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) 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.

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

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