By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada
This e-book will convey the sweetness and enjoyable of arithmetic to the school room. It deals severe arithmetic in a full of life, reader-friendly kind. integrated are workouts and lots of figures illustrating the most strategies.
The first bankruptcy offers the geometry and topology of surfaces. between different subject matters, the authors speak about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of features of the concept that of measurement, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of 3-dimensional manifolds. particularly, it's proved that the three-d sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given via the authors at Kyoto college (Japan).
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Extra resources for A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra
A . . a . , rather than . . a . . a−1 . , since a is a closed curve it may be seen that moving a ‘clock’ once around this curve reverses its orientation. Since S is orientable, this cannot happen, so if a symbol a appears in the identifying sequence, so does its inverse a−1 . 60 2. COMBINATORIAL STRUCTURE AND TOPOLOGICAL CLASSIFICATION Returning to our proof of the theorem, we now have a surface whose 1-skeleton is a bouquet of circles a, b, c, . , which we draw as a polygonal map with certain identifications a ∼ a−1 , etc.
In this way we obtain a single face in place of the two which were there before, decreasing the number of faces in the map by one. The result follows by induction. This leads us to the following result which will prove very valuable in our classification of surfaces: Corollary 1. Every compact triangulable surface is homeomorphic to a polygon with pairs of sides identified (which must therefore have an even number of sides). Remark . The process of investigating higher-dimensional manifolds via the analogue of triangulation, known as simplicial decomposition, is in general much more difficult.
Further, because S is orientable, the direction of each identification is specified for us. Indeed, if any symbol a appears twice (as . . a . . a . , rather than . . a . . a−1 . , since a is a closed curve it may be seen that moving a ‘clock’ once around this curve reverses its orientation. Since S is orientable, this cannot happen, so if a symbol a appears in the identifying sequence, so does its inverse a−1 . 60 2. COMBINATORIAL STRUCTURE AND TOPOLOGICAL CLASSIFICATION Returning to our proof of the theorem, we now have a surface whose 1-skeleton is a bouquet of circles a, b, c, .