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AuthorCohen, Marshall M. author
TitleA Course in Simple-Homotopy Theory [electronic resource] / by Marshall M. Cohen
ImprintNew York, NY : Springer New York : Imprint: Springer, 1973
Connect tohttp://dx.doi.org/10.1007/978-1-4684-9372-6
Descript XI, 116 p. online resource

SUMMARY

This book grew out of courses which I taught at Cornell University and the University of Warwick during 1969 and 1970. I wrote it because of a strong belief that there should be readily available a semi-historical and geoยญ metrically motivated exposition of J. H. C. Whitehead's beautiful theory of simple-homotopy types; that the best way to understand this theory is to know how and why it was built. This belief is buttressed by the fact that the major uses of, and advances in, the theory in recent times-for example, the s-cobordism theorem (discussed in ยง25), the use of the theory in surgery, its extension to non-compact complexes (discussed at the end of ยง6) and the proof of topological invariance (given in the Appendix)-have come from just such an understanding. A second reason for writing the book is pedagogical. This is an excellent subject for a topology student to "grow up" on. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple-homotopy theory. The subject is accessible (as in the courses mentioned at the outset) to students who have had a good oneยญ semester course in algebraic topology. I have tried to write proofs which meet the needs of such students. (When a proof was omitted and left as an exercise, it was done with the welfare of the student in mind. He should do such exercises zealously


CONTENT

I. Introduction -- ยง1. Homotopy equivalence -- ยง2. Whiteheadโ{128}{153}s combinatorial approach to homotopy theory -- ยง3. CW complexes -- II. A Geometric Approach to Homotopy Theory -- ยง4. Formal deformations -- ยง5. Mapping cylinders and deformations -- ยง6. The Whitehead group of a CW comple -- ยง7. Simplifying a homotopically trivial CW pair -- ยง8. Matrices and formal deformations -- III. Algebra -- ยง9. Algebraic conventions -- ยง10. The groups KG(R) -- ยง11. Some information about Whitehead groups -- ยง12. Complexes with preferred bases [= (R,G)-complexes] -- ยง13. Acyclic chain complexes -- ยง14. Stable equivalence of acyclic chain complexes -- ยง15. Definition of the torsion of an acyclic comple -- ยง16. Milnorโ{128}{153}s definition of torsion -- ยง17. Characterization of the torsion of a chain comple -- ยง18. Changing rings -- IV. Whitehead Torsion in the CW Category -- ยง19. The torsion of a CW pair โ{128}{148} definition -- ยง20. Fundamental properties of the torsion of a pair -- ยง21. The natural equivalence of Wh(L) and ? Wh (?1Lj) -- ยง22. The torsion of a homotopy equivalence -- ยง23. Product and sum theorems -- ยง24. The relationship between homotopy and simple-homotopy -- ยง25. Tnvariance of torsion, h-cobordisms and the Hauptvermutung -- V. Lens Spaces -- ยง26. Definition of lens spaces -- ยง27. The 3-dimensional spaces Lp,q -- ยง28. Cell structures and homology groups -- ยง29. Homotopy classification -- ยง30. Simple-homotopy equivalence of lens spaces -- ยง31. The complete classification -- Appendix: Chapmanโ{128}{153}s proof of the topological invariance of Whitehead Torsion -- Selected Symbols and Abbreviations


Mathematics Algebraic topology Mathematics Algebraic Topology



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