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 might be available a semi-historical and geo metrically encouraged exposition of J. H. C. Whitehead's attractive thought of simple-homotopy kinds; that tips on how to comprehend this idea is to understand how and why it was once outfitted. This trust is buttressed by means of 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 realizing. A moment explanation for writing the booklet is pedagogical. this is often a great topic for a topology scholar to "grow up" on. The interaction among geometry and algebra in topology, each one enriching the opposite, is fantastically illustrated in simple-homotopy conception. the topic is available (as within the classes pointed out on the outset) to scholars who've had a very good 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 used to be performed with the welfare of the scholar in brain. He may still do such routines zealously.
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Additional resources for A Course in Simple-Homotopy Theory
Then the 1 x 1 matrix ( l - ab) is not an eiementary matrix, but since it represents the same element in Wh(G) as 0 -1 1 a -1 1 0 1 a I ab 0 2 (- ) ( )( )( ) ( ) 0 its torsion is O. ' = b i O I ' b I O I ' Algebra 42 §11. Some information about Whitehead groups In the last section we defined, for any group G, the abelian group Wh(G) given by Wh(G) = KTCIG) where T is the group of trivial units of lLG. The com putation of WHITEHEAD groups is a difficult and interesting task for which there has developed, in recent years, a rich literature.
L x [ - I , O] = [L, L] = since L x [ - I , I ] '\( L == L x I . 0 In pictures, these equations represent \ This completes the proof. 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 ......
We call this last complex M. 7), these 'numbers are precisely equal to the ranks of the free (integral) homology modules H (M r U L, L) r and H (M, M r U L). (M, L) 1 ranks are equal. 0 = O. Thus d is an isomorphism and these §S. 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 .
A Course in Simple-Homotopy Theory by M.M. Cohen