11c. Algebraic manifolds
We have seen so far the mathematical and physical meaning of the conics, which are the simplest curves around, with in both cases, very simple answers. The continuation of the story, however, is more complicated, because beyond conics, things ramify. A first idea, in order to generalize the conics, is to look at zeroes of arbitrary polynomials:
More generally, we can look at zeroes of polynomials in arbitrary [math]N[/math] dimensions:
Observe that, at [math]N\geq3[/math], what we have is not exactly a curve, but rather some sort of [math](N-1)[/math]-dimensional surface, called algebraic hypersurface. Due to this, in order to have a full collection of beasts, of all possible dimensions, we must intersect such algebraic hypersurfaces. We are led in this way to zeroes of families of polynomials, as follows:
An algebraic manifold is a space of the form
And, good news, this is the good definition, and with the branch of mathematics studying such manifolds being called algebraic geometry. However, it is in fact possible to do even more generally, by looking at algebraic manifolds defined over an arbitrary field [math]F[/math], by using a family of polynomials [math]P_i\in F[x_1,\ldots,x_N][/math], as follows:
These ideas are very old, going back to the old Greeks, and there are many things that can be said about algebraic geometry, especially in its “arithmetic” version, over arbitrary fields [math]F[/math], where the theory really shines, with many known advanced results in number theory having been obtained in this way, via algebraic geometry.
Instead of pursuing with usual, affine geometry, which can quickly escalate into fairly complicated things, let us take a look at projective geometry too, which is something fun, and interesting, and quite often more fun and interesting than affine geometry itself.
You might have heard or not of projective geometry. In case you didn't yet, the general principle is that “this is the wonderland where parallel lines cross”. Which might sound a bit crazy, and not very realistic, but take a picture of some railroad tracks, and look at that picture. Do these parallel railroad tracks cross, on the picture? Sure they do. So, we are certainly not into abstractions here, but rather into serious science. QED.
Mathematically now, here are some axioms, to start with:
A projective space is a space consisting of points and lines, subject to the following conditions:
- Each [math]2[/math] points determine a line.
- Each [math]2[/math] lines cross, on a point.
As a basic example we have the usual projective plane [math]P^2_\mathbb R[/math], which is best seen as being the space of lines in [math]\mathbb R^3[/math] passing through the origin. To be more precise, let us call each of these lines in [math]\mathbb R^3[/math] passing through the origin a “point” of [math]P^2_\mathbb R[/math], and let us also call each plane in [math]\mathbb R^3[/math] passing through the origin a “line” of [math]P^2_\mathbb R[/math]. Now observe the following:
(1) Each [math]2[/math] points determine a line. Indeed, 2 points in our sense means 2 lines in [math]\mathbb R^3[/math] passing through the origin, and these 2 lines obviously determine a plane in [math]\mathbb R^3[/math] passing through the origin, namely the plane they belong to, which is a line in our sense.
(2) Each [math]2[/math] lines cross, on a point. Indeed, 2 lines in our sense means 2 planes in [math]\mathbb R^3[/math] passing through the origin, and these 2 planes obviously determine a line in [math]\mathbb R^3[/math] passing through the origin, namely their intersection, which is a point in our sense.
Thus, what we have is a projective space in the sense of Definition 11.12. More generally now, we have the following construction, in arbitrary dimensions:
We can define the projective space [math]P^{N-1}_\mathbb R[/math] as being the space of lines in [math]\mathbb R^N[/math] passing through the origin, and in small dimensions:
- [math]P^1_\mathbb R[/math] is the usual circle.
- [math]P^2_\mathbb R[/math] is some sort of twisted sphere.
We have several assertions here, with all this being of course a bit informal, and self-explanatory, the idea and some further details being as follows:
(1) To start with, the fact that the space [math]P^{N-1}_\mathbb R[/math] constructed in the statement is indeed a projective space in the sense of Definition 11.12 follows from definitions, exactly as in the discussion preceding the statement, regarding the case [math]N=3[/math].
(2) At [math]N=2[/math] now, a line in [math]\mathbb R^2[/math] passing through the origin corresponds to 2 opposite points on the unit circle [math]\mathbb T\subset\mathbb R^2[/math], according to the following scheme:
Thus, [math]P^1_\mathbb R[/math] corresponds to the upper semicircle of [math]\mathbb T[/math], with the endpoints identified, and so we obtain a circle, [math]P^1_\mathbb R=\mathbb T[/math], according to the following scheme:
(3) At [math]N=3[/math], the space [math]P^2_\mathbb R[/math] corresponds to the upper hemisphere of the sphere [math]S^2_\mathbb R\subset\mathbb R^3[/math], with the points on the equator identified via [math]x=-x[/math]. Topologically speaking, we can deform if we want the hemisphere into a square, with the equator becoming the boundary of this square, and in this picture, the [math]x=-x[/math] identification corresponds to a “identify opposite edges, with opposite orientations” folding method for the square:
(4) Thus, we have our space. In order to understand now what this beast is, let us look first at the other 3 possible methods of folding the square, which are as follows:
Regarding the first space, the one on the left, things here are quite simple. Indeed, when identifying the solid edges we get a cylinder, and then when further identifying the dotted edges, what we get is some sort of closed cylinder, which is a torus.
(5) Regarding the second space, the one in the middle, things here are more tricky. Indeed, when identifying the solid edges we get again a cylinder, but then when further identifying the dotted edges, we obtain some sort of “impossible” closed cylinder, called Klein bottle. This Klein bottle obviously cannot be drawn in 3 dimensions, but with a bit of imagination, you can see it, in its full splendor, in 4 dimensions.
(6) Finally, regarding the third space, the one on the right, we know by symmetry that this must be the Klein bottle too. But we can see this as well via our standard folding method, namely identifying solid edges first, and dotted edges afterwards. Indeed, we first obtain in this way a Möbius strip, and then, well, the Klein bottle.
(7) With these preliminaries made, and getting back now to the projective space [math]P^2_\mathbb R[/math], we can see that this is something more complicated, of the same type, reminding the torus and the Klein bottle. So, we will call it “sort of twisted sphere”, as in the statement, and exercise for you to figure out how this beast looks like, in 4 dimensions.
All this is quite exciting, and reminds childhood and primary school, but is however a bit tiring for our neurons, guess that is pure mathematics. It is possible to come up with some explicit formulae for the embedding [math]P^2_\mathbb R\subset\mathbb R^4[/math], which are useful in practice, allowing us to do some analysis over [math]P^2_\mathbb R[/math], and we will leave this as an instructive exercise.
There is some linear algebra to be done here too, by identifying the lines in [math]\mathbb R^N[/math] with the corresponding rank 1 projections, along with many other things, and we have:
The projective space [math]P^{N-1}_\mathbb R[/math] can be thought of as being the space of rank [math]1[/math] projections in the matrix algebra [math]M_N(\mathbb R)[/math], given by
There are several things going on here, the idea being as follows:
(1) The main assertion is more or less clear from definitions, the point being that the lines in [math]\mathbb R^N[/math] passing through the origin are obviously in bijection with the corresponding rank [math]1[/math] projections. Thus, we obtain the interpretation of [math]P^{N-1}_\mathbb R[/math] in the statement.
(2) Regarding now the formula of the rank 1 projections, which is a must-know, for this, and in everyday life, consider a vector [math]y\in\mathbb R^N[/math]. Its projection on [math]\mathbb Rx[/math] must be a certain multiple of [math]x[/math], and we are led in this way to the following formula:
(3) But with this in hand, we can now compute the entries of [math]P_x[/math], as follows:
Thus, we are led to the formula in the statement.
All this is very interesting, but we will pause our study here, because we still have many other things to say. Getting now to finite fields, we have:
Given a field [math]F[/math], we can talk about the projective space [math]P^{N-1}_F[/math], as being the space of lines in [math]F^N[/math] passing through the origin. At [math]N=3[/math] we have
This is indeed clear from definitions, with the cardinality coming from:
Thus, we are led to the conclusions in the statement.
As an example, let us see what happens for the simplest finite field that we know, namely [math]F=\mathbb Z_2[/math]. Here our projective plane, having [math]4+2+1=7[/math] points, and 7 lines, is a famous combinatorial object, called Fano plane, which is depicted as follows:
Here the circle in the middle is by definition a line, and with this convention, the basic axioms in Definition 11.12 are satisfied, in the sense that any two points determine a line, and any two lines determine a point. And isn't this beautiful. Let us record:
\begin{conclusion}
For getting started with geometry, all you need is [math]7[/math] points.
\end{conclusion}
So long for algebraic geometry, real, complex or over arbitrary fields, and affine or projective. For more, a good reference here is the book by Harris [1].
Finally, no discussion about algebraic geometry would be complete without a look into algebraic topology. We have already seen, in the proof of Theorem 11.13, that “shape”, taken in a basic topological sense, matters. So, let us further explore this.
Forgetting about manifolds, let us start with something that we know, namely:
A topological space [math]X[/math] is called connected when any two points [math]x,y\in X[/math] can be connected by a path. That is, given any two points [math]x,y\in X[/math], we can find a continuous function [math]f:[0,1]\to X[/math] such that [math]f(0)=x[/math] and [math]f(1)=y[/math].
The problem is now, given a connected space [math]X[/math], how to count its “holes”. And this is quite subtle problem, because as examples of such spaces we have:
(1) The sphere, the donut, the double-holed donut, the triple-holed donut, and so on. These spaces are quite simple, and intuition suggests to declare that the number of holes of the [math]N[/math]-holed donut is, and you guessed right, [math]N[/math].
(2) However, we have as well as example the empty sphere, I mean just the crust of the sphere, and while this obviously falls into the class of “one-holed spaces”, this is not the same thing as a donut, its hole being of different nature.
(3) As another example, consider again the sphere, but this time with two tunnels drilled into it, in the shape of a cross. Whether that missing cross should account for 1 hole, or for 2 holes, or for something in between, I will leave it up to you.
Summarizing, things are quite tricky, suggesting that the “number of holes” of a topological space [math]X[/math] is not an actual number, but rather something more complicated. Now with this in mind, let us formulate the following definition:
The homotopy group [math]\pi_1(X)[/math] of a connected space [math]X[/math] is the group of loops based at a given point [math]*\in X[/math], with the following conventions,
- Two such loops are identified when one can pass continuously from one loop to the other, via a family of loops indexed by [math]t\in[0,1][/math],
- The composition of two such loops is the obvious one, namely is the loop obtaining by following the first loop, then the second loop,
- The unit loop is the null loop at [math]*[/math], which stays there, and the inverse of a given loop is the loop itself, followed backwards,
with the remark that the group [math]\pi_1(X)[/math] defined in this way does not depend on the choice of the given point [math]*\in X[/math], where the loops are based.
Here the fact that [math]\pi_1(X)[/math] defined in this way is indeed a group is obvious, and obvious as well is the fact that, since [math]X[/math] is assumed to be connected, this group does not depend on the choice of the given point [math]*\in X[/math], where the loops are based.
As basic examples, for spaces having “no holes”, such as [math]\mathbb R[/math] itself, or [math]\mathbb R^N[/math], and so on, we have [math]\pi_1=\{1\}[/math]. In fact, having no holes can only mean, by definition, that [math]\pi_1=\{1\}[/math]. As further illustrations, here are now a few basic computations:
We have the following computations of homotopy groups:
- For the circle, we obtain [math]\pi_1=\mathbb Z[/math].
- For the torus, we obtain [math]\pi_1=\mathbb Z\times\mathbb Z[/math].
- For the disk minus [math]2[/math] points, we have [math]\pi_1=\mathbb Z*\mathbb Z[/math].
These results are all standard, as follows:
(1) The first assertion is clear, because a loop on the circle must wind [math]n\in\mathbb Z[/math] times around the center, and this parameter [math]n\in\mathbb Z[/math] uniquely determines the loop, up to the identification in Definition 11.18. Thus, the homotopy group of the circle is the group of such parameters [math]n\in\mathbb Z[/math], which is of course the group [math]\mathbb Z[/math] itself.
(2) In what regards now the second assertion, the torus being a product of two circles, we are led to the conclusion that its homotopy group must be some kind of product of [math]\mathbb Z[/math] with itself. But pictures show that the two standard generators of [math]\mathbb Z[/math], and so the two copies of [math]\mathbb Z[/math] themselves, commute, [math]gh=hg[/math], and so we obtain the product of [math]\mathbb Z[/math] with itself, subject to commutation, which is the usual product [math]\mathbb Z\times\mathbb Z[/math].
(3) This is quite clear, because the homotopy group is generated by the 2 loops around the 2 missing points, which are obviously free, algebrically speaking. Thus, we obtain a free product of the group [math]\mathbb Z[/math] with itself, also known as free group on 2 generators.
There are many other interesting things that can be said about homotopy groups. Also, another thing that can be done with the arbitrary spaces [math]X[/math], again in relation with studying their “shape”, is that of looking at the fiber bundles over them, again up to continuous deformation. We are led in this way into a group, called [math]K_0(X)[/math]. Moreover, both [math]\pi_1(X)[/math] and [math]K_0(X)[/math] have in fact higher analogues [math]\pi_n(X)[/math] and [math]K_n(X)[/math] as well, and the general goal of algebraic topology is that of understanding all these groups.
But all this, obviously, starts to become too complicated. So, leaving now the general manifolds and topological spaces aside, let us focus now on the simplest objects of topology, namely the knots, with this meaning the smooth closed curves in [math]\mathbb R^3[/math]:
A knot is a smooth closed curve in [math]\mathbb R^3[/math], regarded modulo smooth transformations.
And isn't this a beautiful definition. We are here at the core of everything that can be called “geometry”, and in fact, thinking a bit on how knots can be knotted, in so many fascinating ways, we are led to the following philosophical conclusion: \begin{conclusion} Knots are to geometry what prime numbers are to number theory. \end{conclusion} At the level of questions now, once we have a closed curve, say given via its algebraic equations, can we decide if is tied or not, and if tied, how complicated is it tied, how to untie it, and so on? But these are, obviously, quite difficult questions.
Perhaps simpler now, experience with cables and ropes shows that a random closed curve is usually tied. But can we really prove this? Once again, difficult question. So, we will stop here, and exercise for you, to say something non-trivial about knots.
General references
Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].