1-Cobordisms, Part 2

To understand the material in Kock about orienting 0 and 1 manifolds and their in and out boundaries I needed to take a detour to a dimension higher.  In this post we’ll look at 2Cob whose objects are disjoint unions of circles.  The arrows are cobordisms between these unions.  To have an arrow we need a direction and so we will have to understand in what way a 2-cobordism points from a domain to a codomain. An oriented cobordism isn’t quite enough we’ll have to define in-boundaries and out-boundaries.

The concept of an in-boundary and out-boundary are a little vague too. Here’s the idea. I’ll quote Kock, pg 14

1.1.11 In-boundaries and out-boundaries. Let $\Sigma$ be a closed submanifold of M of codimension 1.  Assume both are oriented.  At a point $x\in\Sigma$, let $[v_1,...,v_{n-1}]$ be a positive basis for $T_x\Sigma$.  A vector $w\in T_x M$ is called a positive normal if $[v_1,..., v_{n-1},w] is a positive basis for$latex T_x M$. Now suppose $\Sigma$ is a connected component of the boundary of M; then it makes sense to ask whether the positive normal $w$ points inwards or outwards compared to M – locally the situation is that of a vector in $\mathbb{R}^n$ for which we ask whether it points in or out from the half space $H^n$. If a positive normal points inwards we call $\Sigma$ an in-boundary, and if it points outwards we call it an out-boundary. To see that this makes sense we have to check that this does not depend on the choice of positive normal (or on the choice of point $x\in\Sigma$). If some positive normal points inwards, it is a fact that every other $y\in\Sigma$ points inwards as well. This follows from the fact that the normal bundle $TM|_{\Sigma}/T\Sigma$ is a trivial vector bundle on$latex\Sigma$. Thus the boundary of a manifold M is the union of various in-boundaries and out-boundaries The in-boundary of M may be empty, and the out-boundary may also be empty. Note that if we reverse the orientation of both M and its boundary$latex\Sigma\$, then the notion of what is in-boundary or out-boundary is still the same.

There are a couple ideas floating around and it will be helpful to distinguish them.  We’re starting with an oriented manifold.  In the coming examples this will be an oriented cylinder.  We’re then looking at points on the boundary and the basis of their tangent spaces.  Given that a particular basis is positive, what would we add to it to create another positively oriented basis but of a tangent space to the manifold as opposed to the boundary?  What we call positively and negatively oriented is arbitrary as we shall see.  Don’t worry though, it doesn’t affect the mathematics.  What will happen is that half of the circles in this category will be oriented one way and the other half the other way and which is positive or negative is just a matter of labeling.

Let’s start with an oriented cylinder which will play the part of a 2-cobordism (and an arrow in 2Cob).  I’ve drawn a basis at a point on the cylinder and have dubbed this positively oriented.  We’ll use this orientation for a while.  Now consider a point on the left boundary.  There’s a circle to the left of the cylinder to clarify.  Let $\vec{u}$ be a positively oriented basis for the boundary (the circle).  So we’re considering the circle as a 1D submanifold of a 2-manifold.  But all the points of that submanifold are also points in a 2-manifold so they all have 2D tangent spaces.  What vector could we add to that 1D basis to create a positively oriented 2D basis?  Let’s add $\vec{v}$.  That works and so $\vec{v}$ is an in-boundary.  If we make the same arguments at the right end we’ll have $\vec{v}$ as an out-boundary.

It would be reasonable to protest here.  I will eventually put up a post where I work out a calculation explicitly (well at least more so than here) on why these are in and out boundaries.  But that would require charts and looking at half-planes.  The in and out boundaries are important if we’re to treat these 2-manifolds as arrows in 2Cob.  An arrow points from a domain to a codomain.  The oriented cylinder above has an in and an out! This particular situation is an arrow from the positively oriented circle to itself.  We’ve constructed the identity cobordism.

Let’s try another,

We start with the same orientation for the cylinder and the same point on the boundaries and the same tangent basis.  But this time we say this is a negative basis.  We need a positive basis for a tangent space at a point on the boundary as a submanifold and then need to find another element to extend it to a positively oriented basis for the point on the boundary as an element of the manifold.  If $\vec{u}$ is a negatively oriented basis then $\vec{-u}$ is positively oriented (only two choices).  Then we can extend this by adding $\vec{-v}$ which is a positively oriented basis for a tangent space on the cylinder.  The same holds for the other end, the right.  Here we have a circle with a negatively oriented basis which is a roundabout way of saying it’s positively oriented basis is the other way.  Again $\vec{v}$ is the positive normal, creating an in-boundary.  This cylinder ends up being an identity arrow on the other orientation of the circle.

It may be tempting to say we’ve done nothing new here.  That the above is just the same arrow as before, just reflected.  However we’ll see now that there isn’t a cobordism between these two oppositely oriented circles (or if there is it isn’t what might think).

The picture above has two oppositely oriented circles in the sense that what is considered ‘positively oriented’ is opposite.  As a result, at the left end $\vec{v}$ is positive normal creating an in-boundary while at the right end $\vec{-v}$ is positive normal creating another in-boundary.  In-boundaries take domains to codomains.  What’s the out-boundary?  The in-boundary is the disjoint union of two circles, oriented oppositely.  The out-boundary is empty. So this is a cobordism from such a union of circles to the empty manifold.  Redrawn below:

Pretty neat eh?  Identity arrows will end up meaning that nothing’s happened as far as evolution in time but cobordisms like this represent some interesting physics.  The topology changes (a pair of circles isn’t remotely topologically equivalent or similiar to the empty set). It’s particularly nice to see some sort of sign conservation.  If you reverse the ends of this cylinder you’ll end up with two out-boundaries and an empty in-boundary which is a cobordism from the empty set to the disjoint union of two circles, oppositely oriented. But before we get too happy about sign conservation I should show you this:

I can’t wait to learn what this is!  In any event, for all the cylinders previous to the pair of pants above notice that they are all oriented the same way, there’s nothing fishy about their orientation.  We have just been talking about how to orient tangent spaces and extend those orientations.  This gives us a way of creating arrows in our category.  There are certainly some loose ends (that quip about normal bundles will require a sequence of posts on fiber bundles for example) but it’s coming together.

Let’s finish this off by looking at the unit interval, which was my original intention from part 1 but I was sidetracked by questions made clear by the cylinders above.  The interval has two orientations, the unit vector can point right or left (from 0 to 1 or reverse). Let’s call left to right positively oriented.  The boundary of the interval is the disjoint union of two 0-manifolds, i.e. points.  These have the trivial tangent space, {0} which has an empty basis.  We can call this basis a positive or negative orientation.  Sort of odd eh? Let’s look at an example,

Let’s start with the left.  I’m going to try and write this just like with the cylinders above to help this seem more analogous. Consider a point in the left boundary and look at the basis for its tangent space.  The picture above shows an example of such a tangent vector (*wink*) which is considered a positively oriented basis.  What could we add to extend this basis to a positively oriented basis for a tangent space of the interval?  I forgot to label the arrow on the interval as a positive choice of basis orientation.  In any case, that arrow extends the empty basis at either end to a postively oriented basis on the interval’s tangent spaces.  So it creates an in-boundary on the right and an out-boundary on the left.  This is a cobordism, an arrow in 1Cob, from a positively oriented connected component of a 0-manifold to itself.

We can do the same to get an identity on the oppositely oriented version.  We can also choose opposite orientations for the end points,

which will result in two in-boundaries.  How?  Here’s my thought, the positive basis for the interval is not going to extend the negatively oriented empty basis on the right hand boundary.  So it must be the case that the oppositely oriented basis for tangent spaces on the interval will do the trick, which will give us a positive normal pointing from right to left, or inwards on the right hand end.  So this is an arrow from a disjoint union of points, or a two component 0-manifold, where the two points have opposite orientation.

The codomain of this arrow is the empty set since the out-boundary is empty.  I hope this post helps in the understanding of the role of orientation of cobordisms for TQFT.  I’ve written this post over a couple of times so it’s certainly helped me.