1. Quotient Spaces and the Shape of the Universe
A Topological Exploration of
3-manifolds.
Dorothy Moorefield
Mat 4710
Dr. Sarah Greenwald
10 December 2001

2. Geometric 3-manifold
A geometric three-space manifold is a space in which each point has a neighborhood that is isometric with a neighborhood of either Euclidean 3-space, a 3-sphere or a hyperbolic 3-space.

3. Metric Space A metric on a set X is a function d: x  such that:
A) d( x, y)  0  x, y .
B) d( x, y) = 0  x = y.
C) d( x, y) = d( y, x) x, y  .
D) d( x, y) + d( y, z)  d( x, z) x, y, z  .
If d is a metric on a set x, the ordered pair (x,d) is called a metric space.

4. Isometric
An isometry is a one-to-one mapping, f: (X, d)  (Y, d’) of a metric space (X, d) onto (Y, d’) such that distance is preserved. In other words for all x1 and x2 in X we have:
D( x1, x2) = d’( f(x1), f(x2)).
Two spaces are isometric if there exists an isometry from one space onto the other.

6. Cosmic Microwave Background

7. Quotient Spaces
Let (X, ) be a topological space, let Y be a set and let  be a function that maps X onto Y. Then U = { U P(Y): -1(U)   } is called the quotient topology on Y induced by . (Y, U ) is a quotient space of X.

8. Fundamental Domain
The fundamental domain is the simplest space that can be used to form the quotient spaces that form our manifolds.
This a 2-torus which is a 2-manifold. The fundamental domain in the rectangle. The 2-torus is the quotient space formed from the fundamental domain.

9. Euclidean 3-manifolds There are exactly 10 Euclidean 3-manifolds.
Four are non-orientable.
The remaining six are orientable.
Of the orienable there are the 3-torus, 1/4 turn manifold, 1/2 turn manifold, 1/6 turn manifold and the 1/3 turn manifold.

10. Path on a 3-torus

19. The Quarter-turn Manifold

21. Sources
David, W. Henderson. Experiencing geometry: in Euclidean, spherical, andHyperbolic spaces. 2nd ed. Prentice hall; Upper saddle river, NJ
Patty, C. Wayne. Foundations of topology. PWS-KENT publishing co.; Boston
Arkhangel’skii, A.V.; Ponomarev, v.I.. Fudamentals of general topology. D. Reidel publishing co.; Boston
Adams, Colin; Shapiro, Joey. The shape of the universe: ten possibilities. American scientist. V. 89. No. 5. P
Thurston, William P. ; Weeks, Jeffrey r. The mathematics of three-dimensional manifolds. Scientific American v. 251 (July '84) p
Gribbon, john. Astronomers chew on Brazilian doughnut. New scientist. V Jan. 28. P.34.