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 carry the sweetness and enjoyable of arithmetic to the study room. It bargains severe arithmetic in a full of life, reader-friendly type. integrated are routines and lots of figures illustrating the most suggestions.
The first bankruptcy offers the geometry and topology of surfaces. between different issues, the authors speak 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 points of the concept that of size, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of third-dimensional manifolds. particularly, it truly 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 through the authors at Kyoto college (Japan).

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

Example text

R' X be two fibrant models of X. 4. 0 We conclude this paragraph with an example of a cofibration category where all objects are not fibrant, namely CW

36 Chapter 3: Colibration Categories Table 1: Topological examples of cofibration categories Notation Top ToPCW· CW+ Objects topological spaces connected pointed top. *L) ~ H*(Y,L) HEP = homotopy extension property inductive attachments of cells HEP all objects all objects objects of the form (X, 0); the fibrant model of (X,Nx) is the Quillen (+ )-construction (X+, 0) Morphisms we co! fibrant We denote by k* a generalized homology theory defined on CW-pairs with the limit axiom; that is this theory satisfies: lim k*Xa = k*X, where Xa is a finite -+ sub complex.

2) in DA of C*(OX). Ax is called the Adams-Hilton model of X. 3 Definition. A map i: B --+ A in DA is a cofibration if and only if there is a free subspace V of A and an isomorphism of algebras AS:! i(B) EB T(V). Let i: B --+ (B II T(V), d) be a cofibration and f: B --+ C any map in DA. Then we define a cocartesian diagram f B-----+)y iI B II (T(V) , d) =A ------+) A UB Y(Y II T(V), d) ] by] = f II T(V). The differential on C II T(V) is defined by requiring] and be differential morphisms. 4 Acyclicity.

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