By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

The authors current introductory fabric in algebraic topology from a unique viewpoint in utilizing a homotopy-theoretic procedure. This conscientiously written e-book could be learn via any scholar who is aware a few topology, offering an invaluable technique to fast study this novel homotopy-theoretic perspective 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 prompt to anyone wishing to profit the sector. The reader can locate many examples, calculations, and in addition a couple of workouts. … The booklet has appendices, … references together with eighty three goods and a protracted checklist of symbols. there are numerous 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, akin to teams or jewelry, to a topological house in this type of approach that identical areas show isomorphic invariants; right here, ‘equivalent’ will be selected to slot the geometry of the matter. during this ebook, ‘equivalent’ ability homotopy an identical … . For its readability and directness, a great addition to complex arithmetic collections. Upper-division undergraduates via faculty." (J. McCleary, selection, December, 2002)

"The goal of this ebook is to introduce algebraic topology utilizing the radical method of homotopy concept … . the fundamental options of homotopy conception … are used to build singular homology and cohomology, in addition to K-theory. … in the course of the textual content many different basic options are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)

“The booklet of Aguilar, Gitler and Prieto provides an attractive new process for a primary direction in algebraic topology. … what's additionally very fascinating is the truth that the ebook encompasses a specific presentation of a few deep result of algebraic topology no longer often coated by means of a primary ebook in algebraic topology … . The textual content is thoroughly good written. … the coed 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 objective of this ebook is to introduce algebraic topology utilizing the radical procedure of homotopy idea, an procedure with transparent purposes in algebraic geometry as understood through Lawson and Voevodsky. this technique permits the authors to hide the fabric extra successfully than the extra universal technique utilizing homological algebra. the elemental suggestions of homotopy conception, similar to 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 techniques are brought, together with the development of the attribute periods of vector bundles. even if functors look regularly in the course of the textual content, no wisdom approximately type conception is anticipated from the reader. This publication is meant for complex undergraduates and graduate scholars with a simple wisdom of element 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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**Extra info for Algebraic Topology from a Homotopical Viewpoint (Universitext)**

**Sample text**

2 Differential Forms on a Euclidean Space In the last section we introduced the set 3 (U) of vector fields on an open set U c R", and showed that this notion is "intrinsic," that is, does not depend on the basis used for R". , x"}, which may be more illuminating for the reader. , k," } of V consists of the linear mappings defined by X. (v1) = S =1 if k = j, 0 otherwise. +a"v") = a,. 7) The cotangent space at y e U is defined to be (TyR") *, that is, the n-dimensional vector space of linear forms on the tangent space at y.

Show that if X and Y are the vector fields on U = R3\ { 0} given below, then LXY = 0: X= -{x2+y2+z2}-3/2{xa +ya +za ax ay az }; Y = -ya +xa . -3 Hint: For brevity, take r = {x2 + y2 + z2 } 1 /2 and note that (xar y ax ay -3 y ax ) = 0. 5. 1 for the case of R4 with coordinates {x, y, z, t}. 6. 1. 7. Show that if t) = P1 (dy n dz) + P2 (dz A dx) + P3 (dx A dy) is a 2-form on an open set U g. 1, and if X= a +42a +43a , y = ax ay az t a + r2 a + r3 a ax bay S Wz- ' then the coordinate expression for (2.

60). (iv) Prove that for a smooth function h, LhXw = hLXw + dh n tXw. 25. (Continuation) Suppose U c R" and V c R" are open sets, and tp: U -4 V is a diffeomorphism. Prove that, for every vector field X on U, tX (tP* w) = (p* (t9. Xw). 62) History and Bibliography Differential forms in the sense discussed here originated in 1899 in an article by the Cartan (1869-1951) and in the third volume of Les Methodes Nouvelles de la Mecanique Clleste by Henri Poincard (1854-1912). The program of restating the laws of physics in an invariant notation, including the use of differential forms, was initiated by Gregorio Ricci-Curbastro (1853-1925) and his student Tullio Levi-Civita (1873-1941), and helped to provide the framework in which Albert Einstein (1879-1955) developed the theory of relativity.