Sorry this didn’t go up as scheduled last week.
We must now show that the forms we’ve defined are closed, and thus that they indeed will define homology classes. That is, . We must also show that they cannot be exact, and thus that the homology classes they define are nontrivial.
We defined to be the pullback of the form on the sphere. Thus we can calculate
which follows since is an -form on the -dimensional sphere, and so no nontrivial -form exists on to be equal to .
As for exactness: if for some -form , then Stokes’ theorem tells us that
Since the sphere has an empty boundary.
But it’s also possible to calculate this integral directly.
Now, if is any parameterization of the sphere, we set and calculate
We extend to depend on a new variable which will vary in some small neighborhood of by defining , so that we have
and so we can continue:
Now, what is this integral calculating? The first column of the determinant is the vector which, as the position vector of a point on the sphere, has unit length and points perpendicularly to its surface. The other columns form a basis of the tangent space to the sphere at . At any given point, then, we can change the basis so that the upper-left entry in the matrix is and all other entries in its row and column are zero. The determinant is thus the determinant of the rest of the matrix, which is the Jacobian determinant of the parameterization, and its integral is thus the volume of the image — the volume of the sphere.
Now, we may not have a formula handy for the -dimensional volume of the -sphere, but it’s certainly not zero! Therefore the closed form cannot have been exact, and so it must define a nontrivial homology class, just as asserted.
We continue investigating the differential forms we defined last time. Recall that we started with the position vector field and use the interior product to produce the -form on the punctured -dimensional space . We restrict this form to the -dimensional sphere and then pull back along the retraction mapping to get the form .
I’ve asserted that , and now we will prove it; let be tangent vectors at and calculate
as asserted. Along the way we’ve used two things that might not be immediately apparent. First: the derivative works by transferring a vector from to and scaling down by a factor of , which is a consequence of the linear action of and the usual canonical identifications. Second: the volume form on can be transferred to essentially the same form on itself by using the canonical identification .
We want to exhibit a family of closed -forms that aren’t exact, albeit not all on the same space. In fact, there forms will provide models for every possible way a nontrivial homology class can arise.
For each , we consider the space consisting of the normal -dimensional real affine space with the origin removed. Key to our approach will be the fact that we have a “retract” — a subspace along with a “retraction mapping” such that . That is, the retraction mapping sends every point in to some point in , and the points that were in to begin with stay exactly where they are. Explicitly in this case, the “punctured” -dimensional space retracts onto the -dimensional sphere by the mapping , which indeed is the identity on the unit sphere .
Now, in this space we take the position vector field , which we define by taking the canonical identification and applying it to the vector itself: . We also take the canonical volume form , and we use the interior product to define the -form .
Geometrically, the volume form measures -dimensional volume near any given point . Applying the interior product with is like rewriting the volume form in terms of a different basis so that is the first vector in the new basis and all the other vectors are perpendicular to that one, then peeling off the first term in the wedge. That is, measures -dimensional volume in the space perpendicular to — tangent to the sphere of radius at the point .
Next we restrict this form to , and we pull the result back to all of along the retraction mapping , ending up with the form . I say that the net effect is that , but the proof will have to wait. Still, the form is the one we’re looking for.
Indeed, say that is simply-connected, so that any closed curve can be written as the boundary of some surface . Then we take any closed -form with . Stokes’ theorem tells us that
It will (eventually) turn out that the fact that both and vanish together is not a coincidence, but is in fact an example of a much deeper correspondence between homology and cohomology — between topology and analysis.
Today we prove the assertion from last time: if is a -form on a manifold such that for every closed curve we have
then for some function . As we saw last time, the condition on is equivalent to the assertion that the line integral of over any curve depends only on the endpoints and , and not on the details of the path at all.
So, let’s define a function. In every connected component of , pick some base-point . As an aside, what we really want are the arc components of , but since is pretty topologically sweet the two concepts are the same. Anyway, if is in the same component as the selected base-point , we pick some curve from to and define
Remember here that the choice of doesn’t matter at all, since we’re assuming that is path-independent, so this gives a well-defined function given the choice of .
Incidentally, what would happen if we picked a different base-point ? Then we could pick a path from to and then always choose a path from to by composing and . Doing so, we find
So the only difference the choice of a base-point makes is an additive constant over the whole connected component in question, which will make no difference once we take their differentials.
Anyway, we need to verify that . And we will do this by choosing a vector field and checking that . So, given a point we may as well choose itself as the base-point. We know that we can choose an integral curve of through , and we also know that
for an integral curve. For any , we can get a curve from to by defining . And so we calculate (in full, gory detail):
So, having verified that at any point we have , we conclude that for the given function , and is thus exact.
Now, “conservative” is not meant in any political sense. To the contrary: integration is easy with conservative vector fields. Indeed, if we have a curve that starts at a point and ends at a point then fundamental theorem of line integrals makes it easy to calculate:
I didn’t go into this before, but the really interesting thing here is that this means that line integrals of conservative vector fields are independent of the path we integrate along. As a special case, the integral around any closed curve — where — is automatically zero. The application of such line integrals to calculating the change of energy of a point moving through a field of force explains the term “conservative”; the line integral gives the change of energy, and whenever we return to our starting point energy is unchanged — “conserved” — by a conservative force field.
This suggests that it might actually be more appropriate to say that a vector field is conservative if it satisfies this condition on closed loops; I say that this is actually the same thing as our original definition. That is, a vector field is conservative — the gradient of some function — if and only if its line integral around any closed curve is automatically zero.
As a first step back the other way, it’s easy to see that this condition implies path-independence: if and go between the same two points — if and — then
Indeed, the formal sum is a closed curve, since , and so
Of course, this also gives rise to a parallel — and equivalent — assertion about -forms: if the integral of around any closed -chain is always zero, then for some function . Since we can state this even in the general, non-Riemannian case, we will prove this one instead.
We say that a space is “simply-connected” if any closed curve with is homotopic to a constant curve that stays at the single point . Intuitively, this means that any loop in the space can be “pulled tight” without getting caught up on any “holes”.
It turns out that this is equivalent to saying that every closed curve is the boundary of some parameterized square. Indeed, consider the following diagram I’ve drawn with the help of Geogebra:
This is a picture of the homotopy cylinder. The domain of a curve is the interval , so the domain of the homotopy cylinder is the square . I’ve labeled the sides to describe what the homotopy does to them: the lower edge follows the curve ; the upper edge is the constant point ; the two sides are also constant at , meaning that we’re holding the curve’s ends fixed as we perform the homotopy. And so the homotopy is exactly a continuous (or smooth) map from the square into our space, and the boundary of the parameterized square is exactly the curve . The converse — that any parameterized square can be homotoped to look like this — shouldn’t be hard to see.
So what does this mean for homology? Well, for cubic singular homology it means that is exact if is simply-connected. Indeed, if is a closed -chain, then it must be made up of a formal sum of curves. Any curve which isn’t already closed must have a start and an end, and the end must be the start of another curve, or else the boundary points of wouldn’t cancel off. We can break up — possibly non-uniquely — into a collection of closed curves, each of which is the boundary of some parameterized square, by the above argument. Thus is itself the boundary of this collection of squares; since all closed -chains are exact, the first homology vanishes.
We can use the concept of degree to prove the (in)famous “hairy ball theorem”. That is, any smooth vector field defined on the whole sphere for even must vanish at some point. The name comes from thinking of a “hair” growing out of each point on the sphere, and trying to “comb” them all to lay flat — tangent — against the sphere. If this can be done, then the direction each hair points defines a unit vector at its point, and combing smoothly means we have a smooth vector field. The assertion is that such a “combing” is impossible.
First, I say that if is even then the antipodal map sending a point to is orientation-reversing. Indeed, we first extend to the larger space ; the antipodal map is clearly orientation-reversing, as we can see by taking its Jacobian. This matrix has eigenvalues, all equal to , so the determinant is .
If is the position vector of a point in , then we consider the canonical identification of with and calculate
So if is a positively-oriented basis of , then is a negatively-oriented basis of .
Now, the position vector field is -related to itself; if is used as the first vector in a positively-oriented basis of , then is the first vector in the antipodal basis, which we know is negatively-oriented. But this means that the positively-oriented basis of must be flipped to a negatively-oriented basis of .
So what does this mean? The identity map has degree , while we can calculate that the antipodal map has degree . Since these are different, the two maps must not act identically on homology, and therefore cannot be homotopic.
But now let’s assume that is an everywhere-nonzero vector field which, without loss of generality, we may assume to have constant length — use the inner product we get by taking and divide by the original length to normalize. For each point we can define the great circle
This is a curve that lies on the sphere for every . Indeed, we can check that
So if we define by , then we have a homotopy from the identity map on to the antipodal map, which is exactly what we just showed could not exist. Thus we conclude that no such non-vanishing vector field can exist on an even-dimensional sphere.
As I asserted yesterday, there is a simple formula for the degree of a proper map . Pick any regular value of , some of which must exist, since the critical values have measure zero in . For each we define to be or as is orientation-preserving or orientation-reversing. Then I say that
The key is the inverse function theorem: the Jacobian of must have maximal rank at , so there’s some around on which is a diffeomorphism. This is true for each , so we get an open around each . Since is proper, there are only finitely many such , and they’re all separated from each other, so we can shrink each so that they’re all disjoint. Then we can set to be the intersection of all the , and we can further shrink each until its image is exactly .
That is, we’ve started with a regular value and its preimage , consisting of a bunch of points ; we’ve used the fact that is proper to widen it to an open set and its preimage, consisting of a bunch of disjoint open sets . Since is a diffeomorphism for each , is constantly orientation-preserving or orientation-reversing over all of , though of course each can be different.
Now if we pick some supported in , then is supported in the disjoint union of the . So we can calculate
Since the coefficient is independent of the choice of , this proves the assertion.
This will be a nice, simple definition. As we saw last time, the top-degree compactly-supported de Rham cohomology is easy to calculate for a connected, oriented manifold . Specifically, we know that .
So, let’s say that is a smooth map from one connected, oriented -manifold to another. The pullback induces a map on cohomology: . But since each of these cohomology spaces is (isomorphic to) , must act on them by multiplication by some number. That is, if is a closed -form, and if the integral of over is , then is also a closed -form, and the integral of over is , where is some real number that depends only on the function and not on the form . That is, the same works for all .
We call this number the “degree” of , and write . It turns out that when is a proper map — one for which the preimage of any compact set is compact, which is always the case when is itself compact — that we have a good way of calculating it. If is a regular value of , some of which which will always exist, then the preimage of consists of a finite collection of points . For each one we calculate the “signum” of at — written — to be if is orientation-preserving and if is orientation-reversing. I say that
While we will not prove this until next time, there is one immediate consequence: if is not onto, then any point in has an empty preimage, and so we find that . In a sense, the degree of is counting how many times the image of covers that of , considering orientation.