Notes on Topology and Geometry

This collection of definitions and results is the product of a lengthy fishing trip for interesting and perhaps useful information. No pretense is claimed that it is in any way complete. Any errors are the sole responsibility of the fisherman, and will be cheerfully corrected as they are found.
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:
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.
Zn 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.


On n-Manifolds

This section applies to spaces of arbitrary dimension.




Characteristic Classes




Manifolds of Constant Curvature

In any dimension n, there are only finitely many compact Euclidean manifolds up to homeomorphism; any such manifold can be finitely covered by Tn. (
Thurston, p. 125)
positive (Sn)negative (Hn)
embeddingΣ xi2 = r2 in Rn+1 x02 - Σ xi2 = - r2 in R1,n (one sheet)
embedding mapx1 = r cos θ1x0 = r cosh η
x2 = r sin θ1 cos θ2 x1 = r sinh η cos θ1
xn = r sin θ1 sin θ2 ... sin θn-1 cos θn xn-1 = r sinh η sin θ1 ... sin θn-2 cos θn-1
xn+1 = r sin θ1 sin θ2 ... sin θn-1 sin θn xn = r sinh η sin θ1 ... sin θn-2 sin θn-1
isometry groupO(n+1)PO(1,n) (one sheet)
arc length (ds2)r2 (dθ12 + sin2θ122 + ... + sin2θ1 ... sin2θn-1n2) r2 (dη2 + sinh2 η dθ12 + ... + sinh2 η sin2θ1 ... sin2θn-2n-12)
curvaturen (n-1) / r2-n (n-1) / r2
volume elementrn (sinn-1 θ1) (sinn-2 θ2) ... (sin θn-1) Π dθi rn (sinhn-1 η) (sinn-2 θ1) ... (sin θn-2) dη Π dθi
volume2 π(n+1)/2 rn / Γ((n+1)/2) 2 πn/2 rn / Γ(n/2) * (-1/in) Cosh(η) 2F1(1/2, 1-n/2, 3/2, Cosh2(η))

(η runs from 0 to ∞; all θ coordinates run from 0 to π except the last, which runs to 2π; 2F1 is a hypergeometric function)
In both cases, geodesics are the intersection with a (timelike) plane in the embedding. (Ratcliffe, chs. 2, 3)

Any manifold finitely covered by S2 ⊗ S1 does not admit a metric with constant sectional curvature. (Besse, p. 158)

The Poincare ball model of Hn 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

ds2 = 4 dx2 / (1 - r2)2
where dx2 is the ordinary Euclidean metric.


On Surfaces




On 3-manifolds


Geometrization Conjecture


Link Complements

Hyperbolic Manifolds


An orbifold is a Hausdorff space M which is the quotient of a geometric space X by a discrete subgroup Γ of the similarities of X which do not necessarily act freely (Γ contains elliptic elements). In particular:

Marden, section 4.11)

(Ratcliffe, ch. 13)

(Abikoff, section 1)

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:



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