Oberwolfach Surgery Seminar 2012: Discussion

This page is for discussion following the seminar

• General information
• Exercises
• Glossary
• Surgery on the Manifold Atlas

Tibor's comments on 07/06/2012

Recommended literature

For a quick and very readable basic introduction to algebraic surgery I would very much recommend Andrew's article:

• Foundations of algebraic surgery, Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002), 491–514. [Ranicki2002a]

Here the basic idea of algebraic surgery is explained and the relation to cobordisms.

For a quick introduction to the algebraic surgery exact sequence which involves the structure group $\mathbb{S}_n (X)$$This page is for discussion following the seminar * [[Oberwolfach Surgery Seminar 2012: General information|General information]] * [[Oberwolfach Surgery Seminar 2012: Exercises|Exercises]] * [[Oberwolfach Surgery Seminar 2012: Glossary|Glossary]] * [[:Category:Surgery|Surgery on the Manifold Atlas]] <wikitex>; Tibor's comments on 07/06/2012 '''Recommended literature''' For a quick and very readable basic introduction to algebraic surgery I would very much recommend Andrew's article: * Foundations of algebraic surgery, Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002), 491–514. {{cite|Ranicki2002a}} Here the basic idea of algebraic surgery is explained and the relation to cobordisms. For a quick introduction to the algebraic surgery exact sequence which involves the structure group $\mathbb{S}_n (X)$ I would recommend Andrew's article * The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction, School on High-dimensional Manifold Topology, 519--538, ICTP Trieste (2002). {{cite|Ranicki2002a}} A general note about the notation. There are two notations for the groups that measure the difference between the quadratic and symmetric $L$-groups. One is $\hat{L}^{n}$ and the other $NL^n$. These groups have different definitions, hence the notation, but they are isomorphic, see Proposition 2.11 in {{cite|Ranicki1992}}. Andrew recommended the scanned lecture notes of Adam Mole from Muenster 2008, which I can only agree with. Here you often find more motivation than in published articles. Of those published articles, I would recommend to begin with the first two for the basics of algebraic surgery (but now this is on a higher level than those above): * The algebraic theory of surgery I., Proc. London Math. Soc. (3) 40, 87--192 (1980) {{cite|Ranicki1980}} * The algebraic theory of surgery II., Proc. London Math. Soc. (3) 40, 193--283 (1980) {{cite|Ranicki1980a}} For the algebraic surgery exact sequence and the total surgery obstruction, of course there is the blue book * Algebraic L-theory and Topological Manifolds, Cambridge Tracts in Mathematics 102, CUP (1992) {{cite|Ranicki1992}} which can be accompanied by * The total surgery obstruction revisited by P.Kuehl, T.Macko and A.Mole arXiv:1104.50 (2011) {{cite|Kuehl&Macko&Mole2011}} which is meant partly as a readable summary (mostly without proofs) of the setup of the algebraic surgery exact sequence and partly as a place where some more details in the technical parts of the proof that TSO is TSO are given. Of course there is the original paper of Andrew: * The total surgery obstruction, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 275–316 {{cite|Ranicki1979}} with most of the great ideas already present (although in a slightly different form sometimes, it's 35 years old, after all). Finally, from a more modern point of view (using the language of $\infty$-categories), there are lecture notes of Jacob Lurie: * Algebraic L-theory and surgery, Jacob Lurie's 2011 Harvard course. Links to all the material above can be found on Andrew's webpage. Some of it is under the links "Surgery bits and pieces" -> "Books, papers, theses, notes, links, etc" (in particular Adam Mole's notes and Jacob Lurie's notes). </wikitex> [[Category:Oberwolfach Surgery Seminar 2012]]\mathbb{S}_n (X)$ I would recommend Andrew's article

• The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction, School on High-dimensional Manifold Topology, 519--538, ICTP Trieste (2002). [Ranicki2002a]

A general note about the notation. There are two notations for the groups that measure the difference between the quadratic and symmetric $L$$L$-groups. One is $\hat{L}^{n}$$\hat{L}^{n}$ and the other $NL^n$$NL^n$. These groups have different definitions, hence the notation, but they are isomorphic, see Proposition 2.11 in [Ranicki1992].

Andrew recommended the scanned lecture notes of Adam Mole from Muenster 2008, which I can only agree with. Here you often find more motivation than in published articles.

Of those published articles, I would recommend to begin with the first two for the basics of algebraic surgery (but now this is on a higher level than those above):

• The algebraic theory of surgery I., Proc. London Math. Soc. (3) 40, 87--192 (1980) [Ranicki1980]
• The algebraic theory of surgery II., Proc. London Math. Soc. (3) 40, 193--283 (1980) [Ranicki1980a]

For the algebraic surgery exact sequence and the total surgery obstruction, of course there is the blue book

• Algebraic L-theory and Topological Manifolds, Cambridge Tracts in Mathematics 102, CUP (1992) [Ranicki1992]

which can be accompanied by

• The total surgery obstruction revisited by P.Kuehl, T.Macko and A.Mole arXiv:1104.50 (2011) [Kuehl&Macko&Mole2011]

which is meant partly as a readable summary (mostly without proofs) of the setup of the algebraic surgery exact sequence and partly as a place where some more details in the technical parts of the proof that TSO is TSO are given.

Of course there is the original paper of Andrew:

• The total surgery obstruction, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 275–316 [Ranicki1979]

with most of the great ideas already present (although in a slightly different form sometimes, it's 35 years old, after all).

Finally, from a more modern point of view (using the language of $\infty$$\infty$-categories), there are lecture notes of Jacob Lurie:

• Algebraic L-theory and surgery, Jacob Lurie's 2011 Harvard course.

Links to all the material above can be found on Andrew's webpage. Some of it is under the links "Surgery bits and pieces" -> "Books, papers, theses, notes, links, etc" (in particular Adam Mole's notes and Jacob Lurie's notes).