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

By M.M. Cohen

ISBN-10: 0387900551

ISBN-13: 9780387900551

ISBN-10: 038790056X

ISBN-13: 9780387900568

ISBN-10: 3540900551

ISBN-13: 9783540900559

This booklet grew out of classes which I taught at Cornell collage and the college 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 stimulated exposition of J. H. C. Whitehead's attractive thought of simple-homotopy forms; that find out how to comprehend this concept is to grasp how and why it was once equipped. This trust is buttressed via the truth that the key makes use of of, and advances in, the speculation in contemporary 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 explanation for writing the publication is pedagogical. this can be an exceptional topic for a topology pupil to "grow up" on. The interaction among geometry and algebra in topology, each one enriching the opposite, is superbly illustrated in simple-homotopy idea. the topic is obtainable (as within the classes pointed out on the outset) to scholars who've had an excellent one­ semester direction in algebraic topology. i've got attempted to put in writing 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 scholar in brain. He may still do such routines zealously.

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Example text

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. So Bo is free. Assume inductively that Bi - I is known to be stably free. Theil there is a section s :Bi _ 1 --+ Ci. Because C is acyclic the sequence is thus a split exact sequence.

Hence, since D is a division ring. * (A) is square, implying that A is square. Therefore R satisfies ( *). 2) If G is a group then £'(G) satisfies ( * ). PROOF: The augmentation map A : £'(G) -+ (rationals) given by A(I n,g,) = I n" is a non-zero ring homomorphism. Apply (9. 1). 0 , Matrices : If J: M -+ M is a module homomorphism where M I and M have 2 I 2 ordered bases x = {X l ' . . , xp } and y = {Y l , . . , denotes the matrix (a,) where f(xJ = I aijYj' Thus each row of j

Clearly g is an isomorphism since, by ( 1 3 . 1 C), eSd:oBi_ 1 � eSBi - I ' If Bi and OBi - I were free then we could, by using a basis for Ci which is the union of a basis for Bi and a basis of oBi - l o write down a matrix which clearly reflects the structure of g. This observation motivates the following proof of (A). Bi and oBi - 1 are stably free, so there exist free modules FI and F2 such that FI Et> Bj and oBi - 1 E9 F2 are free. Fix bases for Ft and F2 and take the union of these with a preferred basis for Ci to get a basis c of Ft E9 Ci E9 F2• Let G = I F, E9 g Et> I F, : FI Et> Ci Et> F2 � FI Et> Ci Et> F2 .

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

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