Now I’ll begin to post material from the book proper. Here is the first section of Chapter I. I’m going to try to post stuff in bite-size pieces like this, rather than a whole chapter at a time. Of course, there will probably be an ongoing process of revision as well.
In this installment, we introduce the main objects that surgery theory is about – manifolds, especially (the reason for this will become apparent later) manifolds of high dimension, which typically means dimension at least five. Even as regards the definitions, though, there are important distinctions to be made. The most natural notion of a manifold is simply “a (nice) topological space locally homeomorphic to Euclidean space” – that is, a topological manifold. Historically, however, manifolds arose in connection with problems in differential geometry and analysis and thus possessed not just a topological but a smooth structure (i.e., it makes sense to talk about infinitely differentiable functions on such a manifold, not just about continuous ones.) One might naturally suppose that a topological manifold can always be “smoothed out”, just as a continuous function can always be approximated by a differentiable one. However, this is not the case. In fact, our story really gets its start in the middle fifties, when Milnor discovers that \( S^7 \) admits several distinct differentiable structures.
I am very interested to hear how accessible readers find this chapter. Could you follow the presentation/do the exercises? I should mention that I learned some of the ideas in Chapter 1 from some lecture notes of Tom Farrell, “Introduction to High-Dimensional Manifold Topology”, 2001.
Wolfgang Luck, Diarmuid Crowley and Tibor Macko are organizing a summer school in Bonn on The Topology of High-Dimensional Manifolds. It takes place August 26-30.
Below I reproduce some of the prerequisite and reference material from the event’s web site. Whether or not you choose to attend, this is a useful list of references!
Participants should be familiar with the concepts and ideas covered in the first 7 chapters of the book [Ranicki2002] and the first six chapters of the book [Husemoller1994]. In addition participants should be familiar with the basics of spectra in stable homotopy theory. A good reference here is [Hatcher2002, Section 4.F].
The main references for the material covered in the seminar are [Lück2001], [Crowley2010], [Kühl&Macko&Mole2011], [Lück&Reich2005], [Bartels&Lück&Reich2008], [Bartels&Lück2012].
There is also a reading guide for surgery theory for topological manifolds available at the discussion page of the TSO-at-MFO 2012
- [Bartels&Lück&Reich2008] A. Bartels, W. Lück and H. Reich, On the Farrell-Jones Conjecture and its applications, Journal of Topology 1 (2008), 57–86. MR2365652 (2008m:19001) Zbl 1141.19002
- [Bartels&Lück2012] A. Bartels and W. Lück, The Borel conjecture for hyperbolic and CAT(0)-groups., Ann. of Math. (2) 175 (2012), 631–689. Zbl 06025000
- [Crowley2010] D. Crowley, The smooth structure set of Sp×Sq, Geom. Dedicata 148 (2010), 15–33. MR2721618 (2012a:57041)
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Husemoller1994] D. Husemoller, Fibre bundles, 3rd ed., Springer-Verlag, 1994. MR1249482 (94k:55001) Zbl 0794.55001
- [Kühl&Macko&Mole2011] P. Kuehl, T. Macko and A. Mole, The total surgery obstruction revisited, (2011). Available at the arXiv:1104.5092.
- [Lück&Reich2005] W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-theory. Vol. 1, 2, 703–842, Springer, Berlin, 2005. MR2181833 (2006k:19012) Zbl 1120.19001
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author’s homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001
This is the start screen from the Flash Game Alan Probe: Amateur Surgeon. (I won’t include a link, but Google should find it for you.)
The surgery process is a somewhat drastic one, which can significantly change the homotopy type of a manifold, for example. So how can it be useful for classifying manifolds within a homotopy type? The surgeon has to be kept under control somehow – and this is accomplished by appropriate algebra.
To get started, I am posting the preface to Surgery for Amateurs as I drafted it last time I wrote, in about 2005.
Here is the link.
Comments are welcome!
Posted in Book, math
Tagged plan, preface