Read e-book online A Course in Simple-Homotopy Theory PDF

By M.M. Cohen

ISBN-10: 0387900551

ISBN-13: 9780387900551

ISBN-10: 038790056X

ISBN-13: 9780387900568

ISBN-10: 3540900551

ISBN-13: 9783540900559

This booklet grew out of classes which I taught at Cornell college and the collage of Warwick in the course of 1969 and 1970. I wrote it due to a robust trust that there will be on hand a semi-historical and geo­ metrically prompted exposition of J. H. C. Whitehead's appealing concept of simple-homotopy forms; that tips on how to comprehend this concept is to grasp how and why it used to be equipped. This trust is buttressed through the truth that the most important makes use of of, and advances in, the idea in contemporary times-for instance, the s-cobordism theorem (discussed in §25), using the speculation 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 cause of writing the ebook is pedagogical. this can be an outstanding topic for a topology pupil to "grow up" on. The interaction among geometry and algebra in topology, each one enriching the opposite, is fantastically illustrated in simple-homotopy idea. the topic is offered (as within the classes pointed out on the outset) to scholars who've had an excellent one­ semester direction in algebraic topology. i've got attempted to jot down proofs which meet the desires of such scholars. (When an explanation used to be passed over and left as an workout, it used to be performed with the welfare of the coed in brain. He should still do such routines zealously.

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Thus v,j ! a1' is null homotopic in L and, L being arc-wise connected , it is homotopic to the constant map at eO. Therefore by (7. 1 ) , L u Uej A L u U ej, rei L where the ej are trivially attached at eO. 9) L u U ej u U er + 1 A L u Uej u UfT + 1 • Now let the f:+ 1 have characteristic maps

If (K, L) and (K', L) are homotopically trivial CW pairs, define (K, L) '" (K', L) iff K A K' reI L. This is clearly an equivalence relation and we let [K, L] denote the equivalence class of (K, L). An addition of equivalence classes is defined by setting [K,. of K and K' identified by the identity map L on L. 2 it doesn't matter which "disjoint union of K and K' identified . . " we take. 2) the equivalence classes form a set, since the isomorphism classes of finite CW complexes can easily be seen to have cardinality :s; 2c.

1) Suppose that (P, Po) is a C W pair with P = I a Po u U e'/, where Po is ;=1 connected. Suppose that 'P i : (/"' I" - I , r - I ) � (P, Po ; eO ) are characteristic maps for the e7 and that either: a) n ;::: 3, or b) n = 2 and 'P i(O/") = eO for all i. (P, Po ; eO ) is a free l1T l -module with basis ['Pd, ['P2 ], . . , ['Pa]' PROOF: We claim first that the inclusion map induces an isomorphism i# : 1Tl(PO , eO ) � 1Tl (P, eO), For all n ;::: 2, i# is onto because, by the cellular approximation theorem , any map of (/ 1 , al l ) into (P, eO) can be homotoped rel al l into Po.

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

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