A Mathematical Gift I: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will carry the wonder and enjoyable of arithmetic to the school room. It bargains critical arithmetic in a full of life, reader-friendly type. integrated are workouts and lots of figures illustrating the most ideas.
The first bankruptcy provides the geometry and topology of surfaces. between different themes, 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 variety of facets of the idea that of size, together with the Peano curve and the Poincaré strategy. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it truly is proved that the third-dimensional sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given through the authors at Kyoto collage (Japan).

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Extra resources for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)

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In the case where the maps in the sequence are specified not merely as maps, but as components of particular paracoherent natural transformations, their sources and targets are given an explicit structure as images of iterated prolongations of structure functors. We may thus require t h a t the "padding" maps given in terms of the structural natural transformations be (components of) natural transformations between the appropriate functors. Regardless of whether we avoid ambiguity by modifying the category or by restricting the use of [ ] t o sequences of paracoherent natural transformations, a number of elementary properties of the operators may be deduced in the case of monoidal categories or strong monoidal functors.

29 Structure functors for disjoint unions Given an indexed family of categories {Cj}3^j, the disjoint union U j e j C j is a category with the obvious source, target, identities and compositions. The inclusions ij : Cj —>• UjejCj are functors. Likewise, given functors Fj : Cj —> V, there is a (unique) functor F : JJ e jCj —> V such that F{tj)=Fj for all j . 30 Structure functors for products Given an indexed family of categories {Cj}j£j, the product Ylj^jCj is a category with componentwise source, target, identities and compositions.

Also [29]). For our purposes, this combinatorial notion is less useful than a reduction to diagrams of ambient isotopy of framed oriented links. For this, we need to replace fLl with a substitute move which does respect the framing. 1 change the blackboard framing. 5: The Framed First Reidemeister Move positive crossings change the framing (thought of as an integer) by + 1 , while those with negative crossings change it by —1. l which change the framing by 0 must be admitted as moves. l which decrease the framing in small balls (or disks in the projection).

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