By M.M. Cohen
This publication 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 due to a robust 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 to comprehend this thought is to grasp how and why it used to be equipped. This trust is buttressed through 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 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 figuring out. A moment cause of writing the publication is pedagogical. this can be a very good topic for a topology pupil 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 available (as within the classes pointed out on the outset) to scholars who've had an outstanding one semester path in algebraic topology. i've got attempted to write 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 coed in brain. He should still do such workouts zealously.
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Extra info for A Course in Simple-Homotopy Theory
There is a natural injection of GL(n, R) into GL(n + I , R) given by Using this, the infinite general linear group of R is defined as the direct limit GL(R) = lim GL(n, R). (Alternatively, GL(R) may be thought of as the group -;. ) For notational convenience we shall identify each A E GL(n, R) with its image in GL(R). , j (i i' j) be the n x n matrix with all entries 0 except for a 1 (unity element in R) in the (i, j)-spot. An elementary matrix is a matrix of the form (l. + aEf, j) for some a E R.
Cj ' II d 8 ) Il l lli(Bi - I ) = O. Ci - I II ( d 8 ) Cj - 2 � • • • II � ;;r; B ' 8Bi _ 1 +-;;- IlBj_ 2 +-;;- IlBj _ 3 • Clearly dll + Il d = 1 . This proves (B). Suppose finally we are given Il : C � C such that dll + Ild = 1 . Then dll l Bj _ 1 = (dll + Ild) IBi _ 1 = I B, _ , . 2) If 0 � C' � C � C" � 0 is an exact sequence of chain complexes over R, where C" is free and acyclic, then there exists a section s : C" � C such that s is a chain map and i + s : C' EEl C" � C is a chain isomorphism.
Similarly, for all n ;::: 3, i # is one-one, because any homotopy F: (/ 2 , (1 2) � (P, Po) between maps Fo and FI can be replaced by a map G : /2 � Po such that G l ol 2 = Fl ol 2 . Finally, if n = 2, 'Pi(01 2 ) = eO , by assumption. Let R : P � Po be the retraction such that R ( UeD = e O . Then, if two maps/, g : (I, (1) � (Po, eo) are homotopic in P by the homotopy Ft, they are homotopic in Po by the homotopy RoFt• Hence i# is one-one in this case also. Let p : P � P be the univet:sal covering of P.