By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will carry the wonder and enjoyable of arithmetic to the school room. It deals severe arithmetic in a full of life, reader-friendly variety. integrated are routines and lots of figures illustrating the most innovations.

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 variety of elements of the idea that of measurement, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of third-dimensional manifolds. particularly, it's proved that the third-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a chain of lectures given by way of the authors at Kyoto collage (Japan).

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**Additional info for A mathematical gift, 1, interplay between topology, functions, geometry, and algebra**

**Sample text**

This is the initial object * in the category of graded algebras since we always have is*-+A, i(1)= 1. 2) Definition. An augmentation of an algebra A is a map &:A -* * between graded algebras. Let A = kernel (s) be the augmentation ideal. The quotient module QA = is the module of indecomposables. Here denotes µ(A (D A). An augmentation preserving map f between algebras induces Qf :QA- QB. 3) Definition. For a positive graded module V we have the tensor algebra T(V) _ (@ V®n, n20 where V on = VQx .

3). 2) Theorem. 7), is a fibration category in which all objects are fibrant and cofibrant. Proof. 4). 2) are strictly dual. 3) cof = maps in Top* which are cofibrations in Top, fib = maps in Top* which are fibrations in Top, we = maps in Top* which are homotopy equivalences in Top. 3). 4) Theorem. 3) is a cofibration category in which all objects are fibrant models. The well pointed spaces are the cofibrant objects. Proof. 2). 5) Theorem. 3) is a fibration category in which all objects are fibrant.

We obtain path objects and the notion of homotopy in a fibration category as follows: By (F2) there exist pull backs AxBY )A Y f B in F. 3) (lA, lA):A-+A X BA the diagonal map which is dual to the folding map. 4) A +P-A xBA 1 q 12 I Axioms and examples B. 5) H\ / (a, II) X We call H a homotopy from a to fl over B. Here we assume that X is a cofibrant object in F. 5). 9), and with homotopy equivalences as weak equivalences is a fibration category. This is proved in §4 below. Our definition of a cofibration category (resp.