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

By M.M. Cohen

This e-book grew out of classes which I taught at Cornell collage and the collage of Warwick in the course of 1969 and 1970. I wrote it as a result of a powerful trust that there might be on hand a semi-historical and geo­ metrically prompted exposition of J. H. C. Whitehead's attractive thought of simple-homotopy kinds; that tips on how to comprehend this conception is to grasp how and why it used to be outfitted. This trust is buttressed via 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 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 cause of writing the publication is pedagogical. this is often 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 conception. the topic is obtainable (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 evidence used to be passed over and left as an workout, it was once performed with the welfare of the scholar in brain. He may still do such workouts zealously.

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For notational simplicity we consider RI -,>- RI + pR2 · I. when R I -,>- ± aR I and II. when To realize R I -,>- - R I , set M = K and introduce the new characteristic map

- 1' + 1 by R (X t , X2 , " " xr t > = ( 1 - x I , x2 ' . . pj] ' To realize R I -,>- aR I ' let f: (/', oJ') (K" eO) represent a ' ['P t l I'] E 7Tr(K" L). Extend f trivially to 01' + I . Set = -,>- M = L u U ej u U < + I u e� + t j where ej" + 1 ;> 1 has characteristic map

- f .

7) '\.. Mgo'\.. Ko = K. (b) => (c): Suppose that g is a cellular approximation to f such that Mg A K rei K and that g' is any cellular approximation to f. 5), Mg, A Mg A K rel K. (c) => (a): Let g be any cellular approximation to f. By hypothesis Mg A K, rei K. 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.

C. Alperin , R . K . Dennis and M . R . Stein shows that, even for finite abelian G , SK I (7I.. G) is usually not zero. For example, SK1 (7I.. 2 E9 (71.. 3 ) 3 ) ;;;;; (71.. 3 )6 . From 0 1 . 5) one sees that the functor Wh(G) does not behave very well with respect to direct prod ucts . 6) [STALLINGS I ] E9 Wh(G 2 ) · 0 {f G I and G 2 are any groups then Wh(G j *G 2 ) = Wh(G I ) §12. Complexes with preferred bases [ = (R, G)-complexes] From this point on we assume that G is a subgroup of the units of R which contains the element ( - I ) .

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