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

By M.M. Cohen

This booklet grew out of classes which I taught at Cornell college and the college 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 grasp how and why it used to be equipped. This trust is buttressed by means of the truth that the key makes use of of, and advances in, the speculation 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 knowing. A moment cause of writing the publication is pedagogical. this can be an exceptional topic for a topology scholar to "grow up" on. The interaction among geometry and algebra in topology, every one enriching the opposite, is superbly illustrated in simple-homotopy concept. the topic is on the market (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 put in writing proofs which meet the desires of such scholars. (When an evidence was once passed over and left as an workout, it used to be performed with the welfare of the coed in brain. He may still do such routines zealously.

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6) can , in fact, always be realized geometrically. (G) matrix then ( 1 ) There is a connected CW complex L with 7TI(L, eO) = G. 7) in simplifiedform such that the matrix of (K, L) J'l'ith respect to some set of characteristic maps is precisely A. (K, L) PROOF: Suppose that G is given by generators X I ' . . , Xm and relations R ,�x I ' . . , xm) = 1, (i = 1 , 2, . , n). Let L I = eO V (el V . . V e�), a I wedge product of circles, and let Xj be the element of the free group 7TI(L , eO) 2 represented by a characteristic map for eJ .

Kq = L such that f is homotopic to any deformation associated with this formal deformation. Let g = g q- l . . glgo be such a deformation, where gi: Ki � Ki + l . Notice that, for all i, Mg, '\.. dom gi = Ki. For if Kipl Ki + l> then Mg, = (KixI) U Ki+ l '\.. (KixI) '\.. 4) Mg, '\.. MgdK,+, U Ki = (Ki+l xl) U (KjxO) '\.. KjxO Thus Mg A MgO U ... _, rei Ko, '\.. (MgO U ... _2)'\.. • • • == K,. 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.

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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