Oberwolfach Surgery Seminar 2012: Discussion

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)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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). [Ranicki2002b]

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 lecture notes of Adam Mole from Andrew's 2008 Muenster course, 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) [Kühl&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:

Links to all the material above can be found on Andrew's website. 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 lecture notes may be found here. The easiest way to find items on Andrew's website is to just Google for their titles!