By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

The authors current introductory fabric in algebraic topology from a unique viewpoint in utilizing a homotopy-theoretic process. This rigorously written publication should be learn via any pupil who understands a few topology, offering an invaluable way to quick study this novel homotopy-theoretic viewpoint of algebraic topology.

From the reviews:

"This is a uncomplicated direction on algebraic topology and … it really is excellently written and composed and will be strongly advised to anyone wishing to profit the sphere. The reader can locate many examples, calculations, and in addition a couple of routines. … The e-book has appendices, … references which include eighty three goods and an extended record of symbols. there are various comments 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, resembling teams or jewelry, to a topological area in any such approach that an identical areas convey isomorphic invariants; the following, ‘equivalent’ might be selected to slot the geometry of the matter. during this publication, ‘equivalent’ capability homotopy identical … . For its readability and directness, a welcome boost to complicated arithmetic collections. Upper-division undergraduates via faculty." (J. McCleary, selection, December, 2002)

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

“The publication of Aguilar, Gitler and Prieto offers an attractive new technique for a primary direction in algebraic topology. … what's additionally very fascinating is the truth that the publication includes a distinct presentation of a few deep result of algebraic topology no longer often coated via a primary booklet in algebraic topology … . The textual content is punctiliously good written. … the coed in algebraic topology will locate within the publication loads of attention-grabbing well-exposed material.” (Yves Félix, Bulletin of the Belgian Mathematical Society, 2007)

From the again Cover

The objective of this e-book is to introduce algebraic topology utilizing the unconventional process of homotopy thought, an strategy with transparent functions in algebraic geometry as understood by means of Lawson and Voevodsky. this technique permits the authors to hide the cloth extra successfully than the extra universal technique utilizing homological algebra. the fundamental suggestions of homotopy thought, similar to fibrations and cofibrations, are used to build singular homology and cohomology, in addition to K-theory. during the textual content many different basic suggestions are brought, together with the development of the attribute periods of vector bundles. even though functors look consistently during the textual content, no wisdom approximately type conception is anticipated from the reader. This publication is meant for complicated undergraduates and graduate scholars with a simple wisdom of element set topology in addition to staff conception 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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**Sample text**

We have now obtained a polygon D whose edges have to be identiﬁed in pairs to obtain the given surface S. 16, the identiﬁcations are described by aa‘lfbb“lf—‘e‘lgcc—lg—ldd—le. If the letter designating a certain pair of edges occurs with both exponents, +1 and —1, in the symbol, then we will call that pair of edges a pair of the ﬁrst kind; otherwise, the pair is of the second kind. 16, all seven pairs are of the ﬁrst kind. We wish to show that an adjacent pair of edges of the ﬁrst kind can be eliminated, provided there are at least four edges in all.

2) Let 2) be a vertex of a triangulation. Then we may arrange the set of all triangles with v as a vertex in cyclic order, T0, T1, T2, . - and Ti+1 have an edge in common for 0 g 2' < n — 1. The truth of (1) follows from the fact that each point on the edge in question must have an Open neighborhood homeomorphic to the Open disc U2. If an edge were an edge of only one triangle or more than two triangles, this would not be possible. The rigorous proof of this last assertion would take us rather far aﬁeld; however, its plausibility cannot be diSputed.

CkBkcfl. (b) Normal form for the connected sum of n tori with k holes. 29(a) and (b) show how to proceed when n = 2 and k = 4. It is entirely analogous to the case of a sphere with holes cut in it. The result is a polygon with 4n + 3k sides, which must be identiﬁed in accordance with the following symbol: — — alblallbll — — — anbnanlbnlclBlcl1 ckBkck—l . (c) Normal form for the connected sum of n projective planes with k holes. We leave it to the reader to see that in this case we obtain a polygon with 2n + 3k sides, which are identiﬁed by the symbol a1a1 .