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urs schreiber1
science forum beginner

Joined: 28 Feb 2006
Posts: 3

Posted: Fri Apr 21, 2006 3:45 pm    Post subject: algebraic cobordisms

For any integer n, we define the category nCob to be that whose objects
are diffeomorphism classes of (n-1) dimensional manifolds, and whose
morphisms are n-dimensional manifolds cobording their source and target
objects. Composition is gluing of manifolds at their boundaries.

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.

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 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.

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
science forum Guru

Joined: 28 Apr 2005
Posts: 566

Posted: Fri Apr 21, 2006 8:30 pm    Post subject: Re: algebraic cobordisms

 Quote: 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.

 Quote: 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?

 Quote: 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

 Quote: 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).

 Quote: 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.

 Quote: 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
Tom Leinster
science forum beginner

Joined: 06 Feb 2006
Posts: 6

Posted: Wed Apr 26, 2006 4:32 pm    Post subject: Re: algebraic cobordisms

On Fri, 21 Apr 2006 16:30:02 -0400, Lee Rudolph wrote:

 Quote: "Urs Schreiber" 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? [...] 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.

A good point, but it's not as easily handled as you suggest, I think. The
reason is that we're interested in *two* notions of equivalence of
manifolds: cobordism and diffeomorphism. Any diffeomorphism gives rise to
a cobordism, but not vice versa. So if we do what you suggest then yes,
we no longer commit the sin of identifying "equivalent" objects, but we
lose some information: as far as I know, nothing in your category tells us
which pairs of manifolds are diffeomorphic and which are merely cobordant
(= isomorphic in this category).

One resolution is to use a categorical structure in which there are
objects and *two* types of morphism (and some extra stuff relating them).
This sounds a bit ad hoc, put like this, but in fact this notion arises
very naturally in the development of higher category theory. In the case
at hand, the objects are manifolds, the morphism of one type are smooth
maps, and the morphisms of the other type are cobordisms. See p.138-9 of
my book,

http://arxiv.org/abs/math.CT/0305049

for explanation and a picture.

Best wishes,
Tom
John Baez
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 220

Posted: Tue Jun 27, 2006 5:14 am    Post subject: Re: algebraic cobordisms

In article <e2bfca\$a2u\$1@dizzy.math.ohio-state.edu>,
Lee Rudolph <lrudolph@panix.com> wrote:

 Quote: "Urs Schreiber" 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?

Yes - Urs is not giving the usual definition of nCob here.
The usual definition has (n-1)-dimensional manifolds as objects.

(Usually these manifolds are smooth, compact, and oriented, but
let's take that as given.)

 Quote: 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.

Indeed. Isomorphisms are not bad things. No need to make ones
categories skeletal.

 Quote: 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.

Usually the morphisms

M: S -> S'

in nCob are taken to be certain *equivalence classes* of n-manifolds
with boundary equipped with diffeomorphisms

f: S U S' -> boundary(M)

The equivalence relation is that

f: S U S' -> boundary(M)

and

f': S U S' -> boundary(M')

are equivalent if there is a diffeomorphism

g: M -> M'

making a certain obvious triangle involving f, f' and g commute.

We need this equivalence relation to make composition well-defined!
The reason is that to compose cobordisms, we need to choose a
"collar" for each cobordism and glue by identifying these collars.
This solves the problem Lee is talking about. But, it involves
an arbitrary choice of collars, so we get a well-defined cobordism
only thanks to the equivalence relation, which makes the choice
immaterial.

All this is standard stuff....

(Using equivalence classes of things to be *morphisms* is not
against the spirit of category theory. It's against the
spirit of 2-category theory, but we're not going that far today.)

 Quote: 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?

Aha - see, a collar? For some reason Lee has germs on his collar,
but it's not really necessary to bring germs into it.

 Quote: 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.

It's a pretty straightforward dualized version of the usual thing.
Categorically speaking, a cobordism is a special sort of cospan:

M
/ \
S S'

where the lines denote arrows pointing *up*. When we switch to
using the algebras of smooth functions, we get a span instead,
with arrows pointing *down*.

(A "span" is so called because it's shaped like a bridge.)

A nice example of using spans in an algebraic analogue of cobordism
theory can be found in Derek Wise's work on "chain field theory":

http://xxx.lanl.gov/abs/gr-qc/0510033

A chain field theory is like a topological quantum field theory,
but where nCob is modified by replacing manifolds by chain complexes
This requires that we dualize and use spans of chain complexes
instead of cospans of manifolds. See the stuff starting at page 46.

But, I don't think Derek needs to address the issue of "collaring"
in his work.

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

I'm not sure about that, since all 0-manifolds are empty!
But I think 1Cob will help.
urs schreiber1
science forum beginner

Joined: 28 Feb 2006
Posts: 3

Posted: Wed Jun 28, 2006 9:28 am    Post subject: Re: algebraic cobordisms

John Baez wrote:

 Quote: Categorically speaking, a cobordism is a special sort of cospan: M / \ S S'

and suggested that what I am looking for are spans in algebras.

That sounds good!

In fact, in private conversation with Jens Fjelstad we were talking
about more or less this idea, mainly worrying about the question which
conditions to impose on these algebras in order to assure that the
boundary algebras (at the legs of the span) really correspond to the
boundary of the cobordism algebra (that at the tip of the span).

I am expecting that people must have considerd similar constructions
before, in algebraic geometry or something. Indeed, the term
"algebraic cobordisms" exists and has been studied:

http://www.math.uni-hamburg.de/home/schreiber/AlgebraicCobord.pdf

http://golem.ph.utexas.edu/string/archives/000792.html#c003789.

Given that a Riemann surface (the kind of cobordism we are interested
in for 2D CFT) is just a special sort of scheme, it seems to be the
most straightforward thing on the world to consider cobordisms which
are more general schemes. It seems.

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