T., Gilkey, P. B. and Hanson, A. J., "Gravitation, Gauge Theories and Differential Geometry", Physics Reports 66 pp. 213-393 (1980)
• Gabai, D., Meyerhoff, R. and Milley, P., "Mom technology and volumes of hyperbolic 3-manifolds", "Minimum volume cusped hyperbolic three-manifolds", and Meyerhoff, R. and Milley, P., "Minimum volume hyperbolic 3-manifolds", in preparation.
• Goldberg, S. I., Curvature and Homology, (1962), Dover (1982)
• Gray, B., Homotopy Theory, Academic Press (1975)
• Greenberg, M. J. and Harper, J. R., Algebraic Topology, a First Course, Revised, Benjamin Cummings (1981)
• Gukov, S., "Three-Dimensional Quantum Gravity, Chern-Simons Theory, And the A-Polynomial"
• Haskins, C. N., "On the Invariants of Quadratic Differential Forms", Transactions of the American Mathematical Society 3, 71-91 (1902)
• Henle, M., A Combinatorial Introduction to Topology, W. H. Freeman (1979)
• Hirsch, M. W., Differential Topology, Springer (1976)
• Hu, S.-T., Homotopy Theory, Academic Press (1959)
• Karlhede, A., "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation 12, 693-707 (1980)
• Kauffman, L. H., On Knots, Princeton (1987)
• Kauffman, L. H., "Fourier Knots", in Ideal Knots, World Scientific (1998)
• Koehler, K. R., Computations in Riemann Geometry (2005)
• Kramer, D., Stephani, H., Herlt, H. and MacCallum, M., Exact Solutions of Einstein's Field Equations, Cambridge (1980)
• Lickorish, W. B. R., An Introduction to Knot Theory, Springer (1997)
• Milnor, J. W. and Stasheff, J. D., Characteristic Classes, Princeton (1974)
• Scott, P., "The Geometries of 3-manifolds", Bulletin of the London Mathematical Society 15 pp. 401-487 (1983)
• Thurston, W. P., Notes: "The Geometry and Topology of Three-Manifolds"
• Thurston, W. P., Three-Dimensional Geometry and Topology, Princeton (1997)
• Trautwein, A. K., "An Introduction to Harmonic Knots", in Ideal Knots, World Scientific (1998)
• Wald, R. M., General Relativity, Chicago (1984)

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