8 1. Book Overview

1.4. Covering Spaces

There is a nice way to unwrap the essential loops on a torus. The

idea is that we remember that the square torus is made from a square,

which we think of as the unit square with vertices (0, 0), (0, 1), (1, 0)

and (1, 1). We draw a line segment in the plane that starts out at

the same point as the loop and has the same length. We think of this

path starting at the point (0, 0). Figure 1.6 shows an example. In

this example, the unwrapped path joins (0, 0) to (3, 2).

The process can be reversed. Starting with a line segment that

joins (0, 0) to (m, n), a point with integer coordinates, we can wrap the

segment around the torus so that it makes an essential loop. In fact,

the essential loops that start at (0, 0) are, in the appropriate sense, in

one-to-one correspondence with the points of

Z2,

the integer grid in

the plane. The basic result is that any 2 essential loops L1 and L2,

corresponding to points

(m1,n1) and (m2,n2), can be continuously

moved, one into the other, if and only if (m1,n1) = (m2,n2).

(3,2)

(0,0)

Figure 1.6: Unwrapping a loop on the torus

As we will explain in Chapter 6 and Chapter 7, this unwrapping

construction can be done for any surface. In the case of the torus, we

see that the (equivalence classes of) essential simple loops are in exact

correspondence with the points of the integer grid in the plane. One

might wonder if a similarly nice picture exists in general. The answer

is “yes”, and in fact the picture becomes more interesting when we