Stuff white people like / I need a project
February 20, 2008 at 4:49 am | In Internet, Life in America, Mathematics |6 Comments »
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I can think of a few things in algebraic topology; it really depends what you are after. Do you want to write about an application, a theorem, or techniques/gadgets?
Steenrod operations and the extension problem is a nice topic, although the higher cohomology operations can be a bit yucky to define.
Another possibility is model categories. They provide an approach to formally inverting a class of maps that you want to be isomorphisms (e.g. homotopy equivalences or quasi-isomorphisms of chain complexes) while maintaining some control over your category.
Cobordism theory is another topic, which has the virtue of combining differential geometry and algebraic topology. I don’t know so much about it though except that there is a nice theorem of Quillen’s that complex cobordism is universal for oriented cohomology theories for smooth manifolds (I think that is the correct statement, I could be wrong).
One final suggestion would be to look at some sheaf cohomology. Perhaps looking at the equivalence of sheaf cohomology with coefficients in a constant sheaf with singular cohomology. You need some restrictions on the space (paracompact and Hausdorff?) but it is nice stuff and you can also then go on to consider k-local systems and their equivalence to functors from the fundamental groupoid to the category of k-modules for a 1-connected, path connected, paracompact space (I might have missed a condition).
If you are interested in any of those things I can recommend some references and/or send you some notes I have.
Whenever I try and make a post on your blog it ends up turning into a rant…
Comment by Greg Stevenson — February 20, 2008 #
Thanks for this Greg. Don’t worry about the rant - this post was just a thinly veiled attempt to get you started…
Wikipedia still has some great lines in it: “It has also been suggested[3] that cobordism is “a Danish modern art movement based on gluing cardboard boxes to your face.”"
I’ll have a think about this stuff and get back to you.
Comment by martinleslie — February 20, 2008 #
I’ve done a bit of searching on the internet and (co)bordism sounds interesting - it seems like a nice idea to tie the two halves of the course together (along with De Rham cohomology which hopefully we’ll do in class).
The only problem is the problem common to all these types of project: how to cover the motivation and development of some nontrivial theory in not much more than 5 pages.
These lecture notes look reasonable, and I think go up to the result you mentioned. This presentation and this article give a bit of motivation.
So, barring me not liking it when I start reading it seriously, this sounds pretty good. I know you said you didn’t know much about it but do you know any other references?
Comment by martinleslie — February 20, 2008 #
I have some brief notes from a talk I went to the other day; I might try and get them scanned or if I can be bothered type them up. We’re running a seminar on algebraic cobordism this semester, based on the book by Morel and Levine (it’s available from Levine’s website), which is basically on setting up a universal oriented cohomology theory for smooth quasi-projective schemes. If you end up being interested I can keep you posted on the talks.
Other than that I don’t know of anything in particular; if you were feeling brave you could always look at Quillen’s original papers I guess.
If you are just required to do a somewhat technical survey you should be ok. The actual definition of cobordism is pretty elementary. It is possible to set up all the fancy conditions on the normal bundle at once using the classifying spaces of the classical groups (e.g. homotopy classes of maps from a manifold M to BSO(k) over BO(k) classify isomorphism classes of oriented O(k)-principal vector bundles). Then the most important stuff to get in I guess is the Thom isomorphisms which relate oriented cobordism to the stable homotopy groups of spheres (and possibly their generalizations to complex cobordism etc…
and that universality theorem of Quillen.
Have fun. I’d be interested to have a look at it when you’re done if you don’t mind. I could give it a proof read if you wanted.
Comment by Greg Stevenson — February 20, 2008 #
Just wanted to correct a dumb mistake in the above post. The correct cobordism ring to have graded pieces isomorphic to the stable homotopy groups of spheres is framed cobordism, i.e. everything is required to preserved a given framing of each manifold. I think in the literature this isomorphism is called the Thom-Pontrjagin map.
Comment by Greg Stevenson — February 20, 2008 #
No worries. Don’t bother sending me those notes unless you really want to.
I’ll send you a copy when I’m done. It’s not due til the end of semester so we’ll see when I get around to it…
Comment by martinleslie — February 21, 2008 #