Andrew H Wallace's An introduction to algebraic topology PDF

By Andrew H Wallace

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The next three pictures are a computer-assisted exploration of how to cancel a pair of pinch points. The program uses Morin's parametrization of the homotopy, (X, Y,Z) = (x,1 + (t - r)y,y). First come three wireframe stereographs corresponding to moving t from negative to positive through zero. You can isolate the parabolic double curve by factoring the equation Y = 0 into y = 0 and y = r - t. In the space coordinates these become Z = 0 and Z = X2 - t. Thus the vertex of the parabola is at (O,O,t).

Move the tip closer or further away until two of the four fuzzy images fuse. The center image will still be fuzzy at first. Don't try to focus on the page. Continue to look at the tip of the pencil, and think about the surface until the middle image becomes sharp. With a little practice, you should be able to visualize reversed stereographs on paper or the computer screen quite easily. If you have a stereoscope or get a headache from crossing your eyes, just xerox, cut and paste the stereograph, reversing the right and left sides.

What the dunce hat might look like is not at all obvious from this prescription, nor that it lives in 3-space at all. Our first visualization uses only common topological ingredients. Suppose you tilt the seam, 5( 13), into the same plane as the rim, thereby drawing out a triangular flap or sail. Wrap this around the cone, 5(14), and finish the gluing, 5(21). The resulting 2-complex, 5(22), is not the dunce hat. It has two extra edges: where the sail meets the cone, and the free edge of the sail.

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An introduction to algebraic topology by Andrew H Wallace

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