Quotient Spaces and the Shape of
the Universe
A Topological Exploration of
3-manifolds.
Dorothy Moorefield
Mat 4710
Dr. Sarah Greenwald
10 December 2001
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.
Metric Space
• A metric on a set X is a function d: x
 such that:
•
•
•
•
A)
B)
C)
D)
d( x, y)  0  x, y .
d( x, y) = 0  x = y.
d( x, y) = d( y, x) x, y  .
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.
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.
Cosmic Microwave Background
•
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.
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.
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.
Path on a 3-torus
•
The Quarter-turn Manifold
Sources
• David, W. Henderson. Experiencing geometry: in Euclidean, spherical,
andHyperbolic spaces. 2nd ed. Prentice hall; Upper saddle river, NJ.
2001.
• Patty, C. Wayne. Foundations of topology. PWS-KENT publishing co.;
Boston. 1993.
• Arkhangel’skii, A.V.; Ponomarev, v.I.. Fudamentals of general topology.
D. Reidel publishing co.; Boston. 1984.
• Adams, Colin; Shapiro, Joey. The shape of the universe: ten
possibilities. American scientist. V. 89. No. 5. P. 443-53.
• Thurston, William P. ; Weeks, Jeffrey r. The mathematics of threedimensional manifolds. Scientific American v. 251 (July '84) p. 108-20.
• Adams, Colin; Shapiro, Joey. The shape of the universe: ten
possibilities. American scientist. V. 89. No. 5. P. 443-53.
• Gribbon, john. Astronomers chew on Brazilian doughnut. New scientist.
V. 117. Jan. 28. P.34.
• http://darc.obspm.fr/~luminet/etopo.html.
• www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.html.
• www.iap.fr/user/roukema/top/top-easyE.html.