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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1
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Figure 1.2. Another torus
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