Robert ghrist home page. All notes copyright dr.r.w.ghrist. all rights reserved.
ELEMENTARY APPLIED TOPOLOGY TEXT DRAFT (in progress; revised 3/2014) the following are rough draft versions of a text-to-be on applied algebraic topology, all in pdf. enjoy! The bibliographic entries are not yet added, and some of the cross-references and pictures are muddled...sorry! Preface Chapter 1: Manifolds Chapter 2: Complexes Chapter 3: Euler Characteristic Chapter 4: Homology Chapter 5: Sequences Chapter 6: Cohomology Chapter 7: Morse Theory Chapter 8: Homotopy (new!)
Chapter 9: Sheaves Chapter 10: Categorification all notes below were created with a fujitsu sytlistic tablet via microsoft's journal software: [2010] “Applied Algebraic Topology & Sensor Networks” - caveat! (see the Funny Little Calculus Text - FLCT link above) Topological space. Definition[edit] Neighbourhoods definition[edit] The first three axioms for neighbourhoods have a clear meaning.
The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, where a subset N of R is defined to be a neighbourhood of a real number x if there is an open interval containing x and contained in N. Open sets definition[edit] Four examples and two non-examples of topologies on the three-point set {1,2,3}.
Given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. Examples[edit] Closed sets definition[edit] Manifold. The surface of the Earth requires (at least) two charts to include every point.
Here the globe is decomposed into charts around the North and South Poles. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. Motivational examples[edit] Circle[edit] Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.
Algebraic Topology Book. What's in the Book?
To get an idea you can look at the Table of Contents and the Preface. Printed Version: The book was published by Cambridge University Press in 2002 in both paperback and hardback editions, but only the paperback version is currently available (ISBN 0-521-79540-0). I have tried very hard to keep the price of the paperback version as low as possible, but it is gradually creeping upward after starting at $30 in 2002. Less expensive printings have been made for sale in China (Tsinghua University Press) and South Asia. A Russian translation was published in 2011. Electronic Version: By special arrangement with the publisher, an online version will continue to be available for free download here, subject to the terms in the copyright notice.
The whole book as a single rather large pdf file of about 550 pages. Note: Section 3.2 has been revised from the original version, necessitating a renumbering of items 3.11-3.21.