Surgery Obstruction Groups

Euclid

I need a crash course in surgery obstruction groups. Where should I go to find information on this topic, or does anyone know something about it?
Thanks.

mathwonk

I'm not sure what you mean. To me there is no obstruction to performing a surgery.

there are groups that measure obstructions to extending maps, or extending homotopies.

lets take a very simple case: suppose you have a continuous map from the circle to the punctured plane, and you want to extend it to a continuous map from the disk to the punctured plane.

to do this you need the induced map from the fundamental group of the circle to the fundamental group of the punctured plane to be zero. if this is not true, then you can perform a surgery on the punctured plane, by sewing in a copy of the disk, along the image of its boundary circle by the first map.

then the new space obtained by attaching a cell to the punctured plane will admit an extension of the original map, tautologically.

there were some nice notes by griffiths, morgan, and sullivan on this topic, keywords "postnikov towers",...

try a book on homotopy theory. or rea`d about attaching cells in any algebraic topology book, or book on cellular homology. i think alan hatcher's book is free on the web.

mathwonk

here is one of the simplest surgeries: it occurs when a complex curve, i.e. locally a cylinder, acquires a singualr point and then becomes desingularized, i.e. thew cylinder degenerates to a cone, and then the cone separates into two smooth nappes of a hyperboloid. this happens when a hyperboloid of one sheet passes trough a cone and becomes a hyperboloid of two sheets.

the surgery point of view is the following: look for two manifolds, or bordewred surfaces which have the same border. then if you find a surface with this same border, you can sew in either one of your two surfaces and both will fit just fine. passing from the figure obtained by sewing in one of them, to that obtained by sewqing in the oither one is called a surgery.

e.g. the product D^1 x S^1 of an interval and a circle, has boundary equal to two copies of the circle {0} x S^1 + {1} x S^1, i.e. the product of the two point boundary {0,1} of D^1, with the circle S^1.


but also S^0 x D^2, the product of the two point set {0,1} with the disc, has the same boundary, namely S^0 x S^1 = {0,1} x S^1.


now consider a cone, and cut out that part of the cone inside a sphere centered at the vertex, and thorw it away. you leave a surface having as boundary, two circles.

thus you can sew in either of the two surfaces above, either sew in a cylinder

D^1 x S^1, giving a hyperboloid of one sheet, or sew in a disjoint union of two discs,

i.e. S^0 x D^2, giving a hyperboloid of two sheets.


this is a simple surgery. thus the process of a complex curve, or hyperboloid of one sheet, acquiring a singular point (vertex) then being desingularized by separating that vertex into two nappes, coulod be achieved by a simple surgery.



obstruction theory is another matter, and a good source for it might be, if you have time, steenrod's great book, topology of fiber bundles.

i have not yet searched the web.

mathwonk

are you trying to read some of those papers on surgery theory and quantum theory? what are they doing, studying the geometry and topology of space time?


you might take a look at the bibliography of the paper you are trying to read, or maybe look at the websiet of msri, the math science research institute, where ther are some notes by thurston available for downloading on a slightly different topic, 3 dimensional hyperbolic geometry.


the word "cobordism" also came up in some fo the physics papers i ran across just now, and that subject was pioneered alrgely by rene thom in the 50 and 60's and is the topic of some famous notes by milnor from princeton, on differential topology.

mruncleramos

You should check out Survey's on Surgery Theory (part of the annals of mathematics studies series of princeton university press) They have a nice little article there on calculating surgery obstruction groups and etc.

Also recommended is C. T. C. Wall's Classic Surgery on Compact Manifolds published by the AMS

mathwonk

you mean there is sucha thing as surgery obstruction groups?

if you mean the article of hambleton and taylor, are you implying you actually read that article?

that is about the least readable (by me) article I have seen in a long time.

mruncleramos

yes, i myself am not intimately familiar with the subject, but I'm sure it is included in most advanced surgery books.

mruncleramos

Surgery on Compact Manifolds: "Part 1 consists of the statement and proof onf our main result, namely that the possibility of successfully doing surgery depends on an obstruction in a certain abelian group, and that these 'surgery obstruction groups' depend only on the fundamental groups involved and on dimension modulo 4" Available online at http://www.maths.ed.ac.uk/~aar/books/scm.pdf#search='Surgery%20on%20Compact%20Manifolds '
I have not read the article that I mentioned earlier, but it seemed convenient. I am, however, fairly aquainted with this book.

Euclid

Thanks, everyone, for your replies. This will help, but I think I'm a bit in over my head. What would be the first thing to read, before reading about surgery theory?

mruncleramos

First of all, it would be helpful to know what you have learned so far. There are quite a few prerequisites before surgery. You should have had a good graduate level classes in Algebra, Algebraic Topology, and some Differential Topology at very least. If you have enough Alg. Top. under your belt, you might try Algebraic and Geometric Surgery by Ranicki.