By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This publication will convey the sweetness and enjoyable of arithmetic to the school room. It deals critical arithmetic in a full of life, reader-friendly type. integrated are workouts and plenty of figures illustrating the most recommendations.

The first bankruptcy provides the geometry and topology of surfaces. between different themes, 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 quite a few features of the idea that of size, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of 3-dimensional manifolds. particularly, it really 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 collage (Japan).

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

**Best topology books**

**Topology and Geometry (Graduate Texts in Mathematics, Volume 139)**

Uploader's observe: Ripped from SpringerLink.

This e-book deals an introductory direction in algebraic topology. beginning with normal topology, it discusses differentiable manifolds, cohomology, items and duality, the basic crew, homology thought, and homotopy idea.

From the reports: "An fascinating and unique graduate textual content in topology and geometry. .. a very good lecturer can use this article to create a great path. .. .A starting graduate pupil can use this article to profit loads of arithmetic. "—-MATHEMATICAL reports

**Central Simple Algebras and Galois Cohomology **

This ebook is the 1st complete, smooth advent to the idea of relevant basic algebras over arbitrary fields. ranging from the fundamentals, it reaches such complicated effects because the Merkurjev-Suslin theorem. This theorem is either the end result of labor initiated by means of Brauer, Noether, Hasse and Albert and the start line of present examine in motivic cohomology conception via Voevodsky, Suslin, Rost and others.

**Introduction to Topology: Third Edition**

Extremely popular for its unheard of readability, inventive and instructive routines, and advantageous writing type, this concise ebook deals a great introduction to the basics of topology. It presents an easy, thorough survey of trouble-free subject matters, beginning with set conception and advancing to metric and topological spaces, connectedness, and compactness.

- Category Theory: Proceedings of the International Conference Held in Como, Italy, July 22-28, 1990
- Explorations in topology: map coloring, surfaces, and knots
- A mathematician and his mathematical work : selected papers of S.S. Chern
- Algebraic Topology (EMS Textbooks in Mathematics)

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

**Sample text**

The problem is to ﬁnd the extremal values of this function on the constraint space G given by a number of equations gi (x1 , . . , xn ) = 0, i = 1, . . , k. The LMM says that the points at which extrema of f may occur are contained in the space of solutions of ∂L ∂L = 0, i = 1, . . , n; = 0, j = 1, . . 60) where L is the Lagrange multiplier function deﬁned by k L := L(x, Λ) = f (x) + λi gi (x). 61) i=1 ∂L When we equate the partial derivatives ∂λ to zero, we get back the constraint space i itself.

This is what we need in the following theorem. 2 Recall that a metric space X is complete if every Cauchy sequence in X is convergent in X. 2 Inverse Function Theorem (IFT): Let E ⊂ Rn be an open set, 0 ∈ E and let f ∈ C 1 (E, Rn ) be such that Df (0) is invertible. Then there exist open neighborhoods U of 0 and V of f (0) such that (i) f : U → V is a bijection; (ii) g = f −1 : V → U is diﬀerentiable. (iii) D(f −1 ) is continuous on V. Proof: (i) Put A = Df (0) and consider fˆ = A−1 ◦ f. Then fˆ ∈ (E; Rn ) and D(fˆ)(0) = Id.

There exists a unique point y ∈ X such that φ(y) = y. Proof: Start with any point x0 . Deﬁne x1 = φ(x0 ), x2 = φ(x1 ), . . , xn = φ(xn−1 ), . . Observe that d(xn+1 , xn ) ≤ cn d(x1 , x0 ) for some 0 < c < 1. Since that given > 0 we can ﬁnd n0 such that for m > n > n0 : n cn < ∞ it follows m−1 d(xm , xn ) ≤ ck d(x1 , x0 ) < d(x1 , x0 ). k=n Therefore {xn } is a Cauchy sequence. Since X is complete, this sequence has a limit point y ∈ X. Also observe that any contraction is a continuous function.