Marcelo Aguilar, Samuel Gitler, Carlos Prieto's Algebraic Topology from a Homotopical Viewpoint PDF

By Marcelo Aguilar, Samuel Gitler, Carlos Prieto

ISBN-10: 0387954503

ISBN-13: 9780387954509

The authors current introductory fabric in algebraic topology from a unique viewpoint in utilizing a homotopy-theoretic process. This rigorously written publication could be learn by means of any pupil who is familiar with a few topology, offering an invaluable option to quick study this novel homotopy-theoretic viewpoint of algebraic topology.

From the reviews:

"This is a simple direction on algebraic topology and … it really is excellently written and composed and will be strongly urged to anyone wishing to benefit the sphere. The reader can locate many examples, calculations, and in addition a couple of workouts. … The publication has appendices, … references along with eighty three goods and an extended checklist of symbols. there are numerous 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, reminiscent of teams or jewelry, to a topological house in the sort of manner that identical areas show isomorphic invariants; the following, ‘equivalent’ will be selected to slot the geometry of the matter. during this publication, ‘equivalent’ skill homotopy an identical … . For its readability and directness, a welcome boost to complicated arithmetic collections. Upper-division undergraduates via faculty." (J. McCleary, selection, December, 2002)

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

“The booklet of Aguilar, Gitler and Prieto offers an engaging new strategy for a primary direction in algebraic topology. … what's additionally very attention-grabbing is the truth that the publication includes a targeted presentation of a few deep result of algebraic topology now not often coated by way of a primary publication in algebraic topology … . The textual content is punctiliously good written. … the scholar in algebraic topology will locate within the publication loads of fascinating well-exposed material.” (Yves Félix, Bulletin of the Belgian Mathematical Society, 2007)

From the again Cover
The function of this e-book is to introduce algebraic topology utilizing the unconventional procedure of homotopy idea, an procedure with transparent functions in algebraic geometry as understood through Lawson and Voevodsky. this technique permits the authors to hide the fabric extra successfully than the extra universal process utilizing homological algebra. the fundamental thoughts of homotopy idea, resembling fibrations and cofibrations, are used to build singular homology and cohomology, in addition to K-theory. through the textual content many different primary thoughts are brought, together with the development of the attribute periods of vector bundles. even though functors look regularly 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 simple wisdom of element set topology in addition to workforce thought 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)

Example text

Therefore, it follows from the following exact sequence that nt(s)EFl H:J(X). Then, it follows easily that which clearly implies S-lH:J(X)=O (forsN+1ES). The transition from the case XS = 0 to the case XS i= 0 is the same as in the compact case. We shall show that is also an isomorphism under the assumption of finite orbit types. Let G 1 , G z ,"" Gn be the orbit types in X _Xs and SiES maps to zero in H*(B G ). Then it is clear that and hence the above proof applies. 0 Remark. (i) The above proof in fact shows that which is sometimes a useful fact.

12=gl~g2 (orresp. glc~g2cl(iii) Since the existence of Lie algebras of the classical types, i. , An' Bn, Cn' D,P is a well known fact, one need only to show the existence of a simple Lie algebra; for each of the five exceptional types. In view of the above explicit basis and structural constants, it is a matter of straightforward verification. Proof of the Chevalley theorem. hat Na,li ~re all real numbers. Note that two pairs {Xa' X -a} and {X~, X'-a} with Xa =X -a' X~ = X'-x and (Xa' X -a) =(X~, X'-a) =2/10:1 2 differ by a factor of eiO, i.

U) + ... }. ,H)(t) + possible more terms. Now the irreducibility of Ij; (cf. 3'), § 1-C, Ch. I) implies that 1= = fG x",(g)· x",(g)dg = -1fT Ix",(t)Q(t)1 2 dt = -1 ! ,H)(t) + '''IIL2(T) 1f 2 =-\m(A",) ·W+"· } w ~m(A",) 2 2 IIx",(t)· Q(t)IIL2(T) (by Schur orthogonality) . ,H)(t) which is exactly the Weyl character formula. Since the character is a complete invariant and the above Weyl character formula gives an explicit expression of X",(t) in turms of the highest weight A"" it is obvious that Ij; ~

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

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