1.2. Gluing Polygons 3

all glued together. As the bug looks in various directions, he sees 4

little quarter-disks that glue together to form a single disk. Even at

the corner(s), the bug thinks that he is living in the Euclidean plane.

Modulo a ton of details, we have shown that the square torus has

no edges at all. At every point it “looks locally” like the Euclidean

plane. In particular, it is perfectly flat at every point. At the same

time, the square torus is bounded, like the sphere.

The torus is such a great example that it demands a careful and

rigorous treatment. The first question that comes to mind is What do

we mean by a surface? We will explain this in Chapter 2. Roughly

speaking, a surface is a space that “looks like” the Euclidean plane in

the vicinity of each point. We do not want to make the definition of

“looks like” too restrictive. For instance, a little patch on the sphere

does not look exactly like the Euclidean plane, but we still want the

sphere to count as a surface. We will make the definition of “looks

like” flexible enough so that the sphere and lots of other examples all

count.

1.2. Gluing Polygons

In Chapter 3 we give many examples of surfaces and their higher-

dimensional analogues, manifolds. One of the main tools we use is

the gluing construction. The square torus construction above is the

starting point for a whole zoo of related constructions.

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Figure 1.2. Another torus