By Hajime Sato
The one such a lot tough factor one faces whilst one starts off to profit a brand new department of arithmetic is to get a suppose for the mathematical feel of the topic. the aim of this e-book 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 basic yet significant examples in concrete phrases. in addition, effects usually are not mentioned of their maximum attainable generality, yet by way of the best and so much crucial situations.
In reaction to feedback from readers of the unique variation of this e-book, Sato has extra an appendix of necessary definitions and effects on units, basic topology, teams and such. He has additionally supplied references.
Topics coated comprise basic notions similar to homeomorphisms, homotopy equivalence, primary teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. gadgets and examples thought of within the textual content contain the torus, the Möbius strip, the Klein bottle, closed surfaces, telephone complexes and vector bundles.
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Extra resources for Algebraic Topology: An Intuitive Approach
BlIBis cycles on the polygon must take the quotient group of zm modulo the subgroup generated by the element (2, ... ,2). Consider el - (1,0, ... ,0), ... , em-l = (0, ... ,0,1,0), and em = (1, ... ,1). We have The condition (al, ... , am) E Imih is equivalent to al - am = 0, ... , am-lam = 0 and am == 0 (mod 2). Therefore, the quotient group zm / 1m ih is isomorphic to zm-l EB Z2; the group zm-l is generated by el. , em-I. and the group Z2 is generated by em. As before, for the coefficient group Z2, we have ih = O.
A) For k > 0, H2k(JRpn) = 0, and for 0 ~ k < -2-' H2k+l (lRpn) = Z2. (b) For 0 ~ k :::; n, Hk(JRpn j Z2) = Z2. Proof. pn is endowed with the structure of the CW-complex with cells JRpk \ JR'pk - 1 (see Part I, pp. 118 119). Then Ck = Z for o ~ k ~ n. The central symmetry of the sphere 8 k - 1 preserves orientation for even k and reverses it for odd k. Therefore, the boundary homomorphism 8 k : Ck -+ Ck-l takes 1 to 1 + 1 = 2 for even k and to 1 - 1 = 0 1. Simplicial Homology 42 for odd k. The chain complex for calculating the cellular homology of the complex lRpn for even n has the form Z x2 ---+ 0 Z -- Z x2 ---+ Z ---+ ...
Therefore, the quotient group zm / 1m ih is isomorphic to zm-l EB Z2; the group zm-l is generated by el. , em-I. and the group Z2 is generated by em. As before, for the coefficient group Z2, we have ih = O. 0 Cycles Q and (3 generating the integral homology group for the Klein bottle are shown in Figure 6. At the level of homology, we have 2(3 = 0 because the boundary of the Mobius band hatched in Figure 6a is homologous to 2(3. Problem 29. Calculate the homology groups of closed two-dimensional surfaces with coefficients in Zp for p f:.
Algebraic Topology: An Intuitive Approach by Hajime Sato