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 e-book 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 as a result of a robust trust that there might be available a semi-historical and geo­ metrically prompted exposition of J. H. C. Whitehead's attractive idea of simple-homotopy forms; that tips to comprehend this conception is to understand how and why it was once outfitted. This trust is buttressed by means of the truth that the main makes use of of, and advances in, the idea 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 evidence of topological invariance (given within the Appendix)-have come from simply such an realizing. A moment reason behind writing the ebook is pedagogical. this can be a good 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 out there (as within the classes pointed out on the outset) to scholars who've had a superb one­ semester direction in algebraic topology. i've got attempted to put in writing proofs which meet the desires 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 coed in brain. He may still do such routines zealously.

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We now attach an (11 + I )-cell to L U eo U e ) by 'P to get the CW complex . K = (L U eo U e l ) U (In X J). " x {i} is a characteristic map for ei we have Ko If eo (') e l eo (') (eo U e l ) Then, by oF = L U eo � K '§. L U el = KI > rei L. 0, construct a CW complex K = L U eo such that 0 and such that eo has the same attaching map as eo· the special case above, Ko A Ko A K1 , reI L. 0 = 24 A geometric approach to homotopy theory As an example, (7. 1) may be used to show that the dunce hat D has the same simple-homotopy type as a point.

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. 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.

For n = 2, 3 we use the fact that 0 is an isomorphism because A was assumed non-singular. Thus, for n = 2, we have the sequence 0 ------- o 0 and by exactness it follows that TT2(K, L) = O. ) Finally note that TTiK, L) ;;;; TT3 (K, L ) ;;;; H3 (K, L ), the last isomorphism coming from the Hurewicz theorem which applies because = TT/K, L) ;;;; TTi(K, L ) for i = 1 , 2 and because L is I -connected by (3. 1 3). To see that H3 (K, L ) = consider the commutative diagram 0 0 Clearly a is an isomorphism so that, by exactness of the top line, H3 (K, L) = O.

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

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