A Course in Simple-Homotopy Theory

I. Introduction.- §1. Homotopy equivalence.- §2. Whitehead’s combinatorial approach to homotopy theory.- §3. CW complexes.- II. A Geometric Approach to Homotopy Theory.- §4. Formal deformations.- §5. Mapping cylinders and deformations.- §6. The Whitehead group of a CW comple.- §7. Simplifying a homotopically trivial CW pair.- §8. Matrices and formal deformations.- III. Algebra.- §9. Algebraic conventions.- §10. The groups KG(R).- §11. Some information about Whitehead groups.- §12. Complexes with preferred bases [= (R,G)-complexes].- §13. Acyclic chain complexes.- §14. Stable equivalence of acyclic chain complexes.- §15. Definition of the torsion of an acyclic comple.- §16. Milnor’s definition of torsion.- §17. Characterization of the torsion of a chain comple.- §18. Changing rings.- IV. Whitehead Torsion in the CW Category.- §19. The torsion of a CW pair — definition.- §20. Fundamental properties of the torsion of a pair.- §21. The natural equivalence of Wh(L) and ? Wh (?1Lj).- §22. The torsion of a homotopy equivalence.- §23. Product and sum theorems.- §24. The relationship between homotopy and simple-homotopy.- §25. Tnvariance of torsion, h-cobordisms and the Hauptvermutung.- V. Lens Spaces.- §26. Definition of lens spaces.- §27. The 3-dimensional spaces Lp,q.- §28. Cell structures and homology groups.- §29. Homotopy classification.- §30. Simple-homotopy equivalence of lens spaces.- §31. The complete classification.- Appendix: Chapman’s proof of the topological invariance of Whitehead Torsion.- Selected Symbols and Abbreviations.