Chapter 1. Prelude .............................................. 1
1. Introduction ................................................. 1
2. Conventions and some facts ................................... 4
3. The Basic Homotopy Lemma for dim(X) ≤ 1 ..................... 12
Chapter 2. The Basic Homotopy Lemma for higher dimensional
spaces .............................................. 27
4. X-theory and traces ......................................... 27
5. Some finite dimensional approximations ...................... 32
6. The Basic Homotopy Lemma - full spectrum .................... 39
7. The Basic Homotopy Lemma - finite CW complexes .............. 46
8. The Basic Homotopy Lemma - compact metric spaces ............ 51
9. The constant S and an obstruction behind the measure
distribution ................................................ 56
Chapter 3. Purely infinite simple C*-algebras .................. 65
10. Purely infinite simple C*-algebras ......................... 65
11. Basic Homotopy Lemma in purely infinite simple C*-
algebras ................................................... 72
Chapter 4. Approximate homotopy ................................ 79
12. Homotopy length ............................................ 79
13. Approximate homotopy for homomorphisms ..................... 87
14. Approximate homotopy for approximately multiplicative
maps ....................................................... 95
Chapter 5. Super Homotopy ..................................... 101
15. Super Homotopy Lemma - the purely infinite case ........... 101
16. Super Homotopy Lemma - the finite case .................... 108
Chapter 6. Postlude ........................................... 117
17. Non-commutative cases ..................................... 117
18. Concluding remarks ........................................ 126
17. Bibliography .............................................. 129
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