1-cobordisms, Part 1

Loosely speaking what we’ll be looking at are orientated 0-manifolds and oriented 1-manifolds with boundary (which can be empty).  In terms of category theory the 0-manifolds are the objects and the 1-manifolds are the arrows.  This is why we need oriented manifolds to give sense to how a 1-manifold can be an arrow from one 0-manifold to another. We also want orientation for the physical application of this category which is a topological model of spacetime.  The orientation of the 1-manifolds will play, loosely, the role of temporal evolution.  The 1-manifolds will also be our cobordisms.

My idea is to start with 1Cob (the category of 1-cobordisms, which is what the first paragraph describes) and then move to 2Cob (where the cobordisms will be 2-manifolds with boundary).  This will give us a lot of opportunity to sketch out examples and to illustrate proofs.  It will still be possible to sketch some ideas in 3Cob but it will be much harder.

We’ll start with some loose definitions.  The idea is not to rehash all of differential topology and geometry.  For more details there are many great books.  One of my favorites is an older gem, Differentiable Manifolds by Brickell and Clark.  As we go along we’ll focus on the examples specific to this post.

A manifold is a topological space that ‘locally looks like \mathbb{R}^n.’  If a manifold is an n-dimensional blob that we want to work on we draw little coordinate patches, little grids that will allow us to do analysis on this blob.  To get these coordinate grids we take a collection of functions x:M\to\mathbb{R}^n from M to open sets in euclidean space.  The preimages of these open sets generates a topology on M. Think of it as uploading nice properties of euclidean space to small patches of M. I use the plural on purpose, we will usually need more than one coordinate chart to cover a manifold.  To any point p\in M the functions, which we call charts, assign coordinates.  The point p is likely to be covered by several preimages of different charts and we require that the change of coordinates, x\circ y^{-1} and y\circ x^{-1},  (called transistion functions sometimes) be diffeomorphisms. As these map open sets in euclidean space to other open sets in euclidean space the notion of diffeomorphism makes sense without having talked about differentiation on M.

If a manifold M has boundary then we need a little tweak.  The tweak is to use the half-space H^n=\{x\in\mathbb{R}^n | \pi_i(x)\geq 0\} (where \pi_i(x) is the usual projection of x onto it’s i-th component, \pi_i :\mathbb{R}^n\to\mathbb{R}).  This is what you’d expect.  A point is in the boundary of M if there is a chart mapping it to the boundary of H^n. The boundary of H^n is \mathbb{R}^{n-1} which has no boundary. Let’s look at an example in the case of 1-cobordisms.

Take the unit interval [0,1].  We’ll need two charts to cover this let’s define a couple.  The first will map from [0,3/4] to H^1=[0,\infty)

(1) f: [0,3/4]\to H^1 by f(x)=x We’ll call the preimage of f, U.

(2) g: [1/4, 1]\to H^1 by g(y)=1-y.  We’ll call the preimage of g, V.

I’ve used x as a coordinate for U and y as a coordinate for V just for book keeping.  Usually you wouldn’t also have function names like f and g but I hope to make this a bit clearer for anyone new to manifolds etc.

There’s only one overlap, U\cap V.  Let’s check on the change of coordinates functions.  There are two.  If we change from x\to y we’ll have g\circ f^{-1} which boils down to y=1-x.  Conversely if we change from y\to x we’ll have f\circ g^{-1} or x=1-y.  These are clearly diffeomorphisms.

At each point on a  manifold there is a tangent space.  You can look at this a number of ways and maybe I’ll do a post soon on the formal construction of a tangent space.  I’d like to review the process.  We’ll just cut to the punchline which is that the coordinates induce a basis on the tangent space.  These basis vectors are derivatives in the coordinates.  So the induced basis for point p\in U would be d/dx and the induced basis for a point q\in V would be d/dy.

What about a point p\in U\cap V?  How do we relate the basis of T_p U to T_p V?  The chain rule gives us a clue:




This should also become justified in the post on tangent spaces.  What I’m wondering now is how I can use the transistion maps for this.  I know that maps between manifolds (in particular maps from M to M) induce linear transformations between tangent spaces.  So my first thought was that the transition map would induce a linear transformation T_p U\to T_p V and vice versa.  But these transition maps are from \mathbb{R}^n\to\mathbb{R}^n not from M to M.  I’ll have to think about that one, I’m sure it’s pretty simple. I just don’t want to keep holding up this post.  I’ve rewritten this several times already as I refined the details.

This gives us the basics of manifolds, manifolds with boundary and tangent spaces (minus the change of basis induced by change of coordinates).  I want to introduce the idea of an orientation of a manifold.  To do this we start with a vector space.  Consider a vector space V with a basis {v_i}  Choose some order for these basis elements {v_1, v_2, ...v_n} and call this positive.  Any other basis for V is considered positive if there is a linear transformation taking the first to the second with a positive determinant.  So we can rotate and stretch but not reflect.

We assign an orientation to a manifold by orienting all the tangent spaces in a smooth fashion.  What does that mean?  Well following Kock, “…the differentials of the transition functions  should all preserve the orientations.”

There are some questions I have so far about orienting 0-manifolds and 1-manifolds (I’m going to leave the empty manifold alone for the moment).  Let’s take a 0-manifold which is a collection of points.  If we consider only a connected 0-manifold (i.e. a point) we have one tangent space {0} which has an empty basis.  Since the basis is {} we can assign a + or a – to this ‘order.’  For a 0-manifold with n points we can choose, for each component, an orientation.  Such a 0-manifold will have 2^n orientations.

The situation for the unit interval seems more complicated.  There are two obvious orientations for the unit interval.  There’s one induced from \mathbb{R} which points from 0 to 1.  There’s the reverse as well.  It looks like, from Kock, that there are two other ways as well.  This needs to be understood.  The idea of a cobordism between two manifolds involves embedding them, preserving orientation, into the cobordism and being able to determine which boundary is “in” and which is “out.”

So for a pair of 0-manifolds N and M we’ll need a pair of maps \phi_N and \phi_M that embed N and M into a 1-manifold W.  As a diagram it looks like:

M\overset{\phi_M}{\rightarrow} W\overset{\phi_N}{\leftarrow} N

Which looks like a coproduct, something I’ve recently come across in McLarty’s book on categories.  There’s more to it than that, but something to look forward to. In the next post I’ll focus on orienting the 0-manifolds and 1-manifolds and constructing embeddings.  Then we’ll consider some examples both graphically and computationally.


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