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 collage and the collage of Warwick in the course of 1969 and 1970. I wrote it as a result of a robust trust that there could be on hand a semi-historical and geo­ metrically prompted exposition of J. H. C. Whitehead's appealing thought of simple-homotopy kinds; that how you can comprehend this idea is to grasp how and why it used to be equipped. This trust is buttressed by way of the truth that the most important makes use of of, and advances in, the speculation 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 facts of topological invariance (given within the Appendix)-have come from simply such an knowing. A moment reason behind writing the publication is pedagogical. this is often an outstanding 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 idea. the topic is on the market (as within the classes pointed out on the outset) to scholars who've had an exceptional one­ semester direction in algebraic topology. i've got attempted to jot down proofs which meet the wishes of such scholars. (When an evidence was once passed over and left as an workout, it was once performed with the welfare of the scholar in brain. He should still do such routines zealously.

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For other tacit assumptions the reader is advised to quickly review §9. An (R, G)-module is defined to be a free R-module hi along with a "preferred" or "distinguished" family B of bases which satisfies : 46 Algebra If b and b' are bases of M and if b E B then b' E B -= T«b/b ' ») = 0 E KG(R). If M 1 and Ml are (R, G)-modules and if f: M l -+ Mz is a module iso­ morphism then the torsion ofI-written T(f)-is defined to be T(A) E KG(R), where A is the matrix of I with respect to any distinguished bases of Ml and M1.

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 .

Acyclic chain complexes In this section we develop some necessary background material concern­ ing acyclic complexes. e. a section) s: M --+ A such that js = 1 M . For suppose that M Ei3 F is free. Then jEi3 1 : A Ei3 F --+ MEi3 F is a surjection and there is certainly a section S: MEi3 F --+ A Ei3 F (gotten by mapping each basis element to an arbitrary element of its inverse image). Then s = P 1 Si, i s the desired section , where i 1 : M --+ M Ef> F and 11 1 : A Ei3 F --+ A are the natural maps.

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