By Kenji Ueno, Koji Shiga, Shigeyuki Morita
This booklet will convey the sweetness and enjoyable of arithmetic to the school room. It deals critical arithmetic in a full of life, reader-friendly type. incorporated are routines and plenty of figures illustrating the most innovations.
The first bankruptcy offers the geometry and topology of surfaces. between different issues, 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 idea that of size, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of 3-dimensional manifolds. particularly, it's proved that the 3-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, 1, interplay between topology, functions, geometry, and algebra
We will show that I is a homotopy inverse of g. Let H: (B 1 xB2) xI->-Y H((x 1 ,x 2 ),t) = p(D 1 (x 1 ,t), D2 (x 2 ,t») . 8 Proposition in Switzer  H induces a continuous map H':YxI->-Y such that H' is, of course, base-point preserving. Furthermore, for (x 1 ,x 2 ) EB1xB2: H' (P(x 1 ,x 2 ) ,0) P(x 1 ,x 2 ) H' (P(x 1 ,x 2 ) ,1) p(D 1 (x 1 ,1), D2 (x 2 ,1» It follows that Iog-Id y . Now, since Di(a,t)=a for aEA i , there is a unique continuous map ki:ZixI->-Zi such that ki(qi(x) ,t)=qi (Di(x,t», i=1,2.
H is called a homotopy from f to g. Obviously, - is an equivalence relation on the set of all morphisms from (Y,A) to (Z,B). Moreover, if (W,C) is a third pair and f and g are two morphisms from (Z,B) to (W,C) and both f-g and f-g, then fof-gog. If (Y,A) and (Z,B) are two topological pairs, then (Y,A) is called homotopy equivalent to (Z,B) (we write (Y,A)-(Z,B)) if there are two 27 morphisms f: (Y,A)+(Z,B) and g:(Z,B)+(Y,A) such that fog-1 (Z,B) and gof-1 (Y,A)' where 1 (Y,A) (Y,A) (resp. 1 (Z,B» are the identity morphisms on (resp.
Then G(x)=[D] E u. The proposition is proved. 7 Index and quasi-index pairs. Given an isolating block B, the pair enjoys certain properties which are crucial for the definitions of the Morse and the homotopy indices. These properties are generalized and abstracted in the definitions of index and quasi-index pairs. Intuitively, index and quasi-index pairs are obtained by squeezing or stretching the pair (see Fig. 3). ,," Figure 3 In this section, as usual, X is a metric space and TI is a local semiflow on X.