# Regensburg Surgery Blockseminar 2012: General information

The Regensburg surgery Blockseminar runs March 25 - 30 2012.

## 1 Introduction

The aim of this seminar is to give a detailed treatment of the smooth surgery exact sequence for manifolds of dimension 5 and higher and then to review extensions of this sequence to the piecewise linear and topological categories and to give a range of applications.

The main reference for the seminar is Wolfgang Lück's lecture notes, A basic introduction to surgery [Lück2001] but we will use various other sources.

### 1.1 Prerequisites

The seminar will assume not go into the following important subjects:

• The general theory of smooth manifolds, submanifolds, normal bundles and tubular neighbourhoods: see e.g. [Milnor&Stasheff1974, Ch.11 & Ch 18].

An excellent place to learn the essential ideas for surgery remains [Milnor1961], in particular the first four sections.

## 2 Information for speakers

Each talk will run for 60 minutes with up to 15 minutes for questions.

The talks will be given using white-boards.

### 2.1 Scientific guidelines

Where there is no single way to give a good talk, here are some suggestions:

• Identify the main result(s) of your talk.
• Aim to state your main results as quickly and as clearly as possible at the beginning of your talk: this may require leaving certain concepts to be fully explained in later in your talk.
• Where possible, illustrate the main results with clarifying examples and explain how they fit into the overall development of the seminar: i.e. indicate results from previous talks you build and/or results in future talks that build on your talk.
• Finally move onto the proof of your main results. Where possible, break the proof into a series of clearly stated lemmas and prove each lemma in turn. It is OK to skip the full proofs of technical or very difficult lemmas.

## 3 Schedule

1. The s-cobordism theorem I: [Lück2001, 1.1-1.3]; Farid Madani and Mihaela Pilca, Lecture notes
2. The s-cobordism theorem II: [Lück2001, 1.3 & 1.4]; Regensburg
3. The s-cobordism theorem III: [Lück2001, Ch.2]; Regensburg, Lecture notes
4. Poincaré complexes: [Lück2001][3.1] L; Freiburg, Lecture notes
5. Spherical fibrations and the normal Spivak fibration: [Lück2001, 3.2.2 & 3.2.3]; Freiburg, Lecture notes
6. Normal maps and the Pontrjagin-Thom isomorphism: [Lück2001, 3.1 & 3.3]; Andreas Hermann and Michael Völkl, Lecture notes
7. Surgery below the middle dimension: [Lück2001, 3.4]; Nicolas Ginoux and Carolina Neira-Jiménez, Lecture notes
8. Intersections and self-intersections: [Lück2001, 4.1]; Regensburg
9. Kernels and forms: [Lück2001, 4.2]; Filipp Levikov, Lecture notes
10. Even dimensional surgery obstructions: [Lück2001, 4.3 & 4.4]; Edinburgh, Lecture notes
11. Odd dimensional surgery obstructions: [Lück2001, 4.5 & 4.6]; Edinburgh
12. Manifolds with boundary and simple surgery obstructions: [Lück2001, 4.7]; Bernd Ammann, Lecture notes
13. The structure set and Wall realisation: [Lück2001, 5.1]; Münster, Lecture notes: part (a) Lecture notes: part (b)
14. The smooth surgery exact sequence: [Lück2001, 5.2, 5.3 & 6.1]; Münster, Lecture notes
15. Exotic spheres: [Lück2001, 6.1-6.5 & 6.7]; Sebastian Goette, Lecture notes
16. The surgery exact sequence for TOP and PL: [Lück2001, 5.4 & 6.6]; Bonn, Lecture notes
17. Manifolds homotopy equivalent to CPn: [Wall1999, 14C], [Madsen&Milgram1979, 8C]; Poznan, Lecture notes
18. Fake Tori: [Wall1999, 15A]; Philipp Kühl and Anna Abczynski, Lecture notes

### 3.1 Division of talks

Note that the program naturally splits into various groups of talks. The speakers from these groups should co-operate with one another both in learning the material and in decided how to organise its presentation.

• [A] Talks 1-3: The aims of these talks is to out-line the proof of the s-cobordism theorem assuming the audience is familiar with the proof of the h-cobordism theorem.
• [B] Talks 4-5: Here the key homotopy theoretic aspects of manifolds are identified: Poincar\'e duality and its surprising consequence: the existence of the Spivak normal fibration.
• [C] Talks 6-8: Working below the middle dimension we see how surgery and bordism are intimately related and how to make normal maps highly connected.
• [D] Talks 9-11: These talks present the heart of surgery: the surgery obstruction map arising out of the challenge of performing desired surgeries on middle-dimensional homotopy classes. Subtle topology and complex algebra and elegantly gathered into a powerful synthesis here.
• [E] Talks 12-14: Here we step back a little and see how to assemble all of the previous hard work into the succinct and powerful surgery exact sequence.
• [F] Talks 15-18: Now we give applications and show how to work with the surgery exact sequence. Starting with exotic spheres we see the spaces fundamental spaces $O$$O$, $PL$$PL$, $TOP$$TOP$ and $G$$G$ and their quotients, compute their homotopy groups and sometimes even their homotopy type. As a result we can calculate manifolds homotopy equivalent to $S^n$$S^n$, $\CP^n$$\CP^n$ and $T^n$$T^n$ (in appropriate categories).