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Algebraic Topology from a Homotopical Viewpoint by Marcelo Aguilar, Samuel Gitler, Carlos Prieto

By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

The authors current introductory fabric in algebraic topology from a unique perspective in utilizing a homotopy-theoretic strategy. This rigorously written booklet could be learn by means of any pupil who understands a few topology, delivering an invaluable technique to fast study this novel homotopy-theoretic perspective of algebraic topology.

From the reviews:

"This is a simple direction on algebraic topology and … it truly is excellently written and composed and will be strongly steered to anyone wishing to benefit the sphere. The reader can locate many examples, calculations, and likewise a few workouts. … The publication has appendices, … references which includes eighty three goods and an extended checklist of symbols. there are various comments and reviews making the orientation of the reader within the box of algebraic topology more straightforward … ." (EMS, June, 2004)

"The modus operandi of algebraic topology is to affiliate algebraic invariants, similar to teams or earrings, to a topological house in the sort of method that an identical areas express isomorphic invariants; right here, ‘equivalent’ could be selected to slot the geometry of the matter. during this booklet, ‘equivalent’ ability homotopy an identical … . For its readability and directness, a welcome boost to complex arithmetic collections. Upper-division undergraduates via faculty." (J. McCleary, selection, December, 2002)

"The objective of this publication is to introduce algebraic topology utilizing the radical technique of homotopy concept … . the elemental strategies of homotopy idea … are used to build singular homology and cohomology, in addition to K-theory. … through the textual content many different basic strategies are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)

“The booklet of Aguilar, Gitler and Prieto provides an engaging new procedure for a primary direction in algebraic topology. … what's additionally very fascinating is the truth that the e-book incorporates a precise presentation of a few deep result of algebraic topology no longer frequently lined by way of a primary e-book in algebraic topology … . The textual content is punctiliously good written. … the scholar in algebraic topology will locate within the ebook loads of attention-grabbing well-exposed material.” (Yves Félix, Bulletin of the Belgian Mathematical Society, 2007)

From the again Cover
The goal of this e-book is to introduce algebraic topology utilizing the unconventional process of homotopy conception, an method with transparent functions in algebraic geometry as understood by way of Lawson and Voevodsky. this technique permits the authors to hide the fabric extra successfully than the extra universal strategy utilizing homological algebra. the fundamental techniques of homotopy idea, reminiscent of 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 primary ideas are brought, together with the development of the attribute periods of vector bundles. even if functors seem continuously through the textual content, no wisdom approximately type idea is predicted from the reader. This e-book is meant for complicated undergraduates and graduate scholars with a easy wisdom of aspect set topology in addition to crew 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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Additional info for Algebraic Topology from a Homotopical Viewpoint (Universitext)

Example text

To avoid technical difficulties we assume that if "p takes infinite values, they are all of the same sign. 1) This property corresponds to Borel's postulate (ii). 1) always makes sense under the usual arithmetic rules: r+oo = 00, r+(-oo) = -00, 00+ 00 = 00, (-00)+(-00) = (-00). These are two examples of additive set functions. 1. Let X be an infinite set and let A = {E ~ X: either E is finite or X \ E is finite}.

Now, endowed with this distance R(I) becomes a metric space. In this metric, the sequence {9n} introduced above is Cauchy, but it does not converge. Thus, this metric space is not complete. This shortcoming will also be corrected by the Lebesgue integral. 4. 1 Assume {In} is a sequence of real-valued nondecreasing functions defined on I = [a,b], and suppose I(x) = limn -+ oo In(x) exists for x E I. Is I necessarily nondecreasing? 2 (F. Riesz's Rising Sun Lemma) Assume I is a bounded real-valued function defined on I = [a,b], and let :F = {9:9 is defined on I, 9 is nonincreasing and 9( x) ~ I( x) for x in I}.

Show that P( m) is true for every m EM. 11 Suppose MI is a subset of a well-ordered set M and that the following holds: If m E M and the initial segment 1m ~ MI, then mE MI' Show that MI = M. 12 An interesting application of Transfinite induction is to show the various properties concerning operations with order types. For instance, prove that the addition is associative and left distributive with respect to multiplication. 13 Does the conclusion of the Principle of transfinite induction hold if, instead of (ii) there, we assume: (ii ') If P( m) is true, then P( m + 1) is also true?

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