# Template:Kirby&Siebenmann1977

R. C. Kirby and L. C. Siebenmann, *Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004

## [edit] 1 External links

At Google Books.

Andrew Ranicki's page on the Hauptvermutung.

## [edit] 2 Table of Contents

Essay I on p.3

1. Introduction ... 3

2. Some definitions, conventions, and notations ... 6

3. Handles that can always be straightened ... 15

4. Concordance implies isotopy ... 24

5. The product structure theorem ... 31

A. Collaring theorems ... 40

B. Submerging a punctured torus ... 43

c. Majorant approximation ... 46

Essay II on p. 56

0. Introduction ... 57

1. Bundle theorems ... 59

2. Sliced concordance implies isotopy ... 73

Essay III on p. 80

0. Introduction ... 81

1. Transversality ... 83

2. Handle decompositions ... 104

3. Morse functions, and the TOP functional Smale theory ... 108

4. The simple homotopy type of a topological manifold ... 117

5. Simple types, decompositions and duality ... 126

A. On treating stable manifolds without surgery ... 139

B. Straightening polyhedra in codimension ... 143

C. A transversality lemma useful for surgery ... 145

Essay IV on p. 154

0. Introduction ... 155

1. Recollections concerning microbundles ... 159

2. Absorbing the boundary ... 164

3. Milnor's criterion for triangulating and smoothing ... 166

4. Statements about bijectivity of the smoothing rule and the pull-back rule ... 168

5. The main lemma for bijectivity of the smoothing rule ... 171

6. Bijectivity of the smoothing rule; its inverse ... 173

7. A parallel classification theorem ... 176

8. Classifying spaces obtained by E.H.Brown's method ... 180

9. Classifying reductions ... 187

10. Homotopy classifcation of manifold structures ... 194

A. Normal bundles ... 203

B. Examples of homotopy tori ... 208

Essay V on p. 216

0. Introduction ... 217

1. Classifying CAT structures on mfds by CAT structures on microbundles ... 221

2. CAT structures and classifying spaces ... 233

3. The relative classification theorem ... 240

4. Classifications for manifolds with boundary ... 242

5. The homotopy groups ... 246

A. The immersion theoretic method without handle decompositions ... 256

B. Classification of homotopy tori: the necessary calculation ... 264

C. Some topological surgery ... 273

- Periodicity in topological surgery ... 277
- Applications of periodicity ... 284

Annex 1. Stable heomorphisms and the annulus conjecture ... 291

Annex 2. On the triangulation of manifolds and the Hauptvermutung ... 299

Annex 3. Topological manifolds ... 307

## [edit] Corrections

A generator for : In Proposition 11.2 on p.322, Kirby and Siebenmann state that the characteristic number , where is the Kirby-Siebenmann invariant and is the second Stiefel-Whitney class, defines an isomorphism detecting . This statement is correct. However, the proof that is incorrect. The authors write that in their argument that a fake gives a generator for . But .

On the other hand, and so a similar argument to the one given for shows that a fake has and so gives a generator for .