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

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This publication will deliver the wonder and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly type. incorporated are workouts and plenty of figures illustrating the most ideas.
The first bankruptcy provides the geometry and topology of surfaces. between different themes, 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 quite a few elements of the idea that of measurement, together with the Peano curve and the Poincaré strategy. additionally addressed is the constitution of three-d manifolds. specifically, 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 via the authors at Kyoto college (Japan).

Show description

Read Online or Download A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23) PDF

Best topology books

Geometry of Characteristic Classes (Translations of Mathematical Monographs)

Attribute periods are imperative to the trendy learn of the topology and geometry of manifolds. They have been first brought in topology, the place, for example, they can be used to outline obstructions to the life of convinced fiber bundles. attribute sessions have been later outlined (via the Chern-Weil concept) utilizing connections on vector bundles, therefore revealing their geometric facet.

Parametrized homotopy theory

This ebook develops rigorous foundations for parametrized homotopy conception, that's the algebraic topology of areas and spectra which are regularly parametrized via the issues of a base house. It additionally starts the systematic research of parametrized homology and cohomology theories. The parametrized global presents the common domestic for plenty of classical notions and effects, akin to orientation conception, the Thom isomorphism, Atiyah and Poincaré duality, move maps, the Adams and Wirthmüller isomorphisms, and the Serre and Eilenberg-Moore spectral sequences.

Models for Smooth Infinitesimal Analysis

The purpose of this publication is to build different types of areas which include the entire C? -manifolds, but additionally infinitesimal areas and arbitrary functionality areas. To this finish, the ideas of Grothendieck toposes (and the good judgment inherent to them) are defined at a leisurely speed and utilized. through discussing issues reminiscent of integration, cohomology and vector bundles within the new context, the adequacy of those new areas for research and geometry may be illustrated and the relationship to the classical method of C?

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

Sample text

If G E I and G=UtF, with each F, closed, define F E I X o by F(x,t)ex~F, and notice that F is closed and G(x)w (3t)F(x, t ) . Thus G is Xi, and since G was arbitrary open, s. 2: E Hence Z: = 3” i 2: E s” i 2 : = 2; and inductively, 2 : E Z:+ 1. This establishes 2: E AZ+I for every n, so taking negations, II~E and the remaining inclusions in the diagram are trivial. 2. -X X , is a product space with at least one factor Xi = X and every Xieither o o r X , then I is homeomorphic with x. Hint. Construct homeomorphisms of o x X and X use induction on k.

The section above y. r A pointclass is Y-parametrized if for every product space X there is some G G Y X X which is universal for r l X . Let No, N1, N2Y. Thus 2; is X parametrized and it is trivial to prove from this that all the Bore1 pointclasses 2 : and their duals IIz are JV-parametrized. The next theorem establishes a little more. l. THEPARAMETRIZATION THEOREM FOR2 :. Let N ( Y , 0), N ( Y , 1),... and N(X, 0), N(X, 1),... enumerate bases for the topology of y and a fixed product space X respectively.

Thus G is Xi, and since G was arbitrary open, s. 2: E Hence Z: = 3” i 2: E s” i 2 : = 2; and inductively, 2 : E Z:+ 1. This establishes 2: E AZ+I for every n, so taking negations, II~E and the remaining inclusions in the diagram are trivial. 2. -X X , is a product space with at least one factor Xi = X and every Xieither o o r X , then I is homeomorphic with x. Hint. Construct homeomorphisms of o x X and X use induction on k. 3. Prove that if I = XI X X X, is a product space with at least one factor Xi not w, then I is a perfect Polish space.

Download PDF sample

Rated 5.00 of 5 – based on 26 votes