1.3. Drawing on a Surface 7

Figure 1.5. Loops on the torus

One of the loops, the one represented by the thick vertical line,

is different from the others. Imagine that these loops are made from

rubber bands, and are allowed to compress in a continuous way. The

first 2 loops can shrink continuously to points, whereas the third loop

is “stuck”. It can’t make itself any shorter no matter how it moves.

Such a loop is commonly called essential. There are many essential

loops on the torus. The right side of Figure 1.5 shows another essential

loop. In contrast, the sphere has no essential loops at all.

We will see in Chapter 4 that there is an algebraic object we can

associate to a surface (and many other kinds of spaces) called the fun-

damental group. The fundamental group organizes all the different

ways of drawing loops on the surface into one basic structure. The

nice thing about the fundamental group is that it links the theory

of surfaces to algebra, especially group theory. Beautifully, it turns

out that 2 (compact) surfaces have the same Euler characteristic if

and only if they have the same fundamental group. The Euler char-

acteristic and the fundamental group are 2 entry points into the vast

subject of algebraic topology.

For the most part, studying algebraic topology is beyond the

scope of this book, but we will study the fundamental group and

related constructions, in great detail. After defining the fundamental

group in Chapter 4, we will compute a number of examples in Chapter

5.