Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)

By Milgram R. (ed.)

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In the plane, on the other hand, every triangle has anglesum exactly equal to 180°. And in the saddle surface, all triangles have angle-sum less than 180°. Thus a Flatlander could experimentally determine which surface he lived in: he need only lay out a triangle and measure its angles! These properties of triangles are treated in detail in Chapters 9 and 10. The mathematician Gauss carried out precisely this experiment in our own three-dimensional universe. (Later chapters will explain how a three-dimensional manifold can be curved.

Rather than developing an extensive theory of these manifolds, you'll come to know each of them in a visual and intuitive way. Obviously this is not an easy task. Imagine the difficulties A Square would have in communicating to A Hexagon the true nature of a torus. A Square cannot draw a definitive picture of a torus, being confined to two dimensions as he is. Similarly, we cannot draw a definitive picture of any three-manifold. There is some hope, though. You can use tricks to define various three-manifolds, and as you work with them over a period of time you'll find your intuition for them growing steadily.

1. 3°. 2. An explorer set out to the east and returned from the west, never deviating from a straight route. 3. " For example, a flat torus and a doughnut surface have the same global topology, but different local geometries. A flat torus and a plane, on the other hand, have the same local geometry but different global topologies. A three-torus has the same local geometry as "ordinary" three-dimensional space, but its global topology is different. The Flatlanders in Chapter 1 were discovering various global topologies for Flatland, but when Gauss surveyed the mountain peaks he was investigating the local geometry of our universe in the region of the Earth.

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