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

By M.M. Cohen

This ebook 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 as a result of a powerful trust that there may be on hand a semi-historical and geo­ metrically inspired exposition of J. H. C. Whitehead's attractive conception of simple-homotopy forms; that the way to comprehend this conception is to understand how and why it used to be outfitted. This trust is buttressed through the truth that the main 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 figuring out. A moment cause of writing the publication is pedagogical. this can be a great 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 idea. the topic is available (as within the classes pointed out on the outset) to scholars who've had a great one­ semester path in algebraic topology. i've got attempted to write down proofs which meet the wishes 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 scholar in brain. He may still do such routines zealously.

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1 I Stable equivalence of acyclic chain complexes -'>- otherwise, and a ; = 1 : T; The relevant diagrams are C EB T: ... -'>- C3 To. W e claim that C EB T ---+ C2 EB dl CI d,ED I � Co � EB /0 Co -'>- � 51 C 6 EB T'(�). 0 Co Define f: C EB T -'>- C6 EB T' by fi = 1 , if i + 1 fl (CO + C I ) = 8 I cO + (c!

K" L ; eO) in the homotopy r + exact sequence of the triple (K, K , L). Since, however, freely homotopic r attaching maps give (7. 1 ) the same result up to simple-homotopy type, we do not wish to be bound to homotopies keeping the base point fixed. To capture this extra degree of freedom formally, we shall think of the homotopy A geometric approach to homotopy theory 28 groups not merely as abelian groups, but as mod ules over l(1T ) (L , e O» . 3] that 1T) = 1T ) (PO ' x) acts on 1TnCP, Po ; x) by the condition that [

Thus [K 't K', L] is an element of Wh(L) if [K, L] and [K', L] are. 3') , so [K v K', L] = [iV K', L]. Similarly, if L L [K', L] = [1', L], then [i "t defined. K', L] = [i L "t i', L 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.

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