By M.M. Cohen

This e-book 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 due to a powerful trust that there might be on hand a semi-historical and geo metrically stimulated exposition of J. H. C. Whitehead's appealing conception of simple-homotopy varieties; that tips to comprehend this concept is to understand how and why it used to be equipped. This trust is buttressed through the truth that the foremost makes use of of, and advances in, the speculation in fresh 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 facts of topological invariance (given within the Appendix)-have come from simply such an knowing. A moment reason behind writing the booklet is pedagogical. this is often a very good 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 offered (as within the classes pointed out on the outset) to scholars who've had a great one semester path in algebraic topology. i've got attempted to put in writing proofs which meet the wishes 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 should still do such workouts zealously.

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**Extra resources for A Course in Simple-Homotopy Theory**

**Sample text**

If G is a group then a unit in 7l.. (G) is an element with a two-sided multi plicative inverse. The elements of the group ± G = {gig E G } v { - gig E G} are called the trivial units, and all others are called the non-trivial units of 7l.. ( G) . 6) A . Suppose that G is an abelian group such that 7l.. (G) has non-trivial units . (G) matrix A which cannot be transformed to an identity matrix by any finite sequence of the operations (I)-(V). B. The group G = 7l.. 5 is an abelian group such that 7l..

9), determines the deformation (M, u K) A (M, u K) up L 2 L, L, = K Ll L2 , reI L 2 . J It follows directly from the second definition that f* is a group homo morphism. 6) it follows directly that g* f* = (gf) * . 2) There is a covariant functor from the category offinite CW complexes and cellular maps to the category of abelian groups and group homomorphisms given by L ...... Wh(L) and (f: L I -+ L2) ...... U* : Wh(L 1) -+ Wh(L 2 ))' Moreover iff ':::!. g then f* = g * . PROOF: The reader having done his duty, we need only verify that if f ':::!.

Matrices and formal deformations Given a homotopically trivial CW pair, we have shown that it can be transformed into a pair in simplified form. So consider a simplified pair (K, L) ; K = L u Uej u Uer + 1 where the ej are trivially attached at eO . W + 1 : 81' + 1 --+ L u U ej, where 'Pi is a characteristic map for er+ 1 . (K" L ; eO) in the homotopy r + exact sequence of the triple (K, K , L). Since, however, freely homotopic r attaching maps give (7. 1 ) the same result up to simple-homotopy type, we do not wish to be bound to homotopies keeping the base point fixed.