Introduction ................................................... ix
0.1. Background ................................................ ix
0.2. Main results ............................................. xii
0.3. The connection with generalised Morse functions and
Part II .................................................. xiv
0.4. Acknowledgements ........................................ xvii
Chapter 1. Definitions and Preliminary Results .................. 1
1.1. Isotopy and concordance in the space of metrics of
positive scalar curvature .................................. 1
1.2. Warped product metrics on the sphere ....................... 2
1.3. Torpedo metrics on the disk ................................ 4
1.4. Doubly warped products and mixed torpedo metrics ........... 5
1.5. Inducing a mixed torpedo metric with an embedding .......... 9
Chapter 2. Revisiting the Surgery Theorem ...................... 11
2.1. Surgery and cobordism ..................................... 11
2.2. Surgery and positive scalar curvature ..................... 12
2.3. Outline of the proof of Theorem 2.3 ....................... 14
2.4. Part 1 of the proof: Curvature formulae for the first
deformation ............................................... 15
2.5. Part 2 of the proof: A continuous bending argument ........ 17
2.6. Part 3 of the proof: Isotoping to a standard product ...... 26
2.7. Applying Theorem 2.3 over a compact family of
psc-metrics ............................................... 29
2.8. The proof of Theorem 2.2 (The Improved Surgery Theorem) ... 31
Chapter 3. Constructing Gromov-Lawson Cobordisms ............... 35
3.1. Morse Theory and admissible Morse functions ............... 35
3.2. A reverse Gromov-Lawson cobordism ......................... 40
3.3. Continuous families of Morse functions .................... 41
Chapter 4. Constructing Gromov-Lawson Concordances ............. 45
4.1. Applying the Gromov-Lawson technique over a pair of
cancelling surgeries ...................................... 45
4.2. Cancelling Morse critical points: The Weak and Strong
Cancellation Theorems ..................................... 47
4.3. A strengthening of Theorem 4.2 ............................ 48
4.4. Standardising the embedding of the second surgery
sphere .................................................... 49
Chapter 5. Gromov-Lawson Concordance Implies Isotopy for
Cancelling Surgeries ................................ 51
5.1. Connected sums of psc-metrics ............................. 51
5.2. An analysis of the metric g", obtained from the second
surgery ................................................... 51
5.3. The proof of Theorem 5.1 .................................. 53
Chapter 6. Gromov-Lawson Concordance Implies Isotopy in the
General Case ........................................ 65
6.1. A weaker version of Theorem 0.8 ........................... 65
6.2. The proof of the main theorem ............................. 67
Appendix: Curvature Calculations from the Surgery Theorem ...... 71
Bibliography ................................................... 79
|
