I just went to a talk and there was one tiny example the speaker gave to explain when something is not a manifold. I liked it so much I thought I’d dedicate an entire blog post to what was half a line on a board.
I defined manifolds a long time ago, but here’s a refresher: an n-manifold is a space that locally looks like . By locally I mean if you stand at any point in the manifold and draw a little bubble around yourself, you can look in the bubble and think you’re just in Euclidean space. Here are examples and nonexamples of 1-manifolds:
At any point on the orange circle or red line, if we look locally we just see a line. But the yellow and green both have bad points: at the yellow bad point there are 2 lines crossing, which doesn’t happen in , and in the green bad point there’s a corner.
We call 2-manifolds surfaces, and we’ve played with them a bunch (curves on surfaces, curve complex, etc. etc.). Other manifolds don’t have fun names. In general, low-dimensional topology is interested in 4 or less; once you get to 5-manifolds somehow everything gets boring/collapses. It’s sort of like how if you have a circle in the plane, there’s something interesting there (fundamental group), but if you put that circle into 3-space you can shrink it down to a point by climbing a half-sphere to the North Pole.
The other thing we’ll want to think about are group actions. Remember, a group acts on a set X if there’s a homomorphism that sends a group element g to a map such that the identity group element maps to the identity map, and group multiplication leads to composition of functions: . That is, each group element makes something happen on the set. We defined group actions in this old post. Here’s an example of the integers acting on the circle:
So far we’ve seen groups and manifolds as two different things: groups are these abstract structures with certain rules, and manifolds are these concrete spaces with certain conditions. There’s an entire class of things that can be seen as both: Lie Groups. A Lie group is defined as a group that is also a differentiable manifold. Yes, I didn’t define differentiable here, and no, I’m not going to. We’re building intuitions on this blog; we might go into more details on differentiability in a later post. You can think of it as something smooth-ish without kinks (the actual word mathematicians use is smooth).
So what are examples of Lie groups? Well, think about the real numbers without zero, and multiplication as the group operation. This is a manifold-at any point you can make a little interval around yourself, which looks like . How is it a group? Well, we have an identity element 1, every element has an inverse 1/x, multiplication is associative, and the reals are closed under multiplication.
Here’s another one: think about the unit circle lying in the complex plane. I don’t think we’ve actually talked about complex numbers (numbers in the form x + iy, where i is the imaginary square root of -1) on this blog, so I’ll do another post on them some time. If you don’t know about them, take it on faith that the unit circle in the complex plane is a Lie group under multiplication. Multiplying by any number on the unit circle gives you a rotation, which preserves the circle, again 1 is the identity, elements have inverses, and multiplication is associative. Circles, as we said before, are 1-manifolds.
If you have a group action on a set, you can form a quotient of the set by identifying two points in the set if any group element identifies them: that is, x and y become one point if there’s a group element so that g.x=y. For instance, in the action of the integers on the circle above, every point gets identified with three other points (so 12, 3, 6, and 9 o’clock on a clock all get identified under the quotient). Your quotient ends up being a circle as well. We denote a quotient of a group G acting on a set X by X/G.
So here’s a question: when is a quotient a manifold? If you have a Lie group acting on a manifold, is the resulting quotient always a manifold? Answer: No! Here’s the counterexample from the talk:
Consider the real numbers minus zero using multiplication as the group operation (this is the Lie group ) acting on the real line (this is a manifold). What’s the quotient? For any two non-zero numbers a, b on the real line, multiplication by a/b sends b to a, so we identify them in the quotient. So every non-zero number gets identified to a point in the quotient. What about zero? Any number times zero is zero, so zero isn’t identified with anything else. Then the quotient is two points: one for zero, and one for all other numbers.
If the two points are “far apart” from each other, this could still be a 0-manifold (locally, everything looks like a point). But any open set that contains the all-other-numbers point must contain the 0-point, since we can find real numbers that are arbitrarily close to 0. That is, 0 is in the closure of the non-zero point. So we have two points such that one is contained in the closure of the other, and we don’t have a manifold. In fact our space isn’t Hausdorff, a word I mentioned a while back so I should probably define in case we run into it again. Hausdorff is a serious way of explaining “far apart.” A space is Hausdorff (adjective) if for any two points in the space, there exist disjoint neighborhoods of the two spaces. So the real line is Hausdorff, because even if you take two points that look super close, like 2.000000000001 and 2.000000000002, you can find infinitely many numbers between them, like 2.0000000000015.
If you’re curious as to when the quotient of a smooth Manifold by a Lie Group is a manifold, you should go take a class to fully appreciate the answer (the Quotient Manifold theorem). The phrasing of the Quotient Manifold Theorem below is from a book by John Lee, called Introduction to Smooth Manifolds (the version from wikipedia gets rid of one of the conditions but also gets rid of much of the conclusion). Briefly: a smooth action means the function on M is smooth (see the picture above; we didn’t do an in-depth definition of smooth), a free action means there aren’t any fixed points, and a proper action has to do with preimages of certain types of sets.
Theorem 21.10. Suppose G is a Lie group acting smoothly, freely, and properly on a smooth manifold . Then the orbit space is a topological manifold of dimension equal to , and has a unique smooth structure with the property that the quotient map is a smooth submersion.