By Kenji Ueno, Koji Shiga, Shigeyuki Morita

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

The first bankruptcy provides the geometry and topology of surfaces. between different themes, the authors talk about the Poincaré-Hopf theorem on severe 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 numerous elements of the concept that of size, together with the Peano curve and the Poincaré strategy. additionally addressed is the constitution of 3-dimensional manifolds. particularly, it truly is proved that the 3-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 college (Japan).

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**Additional resources for A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23)**

**Sample text**

Y , . . , z«] i-^ [^^z\,.. •, ^^Zn] gives a diffeomorphism from L(^; / i , . . , /„) to L(k;ai\,.. ,ain)' Lens spaces exhibit rigidity; all diffeomorphisms are generated by the above three types. 10. Let L(k; i\,.. ,in) cind L' = L(/:; / j , . . , /^) be two lens spaces. The following are equivalent.

Serre conjectured and K. Brown proved that in fact n(F) • x (^) ^ ^- This beautiful result furnishes information about the size of the finite subgroups in F, provided the Euler characteristic can be computed. In many instances this is the case; for example, x iSp^(Z)) = —1/1440, from which we deduce that Sp^(L) has subgroups of order 32, 9 and 5. From a more elementary point of view, this result is simply a consequence of the basic fact that the least common multiple of the orders of the isotropy subgroups of a finite-dimensional G-complex Y (with homology of finite type) must yield an integer when multiplied by x (^)/1G |.

Borel then speculated that while group theoretic rigidity sometimes failed, topological rigidity might always hold. Of Topics in transformation groups course, such phenomena were known prior to the work of Malcev and Mostow. Bieberbach showed rigidity for crystallographic groups. , there are discrete, co-compact subgroups of SL2(R) which are abstractly isomorphic, but there is no automorphism of 5L2(M) which carries one to the other. The theory of group theoretic rigidity was investigated further by Mostow [126] and Margulis [113].