Quotient Spaces and the Shape of the Universe

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 x 1 and x 2 in X we have: • D( x 1, x 2) = d’( f(x 1), f(x 2)). • 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, and. Hyperbolic spaces. 2 nd 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/120598 bg/section 7. html. • www. iap. fr/user/roukema/top-easy. E. html.