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 viewpoint in utilizing a homotopy-theoretic method. This conscientiously written booklet should be learn via any pupil who is familiar with a few topology, offering an invaluable strategy to speedy examine this novel homotopy-theoretic viewpoint of algebraic topology.

From the reviews:

"This is a easy path on algebraic topology and … it's excellently written and composed and will be strongly suggested to anyone wishing to profit the sector. The reader can locate many examples, calculations, and in addition a few workouts. … The e-book has appendices, … references which includes eighty three goods and an extended checklist of symbols. there are lots of feedback 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, corresponding to teams or jewelry, to a topological house in any such means that similar areas express isomorphic invariants; the following, ‘equivalent’ can be selected to slot the geometry of the matter. during this e-book, ‘equivalent’ ability 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 objective of this publication is to introduce algebraic topology utilizing the unconventional technique of homotopy concept … . the elemental innovations of homotopy thought … are used to build singular homology and cohomology, in addition to K-theory. … during the textual content many different basic options are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)

“The publication of Aguilar, Gitler and Prieto supplies an attractive new procedure for a primary direction in algebraic topology. … what's additionally very fascinating is the truth that the booklet encompasses a special presentation of a few deep result of algebraic topology now not frequently lined through a primary booklet 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 function of this booklet is to introduce algebraic topology utilizing the unconventional method 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 cloth extra successfully than the extra universal technique utilizing homological algebra. the fundamental recommendations of homotopy concept, equivalent to fibrations and cofibrations, are used to build singular homology and cohomology, in addition to K-theory. through the textual content many different primary innovations are brought, together with the development of the attribute periods of vector bundles. even supposing functors look consistently through the textual content, no wisdom approximately classification conception is anticipated from the reader. This ebook is meant for complicated undergraduates and graduate scholars with a simple wisdom of aspect set topology in addition to workforce 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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Example text

Embed X into the Hilbert cube Q =UJLi i>k [—l,l]y, and for let Pi be the infinite ^-skeleton of [—1,1]* X l l y ^ i {0}y, so that PtCP^C.... Then set Z=X\J U{>k P{. 6. Proposition. Let f,g : X —> Y be two maps between compacta, and assume that (i) Y is LCk~l. The following two statements are equivalent: For any embedding X—*•% of X into an LCk~l-space with dim (5^ — X) < k, and any extensions 7"5

It induces isomorphisms of homotopy groups of dimension < A: — t — 1 and an epimorphism in dimension k — t ). n — k — l)-complex. Also, observe that <^( 0 be as in axiom 1. Find j > i such that mesh Lj < e/6, and let Q' be the handlebody decomposition of Mj with Lj.

It remains to make the A^fyj's "join the crowd". This is done recursively, using the preferred ordering of Aw[j»]'s. We show how to do it for the first oo — cu(l). Others are treated in exactly the same way. ,Q(s) (s = 5(ct{l)) of partitions so that the following is true. ,A[s] f] Mi{j]) 55 is a partition; i/(j) 3 : T(j) : Q(j) —* Q is a one-to-one correspondence, H(j)4: o{j)# : 7r,-(M,-{y}) —* 7r2-(M') is an isomorphism for a < & — 1, and an epimorphism for i = k. U A[j]. The induction starts with the triple (Q(0), T(0),a{0)).

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