ABSTRACT
A cylinder is the product of an interval and a circle because it is both an interval of circles and a circle of intervals (Figure 17.1). (For a review of products, see Chapter 6.)
A Mobius band is also a circle of intervals (Figure 17.2), but it fails to be an interval of circles. It is al most a product, but not quite. It therefore qualifies as an interval bundle over a circle. In general a bundle over a circle is a bunch of things smoothly arranged in a circle, whether or not they form a product. For example, the quarter turn manifold from Exercise 7.3
is a torus bundle over a circle. Figure 17.3 reviews the construction of the quarter turn manifold: the front and back, and left and right, faces of a cube are glued in the straightforward way, but the top is glued to the bottom with a quarter turn. Figure 17.4 will help you understand the manifold’s global topology. Every flat three-manifold in Chapter 7 was either a torus bundle over a circle (T2-bundle over S1) or a Klein bottle bun dle over a circle (K^-bundle over S1).
