By Marcelo Aguilar, Samuel Gitler, Carlos Prieto
The authors current introductory fabric in algebraic topology from a singular perspective in utilizing a homotopy-theoretic technique. This conscientiously written ebook may be learn via any scholar who is aware a few topology, supplying an invaluable way to fast study this novel homotopy-theoretic standpoint of algebraic topology.
From the reviews:
"This is a easy direction on algebraic topology and … it truly is excellently written and composed and will be strongly suggested to anyone wishing to profit the sector. The reader can locate many examples, calculations, and likewise a few routines. … The e-book has appendices, … references inclusive of 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 this kind of method that an identical areas express isomorphic invariants; right here, ‘equivalent’ will be selected to slot the geometry of the matter. during this publication, ‘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 technique of homotopy idea … . the fundamental techniques of homotopy conception … are used to build singular homology and cohomology, in addition to K-theory. … during the textual content many different basic recommendations are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)
“The publication of Aguilar, Gitler and Prieto supplies an engaging new technique for a primary path in algebraic topology. … what's additionally very fascinating is the truth that the publication incorporates a specific presentation of a few deep result of algebraic topology no longer often coated by way of a primary publication in algebraic topology … . The textual content is thoroughly 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 booklet is to introduce algebraic topology utilizing the radical method of homotopy conception, an technique with transparent functions in algebraic geometry as understood through Lawson and Voevodsky. this system permits the authors to hide the cloth extra successfully than the extra universal strategy utilizing homological algebra. the elemental ideas of homotopy idea, 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 recommendations are brought, together with the development of the attribute periods of vector bundles. even though functors look consistently in the course of the textual content, no wisdom approximately class idea is anticipated from the reader. This publication is meant for complex undergraduates and graduate scholars with a easy wisdom of aspect 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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Additional resources for Algebraic Topology from a Homotopical Viewpoint (Universitext)
We shall prove the last assertion of the proposition, which implies the other assertions by reason of the general properties of final structures (Set Theory, Chapter IV, § 2, no. 5, criterion CST 18). It is clear that the fr. ontinuous in the topology 'lO, by the definition of t) (no. I, Theorem (); hence if g is continuous, so is each mapping g ofr. (no. I, Theorem 2). Conversely, suppose that each g 0 fr. is continuous, and let V be an open set in Z; by hypothesis, flrl(V» is open in YI for each ~ e I; hence gl(V) e t), and the proof is complete.
Be a topology on X" such that G~ is finer than the topology induced on X~ by G.. whenever a. =::;;~. If we take f«~ to be the canonical injection X~ _ X .. for a. =::;; ~, then ~ X.. may be identified canonically with the intersection X of the X", with the topology which is the least upper bound (§ 2, no. 3, Example 2) of the topologies induced on X by the G... 5. OPEN MAPPINGS AND CLOSED MAPPINGS 1. OPEN MAPPINGS AND CLOSED MAPPINGS· DEFINITION I. Let X, X' be two topological spaces. A mapping f: X _ X' is open Crespo closed) if the image under f of each open (resp.
The definition of the induced topology as an initial topology (§ 2, no. 3, Proposition 4) shows that f is continuous at x e X if and only if the mapping of X into the subspace B of Y, having the same graph as f, is continuous at x. 37 TOPOLOGICAL STRUCTURES Now let A be a subset of X; if f is continuous at x &I A (resp. continuous on X), its restriction flA is a mapping of the subspace A into Y, which is continuous at x (resp. continuous on A) by Proposition 2 of § 2, no. I. We shall sometimes say that a mapping f: X -+ Y is continuous relative to A at x e A Crespo continuous relative to A) if its restriction flA is continuous at x (resp.
Algebraic Topology from a Homotopical Viewpoint (Universitext) by Marcelo Aguilar, Samuel Gitler, Carlos Prieto