New PDF release: A Mathematical Gift, 1: The Interplay Between Topology,

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

ISBN-10: 0821832824

ISBN-13: 9780821832820

This ebook will convey the sweetness and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly type. incorporated are routines and lots of figures illustrating the most options.

The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors talk 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 method. additionally addressed is the constitution of third-dimensional manifolds. particularly, it really is 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 by way of the authors at Kyoto collage (Japan).

Show description

Read or Download A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra PDF

Similar topology books

Harmonic maps, conservation laws, and moving frames by Frédéric Hélein PDF

This obtainable advent to harmonic map idea and its analytical elements, covers fresh advancements within the regularity idea of weakly harmonic maps. The e-book starts off by way of introducing those strategies, stressing the interaction among geometry, the function of symmetries and susceptible options. It then provides a guided travel into the speculation of thoroughly integrable platforms for harmonic maps, through chapters dedicated to fresh effects at the regularity of susceptible ideas.

Download PDF by Paul A. Smith, Samuel Eilenberg: Point set topology

Compatible for an entire path in topology, this article additionally features as a self-contained remedy for self sustaining examine. extra enrichment fabrics and complicated subject coverage—including vast fabric on differentiable manifolds, summary harmonic research, and glued element theorems—constitute a good reference for arithmetic lecturers, scholars, and pros.

Ian F. Putnam's A homology theory for Smale spaces PDF

The writer develops a homology idea for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it's according to components. the 1st is a stronger model of Bowen's consequence that each such method is similar to a shift of finite kind less than a finite-to-one issue map. the second one is Krieger's size team invariant for shifts of finite variety.

Additional info for A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra

Sample text

Then we attach an index - 1 to the label of any edge whose arrow points in the opposite direction. Applying this procedure to the rectangle of Fig. 2. we obtain the diagram shown in Fig. 3 This is the final form of the plane model of the torus T. 7-1 Fig. 3. Notice that the vertices of this rectangle represent a single point on the torus. 7. With a single cut we can open out this cylinder to form a rectangle. Then suitable labelling gives us the plane model of Κ shown in Fig. 4. Let us see how to get a space model of Κ from this plane model.

By convention. Oris identified with 5. 5 A compact surface Μ with the appropriate edges identified in R"(n>3 if the surface is orientable and n>4 if it is non-orientable) is called a space model of M. e. a generalization to η dimensions of the 'ordinary' 3-dimensionaI Euclidean space of everyday life. 6 THE CLASSIFICATION THEOREM Let us consider again our fundamental theorem in the light of our new definitions. 1 The Classification Theorem. (a) An orientable compact surface is homeomorphic to η Γ for some n > 0 .

The operations by which we perform this manipulation must be reversible and must be such that the word reached at each stage of the manipulation represents the same surface, up to homeomoφhism. The following operations satisfy these critreria. Operation 1. The cyclic permutation of the edges in a word This can be expressed by the identity \AB = ΒA where A and Β denote strings of edges. For example, O i f t i f l f ' 6 f ' and b^^a\bxa^^ represent the same surface. In fact they represent the same plane model up to rotation.

Download PDF sample

A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

by David

Rated 4.46 of 5 – based on 24 votes