Download PDF by M.M. Cohen: A Course in Simple-Homotopy Theory

By M.M. Cohen

ISBN-10: 0387900551

ISBN-13: 9780387900551

ISBN-10: 038790056X

ISBN-13: 9780387900568

ISBN-10: 3540900551

ISBN-13: 9783540900559

This ebook grew out of classes which I taught at Cornell collage and the college of Warwick in the course of 1969 and 1970. I wrote it due to a powerful trust that there could be available a semi-historical and geo­ metrically encouraged exposition of J. H. C. Whitehead's appealing conception of simple-homotopy forms; that how you can comprehend this thought is to understand how and why it used to be outfitted. This trust is buttressed by means of the truth that the foremost makes use of of, and advances in, the idea in contemporary times-for instance, the s-cobordism theorem (discussed in §25), using the idea in surgical procedure, its extension to non-compact complexes (discussed on the finish of §6) and the evidence of topological invariance (given within the Appendix)-have come from simply such an knowing. A moment reason behind writing the publication is pedagogical. this can be a superb topic for a topology pupil to "grow up" on. The interaction among geometry and algebra in topology, every one enriching the opposite, is fantastically illustrated in simple-homotopy thought. the topic is available (as within the classes pointed out on the outset) to scholars who've had an exceptional one­ semester path in algebraic topology. i've got attempted to jot down proofs which meet the desires of such scholars. (When an evidence was once passed over and left as an workout, it was once performed with the welfare of the coed in brain. He should still do such workouts zealously.

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Extra resources for A Course in Simple-Homotopy Theory

Example text

Hence H*CP,Po) is a free Z17" l -module with basis { <� i> }' We com plete the proof by demonstrating that HnCP,Po) is isomorphic to 17"nCP,P ; eO) as a Z17" I -mod ule, by an isomorphism which takes <�;) onto O ['PJ for each i. p*Cw n). In fact, applying the Hurewicz theorem [SPANIER, p. 397], h is an isomorphism because Po and P are connected and simply connected and because, by the cellular approximation theorem, 17";(P,Po) = 0 for i :5 n - 1. Also p # is an isomorphism for all n ;e: 1, by the homotopy lifting property.

The notion of "elementary collapse" is replaced in the non-compact case by "countable disjoint sequence of finite collapses". For a development of the non-compact theory see [SIEBENMANN] and [FARRELL-WAGONER]. Also the discussion in [ECKMANN-MAUMARV] is valid for locally finite complexes. Finally, the author thinks that [COHEN, §8] is relevant and i nteresting. §7. Simplifying a homotopically trivial CW pair In this section we take a CW pair (K, L) such that K 1", L and simplify it by expanding and collapsing reI L.

Thus the addition is well 6 The viewpoint of this section has recently been arrived at by many people indepen­ dently. It is interesting to compare [Stocker], [Siebenmann], [Farrell-Wagoner], [Eckmann­ Maumary] and the discussion here. The Whitehead group of a CW complex 21 That the addition is associative and commutative follows from the fact that the union of sets has these properties. The element [L, L] is an identity, denoted by O. If [K, L] E Wh(L), let D : K -+ L be a cellular strong deformation retrac­ tion.

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A Course in Simple-Homotopy Theory by M.M. Cohen


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