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

By M.M. Cohen

This publication 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 could be available a semi-historical and geo­ metrically prompted exposition of J. H. C. Whitehead's appealing idea of simple-homotopy forms; that how you can comprehend this conception is to understand how and why it used to be outfitted. This trust is buttressed by way of the truth that the foremost makes use of of, and advances in, the idea in fresh 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 realizing. A moment cause of writing the ebook is pedagogical. this can be a very good 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 concept. the topic is obtainable (as within the classes pointed out on the outset) to scholars who've had a very good one­ semester path 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 was once performed with the welfare of the coed in brain. He may still do such workouts zealously.

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T 2 q) + q+ I = l + t�(t) + t �(t) + = + B(t)�(t) (t) t) = + + C(t)�(t)] uv = k=a = = l- 1 _ . == -I . . + (t (h - l ) q + 1 + . . + th q) . + t(h - l ) q + I � (t) I + A (t):E(t) and it follows that Thus V V( [I B(t)�(t)][ 1 �(t) divides V(t) V(t) - I . Thus I. I n the case where j q + I , set k k, j i and a a. The same argument works . 4) can be greatly sharpened . In fact we have from [HIGMAN] , [ B ASS 2; p. 54], and [BASS-MILNOR-SERRE ; Prop. 4. 1 4] , = . 5) l/ 7I.. q) is a free abelian group of rank [qI2] + 1 - S(q) where S(q) is the number of divisors of q.

Denote B! = dC i + l ,for all i. Then (A) B i is stably free for all i. (B) There is a degree-one module homomorphism S : C --+ C such that Sd+ dS = 1 . ] (C) If S: C --+ C is any chain contraction then, for each i, dS IBi_ 1 = 1 and Ci = B i Ef> SBi - l · REMARK: The S constructed in proving (B) also satisfies S 2 = 0, so that there is a pleasant symmetry between d and S. Moreover, given any chain contraction S, a chain contraction S' with (S') 2 = 0 can be constructed by setting S' = SdS. PROOF: Bo = Co because C is acyclic.

Thus the inclusion map i :K c Mg is a deformation. Also the collapse Mg'\.. L determines a deformation P: Mg � L. Since any two strong deforma­ tion retractions are homotopic, P is homotopic to the natural projection ' p: Mg � L. So f � g = pi � Pi = deformation. Therefore f is a simplehomotopy equivalence. 9) (The simple-homotopy extension theorem). Suppose that X < Ko < K is a C W triple and that f: Ko � L o is a cellular simple-homotopy equivalence such that fiX = 1. Let L = K U L o. Then there is a simple-homotopy f equivalence F: K � L such that FIKo = f.

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