Re: algebraic cobordisms

"Urs Schreiber" <urs.schreiber@xxxxxxxxxxxxxx> writes:

For any integer n, we define the category nCob to be that whose objects
are diffeomorphism classes of (n-1) dimensional manifolds,

Isn't taking the objects to be equivalence classes rather against
the spirit of category theory? (So, anyway, said Barry Mazur, I
think, long ago, when recanting of his first approach to something
or other with handlebodies that involved doing just that.) Take
the objects to be (n-1)-manifolds and accept that there will be
lots and lots of isomorphisms in your category; you'll be happier,
really.

and whose
morphisms are n-dimensional manifolds cobording their source and target
objects. Composition is gluing of manifolds at their boundaries.

If "gluing" is taken in the most naive way, then it seems to me that
(whether the objects are equivalence classes or manifolds) this operation
won't respect the algebras of smooth functions. I mean, let's take
1Cob, where there are *no* choices to be made of the diffeomorphisms
used for gluing. Unless you take, instead of "algebras of smooth
functions", something a bit (or much) richer, involving (maybe)
germs in a collar of the boundary, gluing will yield merely piecewise-
smooth functions, right?


I would like to understand the construction of this category in terms
not of the manifolds themselves, but in terms of their algebras of
smooth functions.

However, supposing that you can fix those complaints up somehow,
then as you say

There should be a nice way to read off the algebra of functions
supported on the boundary from the algebra of functions on the entire
manifold with boundary;

and that way should be "the algebra of functions supported on the
boundary is the algebra of functions on the entire manifold with
boundary modulo its subalgebra of functions that vanish on the
boundary" (presumably, when the fixes are made, "vanish" will
mean "vanish to infinite order" and then some, or the like).

and to encode the gluing of manifolds in terms
of a certain gluing operation on their algebras of functions.

Of course the motivation behind all this is that I would like to
understand if there is something like a noncommutative version of nCob.

I really, really do think that a deep meditation on 0Cob will pay
off.

It feels like these questions should have basic, well-known answers,
but I am probably not familiar enough with the relevant literature in
order to know.

I'd be grateful for any comments and pointers to relevant literature.

P.S.
More on the full motivaiton behind this question can be found here:
http://golem.ph.utexas.edu/string/archives/000792.html

Lee Rudolph
.

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