By Hajime Sato

ISBN-10: 0821810464

ISBN-13: 9780821810460

The one so much tricky factor one faces while one starts off to profit a brand new department of arithmetic is to get a consider for the mathematical feel of the topic. the aim of this booklet is to aid the aspiring reader collect this crucial logic approximately algebraic topology in a brief time period. To this finish, Sato leads the reader via uncomplicated yet significant examples in concrete phrases. additionally, effects will not be mentioned of their maximum attainable generality, yet by way of the best and such a lot crucial circumstances. in line with feedback from readers of the unique variation of this ebook, Sato has extra an appendix of beneficial definitions and effects on units, common topology, teams and such. He has additionally supplied references.Topics coated contain basic notions corresponding to homeomorphisms, homotopy equivalence, primary teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. items and examples thought of within the textual content comprise the torus, the Mobius strip, the Klein bottle, closed surfaces, mobilephone complexes and vector bundles.

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**Extra resources for Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs, Volume 183)**

**Sample text**

Define the n-disk in Rn to be Dn = {(x',. . , x,) E RnI C X: 5 l } and note that S"-' G Dn is its boundary. 17 COROLLARY There is no retraction of Dn onto Sn-'. For n = 1 this is obvious since D' is connected and So is not. Suppose n > 1 and f:D n+ Sn-' is a map such that f o i = identity, where i is the inclusion of Sn-' in D". 7 28 1 SINGULAR HOMOLOGY. THEORY homomorphisms is commutative: However, this gives a factorization of the identity on an infinite cyclic group through zero which is impossible.

X,+~) so that d ( g ) = -1. Then PROOF d ( f ) = d ( h o g 0 h ) = d(h)"(g) = (*l)*(-1) = 0 -1. 22 COROLLARY The antipodal map A : Sn + S" A ( x l , .. , x,) = (-xl,. . , - x , ~ ) has d ( A ) = (-lp+I. 21 A is the composition of (n having degree -1. 0 PROOF defined by + 1)-maps, all - Exercise 8 Show that for n > 0 and m any integer, there exists a map Sn of degree m. 23 PROPOSITION If J g : Sn + Sn are maps with f(x) #g(x) for all x in S", then g is homotopic to A of. 32 1 SINGULAR HOMOLOGY THEORY Graphically the idea is as follows: since g(x) f f(x), the segment in Rn+l from Af(x) to g(x) does not pass through the origin.

A vector field # is nonzero if #(x) # 0 for each x in S". 26 COROLLARY There exists no nonzero vector field on S2n. - If # is a nonzero vector field on SZn,then y ( x ) = #(x)/ll #(x) 1 is a vector field on S2" of unit length. Thus, y : SZn SZnis a map for which y ( x ) is orthogonal to x for each x. 25. Hence, no such vector field exists. 0 PROOF Nonzero vector fields always exist on odd-dimensional spheres. A collection of vector fields 4, ,. . , #k on S" is linearly independent if for each x in S" the vectors #,(x), .

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