By Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y. Miyaoka

This quantity documents the complaints of a global convention held in Tokyo, Japan in August 1990 at the matters of algebraic geometry and analytic geometry.

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This is the working end. Bend the working end around to the other end of your work, and begin to knit those stitches onto the working end, but do not slip them off the other end of the needle as you normally would. When you have knitted all 90 stitches in this way, the needle loops the work twice. 33 Figure 12: A Mobius band. Carry on knitting in the same direction, slipping stitches off the needle when you knit them, as normal. The needle will remain looped around the work twice. Knit five ‘rows’ (that is 5 × 90 stitches) in this way.

We think of the letters as figures made from lines and curves, without fancy doodads such as serifs. Problem • Which of the capital letters are topologically the same, and which are topologically different? How many topologically different capital letters are there? 2 Surfaces A surface, or 2-manifold, is a shape any small enough neighborhood of which is topologically equivalent to a neighborhood of a point in the plane. For instance, a the surface of a cube is a surface topologically equivalent to the surface of a sphere.

The pictures are intended to indicate things like doughnuts and pretzels rather than flat strips of paper. Problem • Can you identify these surfaces, topologically? Which ones are topologically the same intrinsically, and which extrinsically? 10 of SS (The Shape of Space). 9. 31 32 Figure 11: Some surfaces Problems 1. Take some strips and join the opposite ends of each strip together as follows: with no twists; with one twist (half-turn); this is called a M¨ obius strip; with two twists; with three twists.