By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

The authors current introductory fabric in algebraic topology from a unique standpoint in utilizing a homotopy-theoretic procedure. This rigorously written ebook will be learn through any scholar who is familiar with a few topology, offering an invaluable way to fast examine this novel homotopy-theoretic viewpoint of algebraic topology.

From the reviews:

"This is a uncomplicated path on algebraic topology and … it truly is excellently written and composed and will be strongly suggested to anyone wishing to profit the sphere. The reader can locate many examples, calculations, and likewise a few routines. … The e-book has appendices, … references including eighty three goods and a protracted record of symbols. there are lots of feedback 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, corresponding to teams or earrings, to a topological house in this type of manner that an identical areas express isomorphic invariants; the following, ‘equivalent’ can be selected to slot the geometry of the matter. during this booklet, ‘equivalent’ skill 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 function of this publication is to introduce algebraic topology utilizing the unconventional process of homotopy idea … . the fundamental innovations of homotopy conception … are used to build singular homology and cohomology, in addition to K-theory. … during the textual content many different primary strategies are brought … ." (L’Enseignement Mathematique, Vol. forty eight (3-4), 2002)

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

From the again Cover

The goal of this ebook is to introduce algebraic topology utilizing the radical process of homotopy concept, an method with transparent functions in algebraic geometry as understood by means of Lawson and Voevodsky. this system permits the authors to hide the fabric extra successfully than the extra universal strategy utilizing homological algebra. the fundamental suggestions of homotopy concept, corresponding to fibrations and cofibrations, are used to build singular homology and cohomology, in addition to K-theory. during the textual content many different basic techniques are brought, together with the development of the attribute periods of vector bundles. even if functors seem continuously during the textual content, no wisdom approximately type conception is predicted from the reader. This publication is meant for complex undergraduates and graduate scholars with a uncomplicated wisdom of aspect set topology in addition to staff 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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**Example text**

3, where, in addition, β is a smooth section. , the evaluation map H : [0, 1]2 × B → E is a smooth map. 9 (Parameters). Let P be a compact Hausdorﬀ space (a parameter space). Let f : P → Γ(E), β : P → Γ(R) be continuous maps such that for all (p, b) ∈ P × B, the following convex hull property obtains: f (p, b) ∈ IntConv(Rb , β(p, b)). 7) There is P -parameter family of C-structures (h, H) over B with respect to f, β. That is, there are continuous maps, h : P × [0, 1] → Γ(R); H : P × [0, 1]2 → Γ(R), such that, for each p ∈ P , (hp , Hp ) is a C-structure over B with respect to the sections f (p), β(p).

Note that the theorem is proved in the case r = 1 by choosing g = . Indeed, in case r = 1, conclusions (i), (ii), of the theorem are equivalent, respectively, to properties (iv), (v) of the section . Suppose now r ≥ 2. The map g is obtained by “(r −1)-fold” integration of the map . In detail, let r−1 = : P → C ∞ (B, Rq ). By downward recursion on k deﬁne, t k−1 g0 (p, u, 0) + k−1 (p, u, t) = ∂t k (p, u, s) ds, 1 ≤ k ≤ r − 1. e. 37)) satisﬁes conclusions (i), (ii), of the theorem. 37) and employing (v), it follows that ∂tr 0 = ∂t : P → Γ(R).

The map g : [0, 1] → Γ(R) satisﬁes the property that g(0) = g(1) = β, and for each b ∈ B the corresponding path gb : [0, 1] → Rb strictly surrounds f (b) in the ﬁber Eb . Consequently (since “strictly surrounding” is an open condition), for each b ∈ B there is a neighbourhood U of b and a partition 0 < s1 < s2 · · · < sq+1 < 1 of the interval [0, 1], such that for all y ∈ U the sequence of points gy (s1 ), gy (s2 ), . . , gy (sq+1 ), spans an aﬃne q-simplex ∆(y) in the ﬁber Ey , and f (y) is an interior point of ∆(y).