By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

ISBN-10: 0387954503

ISBN-13: 9780387954509

The authors current introductory fabric in algebraic topology from a singular viewpoint in utilizing a homotopy-theoretic strategy. This conscientiously written ebook should be learn by means of any pupil who is aware a few topology, delivering an invaluable approach to speedy study this novel homotopy-theoretic standpoint of algebraic topology.

From the reviews:

"This is a easy path on algebraic topology and … it truly is excellently written and composed and will be strongly suggested to anyone wishing to benefit the sphere. The reader can locate many examples, calculations, and likewise a few routines. … The booklet has appendices, … references which includes eighty three goods and an extended record of symbols. there are various feedback and reviews making the orientation of the reader within the box of algebraic topology more uncomplicated … ." (EMS, June, 2004)

"The modus operandi of algebraic topology is to affiliate algebraic invariants, akin to teams or jewelry, to a topological area in this kind of means that an identical areas express isomorphic invariants; the following, ‘equivalent’ can be selected to slot the geometry of the matter. during this booklet, ‘equivalent’ potential homotopy similar … . For its readability and directness, a great addition to complicated arithmetic collections. Upper-division undergraduates via faculty." (J. McCleary, selection, December, 2002)

"The function of this publication is to introduce algebraic topology utilizing the unconventional strategy of homotopy conception … . the fundamental thoughts of homotopy thought … are used to build singular homology and cohomology, in addition to K-theory. … in the course of the textual content many different primary recommendations are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)

“The booklet of Aguilar, Gitler and Prieto offers an enticing new procedure for a primary path in algebraic topology. … what's additionally very fascinating is the truth that the ebook features a targeted presentation of a few deep result of algebraic topology no longer frequently lined by means of a primary ebook in algebraic topology … . The textual content is thoroughly good written. … the scholar in algebraic topology will locate within the ebook loads of fascinating well-exposed material.” (Yves Félix, Bulletin of the Belgian Mathematical Society, 2007)

From the again Cover

The function of this e-book is to introduce algebraic topology utilizing the unconventional strategy of homotopy idea, an technique with transparent purposes in algebraic geometry as understood through Lawson and Voevodsky. this system permits the authors to hide the fabric extra successfully than the extra universal technique utilizing homological algebra. the fundamental strategies of homotopy conception, resembling fibrations and cofibrations, are used to build singular homology and cohomology, in addition to K-theory. in the course of the textual content many different basic ideas are brought, together with the development of the attribute periods of vector bundles. even though functors look continuously in the course of the textual content, no wisdom approximately class thought is predicted from the reader. This publication is meant for complex undergraduates and graduate scholars with a easy wisdom of element set topology in addition to team concept and will be utilized in a semester course.

Marcelo Aguilar and Carlos Prieto are Professors on the Instituto de Matemticas, Universidad Nacional Autónoma de México, and Samuel Gitler is a member of El Colegio Nacional and professor on the Centro de Investigación y Estudios Avanzados del IPN.

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**Extra info for Algebraic Topology from a Homotopical Viewpoint (Universitext)**

**Sample text**

A-u Ò k ò k (6) m where we put A^i ^k = A if Jc = 0. Proof. Let p e A. mi m/ , m . As peA, it follows that mi exists such that V€Äm , whence similarly p€Ämm^, etc. Consequently there is an infinite sequence mly (x) See N. Lusin and W. Sierpinski, Sur quelques propriétés des ensembles (A), Bull. Acad. Se. Cracovie 1918, p . 35 32 INTRODUCTION m 2 , ... such that j k k which proves that p does not belong to the left-hand side of (6). THEOREM 2. ·» = °)> unv/ = nuv,·. (7) (T) Proof. The inclusion obtained from (7') replacing = by c is always true (for non-regular systems also; comp.

Cit. p . 48. §5] CLOSED SETS, OPEN SETS 53 EXAMPLE 1. The family of all open intervals r < x < s, with r and s rational, is a base of the space δ of all real numbers. The family of all open circles in «f2 with rational radius and with centre having rational coordinates is a base for *f2. EXAMPLE 2. The family of rays of the form x > r and of the form x < r, with r rational, is a subbase of ê. EXAMPLE 3. e. the set of all numbers t of the form t = (0, t{1), i(2), . . ) 3 where $(w> = 0 or 2, or, what is equivalent, the set (0, 2)9.

31-35. (2) Notion due to G. Cantor, Math. Ann. 21 (1883), p. 51. 44 [CH. I TOPOLOGICAL SPACES In the space of integers, every set is simultaneously closed and open. e. the set E[y =f(x))) is closed (in the plane) XV if and only if the function / is continuous (see § 20, V, Theorem 8). II· Operations· THEOREM 1. The union of two closed sets is ta closed set. This follows from Axiom 1 if we put X = X, Y = Ύ. THEOREM 2. The intersection (of a finite or infinite number) of closed sets is closed. Proof.

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