Tag Archives: torus

Homeomorphisms of the torus, part IV (topology of the identity component)

30 Dec

See Part I for a definition of homeomorphism and torus and Part II for a bit more linear algebra.  

I still owe a Part III for the explanation of the linear algebraic classification of homeomorphisms.  But let’s take a step away from linear algebra and look at shapes (my favorite!)

We know what homeomorphisms are (continuous functions with continuous inverses), with the famous example in the picture below: a coffee cup turns into a donut and vice-versa.

New meaning to cup of (j)Oe.  From wikipedia
New meaning to cup of (j)Oe. From wikipedia

In fact, this pictures doesn’t just show us the homeomorphism (which says where each point of the coffee cup gets sent to in the torus and vice-versa).  It also shows us a homotopy (remember the definition from this post)- essentially, because we can see it traveling through time and back, it’s a homotopy.  And in fact, this homotopy is an isotopy– a type of homotopy where at each frozen point in time, the image is homeomorphic to what we started with.  An example of a homotopy which is not an isotopy is the map that ends up sending x\mapsto -x, where x is a real number.  Homotopies take place over time, so I would actually write this map as \mathbb{R}\times [0,1] \to \mathbb{R}; (x,t)\mapsto (1-t)x+t(-x).  So when t=0, we have x mapping to x, and when t=1, x maps to -x.  One reason this isn’t an isotopy is because when t=1/2, all of the real line gets mapped to the point 0.  And mapping everything to 0 isn’t a homeomorphism (what would the continuous inverse be?)

A big part of geometric group theory is using shapes to come up with algebraic theorems, and using algebra to come up with shapes.  One thing you can do (which we’re doing RIGHT NOW!) is take a shape, do some algebra, and then make a new shape.  To be specific, our first shape is the torus.  Our algebra was figuring out the group of homeomorphisms of the torus, also written as Homeo(T)- T for torus.  Sometimes you’ll see T^2, to specify that we’re talking about the 2-torus rather than a higher dimension (more on higher dimensional tori later.  Isn’t it cool that the plural of torus is tori?  Pronounced tor-eye.)  Now we’re going to make a new shape from this group of homeomorphisms.

We’ll only consider homeomorphisms isotopic to the identity, written as Homeo_0(T^2).  Starting in 1962 and finishing in 1965, badass Mary-Elizabeth Hamstrom proved in a series of papers that Homeo_0(X) is contractible (homotopic to a point) if X is a two-manifold with a short list of exceptions [torus, sphere, plane, disk, annulus, disk with a hole in it, plane with a hole in it.]

Abstractly, I realize that there are many 10-year olds out there who could make a better picture than this.  But I'm still so proud of myself.  Exceptions to the theorem that the space of homeomorphisms isotopic to the identity is contractible.
Abstractly, I realize that there are many 10-year olds out there who could make a better picture than this. But I’m still so proud of myself. Exceptions to the theorem that the space of homeomorphisms isotopic to the identity is contractible.

Let’s look at our current favorite from this list, $Homeo_0(T^2)$.  If we start from the identity, which homeomorphisms can we isotope to?  Well, I can rotate my torus around its hole-axis, and that ending homeomorphism is definitely isotopic to the identity (the rotation through time is the isotopy; where the points end up is the ending homeomorphism).

Orange dot moves to red dot.
Orange dot moves to red dot.

Since I can rotate by all the degrees up to 360, which brings me back to the identity, this means that Homeo_0(T^2) contains a circle- each point on the circle represents rotating the torus by that many degrees.

What else can I do that’ll be isotopic to the identity?  I can rotate the torus around its center circle (running through the middle of the donut), like if I was wringing out a towel.

Again, orange dot to red dot.
Again, orange dot to red dot.

Again, I can do this for 360 degrees before coming back to the identity.  So there’s another circle, different from the first one, in Homeo_0(T).

I can also do any combination of these two: I can rotate 27 degrees around the hole-axis, and then 78 around the center circle.  This is true for any numbers between 0 and 360, but then 0 and 360 are the same for both.  So far, we have the picture on the left.  I colored it to indicate that 0=360 on both axes.  Look familiar?

torus

Notice that all four corners are the same homeomorphism: the identity can be had by rotating by 360 degrees in either direction, or in both directions, or by doing nothing.  So we’ve shown that Homeo_0(T^2) actually contains a torus!  This is cool because all those other Homeo_0(X) were contractible.  In fact, Homeo_0(T^2) is a torus- we’ve actually described all the homeomorphisms of the torus which are isotopic to the identity- combinations of rotations around the center axis and around the center circle.

Personally I find this much more exciting than classifying the homeomorphisms by trace (yeup that’s happening in Part III, nope Part IV is coming out before Part III and you’re going to like it), probably because it involves shapes rather than numbers.

Update on health: I’m taking antibiotics to help with whooping cough.  So that explains why I’ve been sick for a month.  My boyfriend walked by and asked what I was doing an hour ago, and I told him I was very busy feeling sorry for myself.  Then I wrote this blog post to be less lump-ish.

Homeomorphisms of the torus, part II (matrices and linear algebra)

12 Oct

So we know what a torus is, and what a homeomorphism is, from our first post in this series.  Hopefully you remember this picture, which demonstrates how we can tile the plane using the unfolded torus.  (If not just check out the first post.)

tiletheplane

We also had a picture showing curves on the torus, represented on the grid.

Left: follow the numbers to see the knot.  Right: look at the bottom-most green line.

Left: follow the numbers to see the knot. Right: look at the bottom-most green line.

Remember we said that every curve on the torus could be represented by drawing it on the torus and cutting (like in the left) and then unfolding it into a line (as in the right).  The left picture is what happened when I cut up the torus, while the right picture is six copies of the left pictures, as they appear in the plane.  When I glue all six squares together, I can see the slope of the line, as in the bottom most green line.

Notice that we can characterize this curve by the slope of the line- here’s it’s 2/3.  This is the same as 4/6 or 6/9, and you can see this on an actual torus by pulling the strings so that you get back to 2 times one directions and 3 times the other.  Or just believe me if you aren’t into making models (I build things out of paper and tape and floss very often) [Clean floss].

So we can think of curves on the torus as rational points \frac{p}{q} in the plane.  Great!

Here’s a fact: we can define a homeomorphism of the torus by describing where the (1,0) and (0,1) base curves are sent (in our case the red and blue curves).  Why is this fact true?  Because if I know where the red and blue curves go, I can figure out where any other point of the torus goes by using their coordinates (like if a point is at (1/3, 1/9) in the plane, I can send that point to 1/3(f(red)) + 1/3(f(blue)) and this’ll be continuous).  Curves have to be sent to curves, so I need to assign a rational point \frac{p}{q} to each base curve- remember, we associate a rational number to each curve of the torus.  I can do this using a matrix \left( \begin{array}{cc} p_{red} & p_{blue} \\ q_{red} & q_{blue} \end{array} \right).

In this manner we can describe a homeomorphism of the torus by a matrix that describes where each base curve is sent via its columns, and hence we can figure out where any point is sent by writing its coordinates in terms of the base curves.

Example: the identity homeomorphism sends the red curve back to itself, and the blue curve back to itself.  So we want to say that (1,0) maps to (1,0), and (0,1) maps to (0,1).  This means we have \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right), which is, coincidentally, the identity matrix.  You can check that this does what we want it to by multiplying the matrices \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \left(\begin{array}{c} 1 \\ 0 \end{array} \right)=\left( \begin{array}{c} 1 \\ 0 \end{array} \right).  Sweet!

Remember that our original idea was that somehow we could classify all homeomorphisms of the torus.  So far we’ve made an equivalence between homeomorphisms of the torus and these 2 by 2 matrices.  Next I’m going to lay down some facts from linear algebra, which are given without proof because honestly I forgot how to prove this stuff/all my linear algebra.

  • Trace is invariant under conjugation.  Conjugation is multiplying a matrix by another as well as its inverse: so when we conjugate A by B, we get B^{-1}AB.  Trace is the sum of the diagonal elements of your matrix, so the trace of the identity matrix is 2.  So this fact says that tr(A) = tr(B^{-1}AB).
  • The determinant is the product of the eigenvalues.  Determinant for a matrix A=\left( \begin{array}{cc} a & b \\ c & d \end{array}\right) is given by the formula $det A = ad-bc$.  The eigenvalues of a matrix are the numbers \lambda that solve the equation (a-\lambda)d-b(c-\lambda) =0.  We are just laying down the facts, yo.
  • Eigenvalues scale eigenvectors.  This means for any \lambda that we found earlier, there’s a vector \textbf{v}=\left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) so that A\textbf{v} = \lambda \textbf{v}.  This is sort of the definition of eigenvalue and eigenvector.

Woof.  Lots of words, lots of terms and definitions.  The next post in this series will tie together the picture of the torus and all this linear algebra and actually classify all the homeomorphisms (I’ll give you a hint; it’s by trace) as well as give some properties of them.

Homeomorphisms of the torus, Introduction (definition of the title)

23 Sep

Besides understanding the proof of the fundamental theorem of geometric group theory, figuring out how to classify the homeomorphisms of the torus is one of the first exercises grad students in my field do.  It involves lots of matrix multiplication and remembering some facts from linear algebra, which we’ll sorta brush over (mostly because I don’t remember anything from my linear algebra class besides that Stephen Goode always said bee-ta and once told us that if we left his class saying bee-ta then we’ll have learned something.  I’d never heard it before so I had no idea what he was talking about until a few years later when I said bee-ta and everyone was like uh you’re not European/Australian… Long story short, Americans say bay-ta for \beta).

So.. what’s a homeomorphism of the torus?  Well we know what the torus is: a donut!  Mmmm donuts.  But sometime’s it’s hard to do math on a three-dimensional donut (it’s easy to forget where things connect up on the backside) so we often unfold the torus into a square and draw things on the square instead of on the torus.  We’ve seen this picture before:

torus-quotient

 

And the idea is that it describes the torus.  You glue the red edges together so the arrows line up, and you have a cylinder.  Then glue the blue edges (which are now the ends of the cylinder) together so the arrows line up, and you have a torus.  Here’s a cartoon:

And here’s some pictures of me doing this in real life:

torus

First line up the green arrows, then the blue arrows

I'm a torus!

I’m a torus!

Visualizing abstractions is one of the hardest skills in math (I think so anyway) but once you’ve started it, it’s difficult to remember that people don’t look at flat squares and see donuts right away.  So hopefully this helped you!

Why would we rather use a torus drawn as a square rather than using a 3D model?  Well, for one, it’s way easier to draw a square than a torus.  But it’s also much easier to mathify things we’re interested in on a square than on a torus: for instance, I can draw a curve on my torus…

Taken from George Hart’s brilliant bagel page: the string shows part of the curve, and the dotted lines show where the string continues on the back side.

Or I can use math to describe it.  Notice that the string goes through the hole 3 times (easy to see) and around the hole twice (a little harder to see).  So it goes around the skinny side (the longitude) three times and around the flatter side (the latitude) two times.  Let’s name this the (3,2)-torus knot.  Just like with the curve complex, we don’t care if we wiggle the string a little bit (isotopy).

There’s another cool thing we can do with a square: tile the plane!

tiletheplane

Every square is the torus that we had before

I know it doesn’t seem that cool but it is!  If I identify every single square with the torus I had before, I can actually draw the (3,2) curve.  Here it is drawn on just one square, and then drawn on the big grid:

Left: follow the numbers to see the knot.  Right: look at the bottom-most green line.

Left: follow the numbers to see the knot. Right: look at the bottom-most green line.

It might take a little while to understand this picture.  On the left, start in the lower left corner.  Follow strand 1 and you hit a point about 1/3 the way down the right side.  Since the blue arrows are identified, this is the same as the point about 1/3 the way down the left side, which means we go to strand 2.  Similarly, the middle of the top and middle of the bottom are identified, so we get to strand 3, and finally strand 4 finishes the loop.

Instead of doing that tracing, we could draw the (3,2)-knot on the tiled plane, so that it goes over 3 times and up 2 times.  So we can see this loop as a line with slope 2/3 in the plane, which is way easier to draw than the picture on the left.  I added in all the lines on the right to show how the left and right pictures are related.

To summarize so far: a torus is a donut, and we can think of it as embedded in real space (an actual donut) or more abstractly as a square with the sides identified.  The plus of the flat picture is that it’s easier to draw, and it can tile the plane, which led to us associating loops of the torus with lines of fractional slope in the plane, just like our bagel loop was the line with slope 2/3.

OK, onto the homeomorphisms part.  A morphism is a map f: X\to Y that assigns points in $X$ to points in $Y$.  For instance, the yellow pages describe a morphism from X=People to Y=Addresses.  Yes, I did just refer to the yellow pages.  No, I don’t know if they still exist.  Deal with it.

A homeomorphism, then, is a special kind of morphism.  It goes from one topological object back to itself, so Y=X, and it has an inverse.  In our example, there’s no inverse because if I give you an address, you might give me five people back, while if you give me a person, I’ll give you just one person back.  To have an inverse, your map has to be one-to-one, which means exactly what it sounds like: each person in X gets exactly one address in Y.

There’s one more property that homeomorphisms have to have: continuity.  Roughly, continuity says that if you wiggle around a little bit in $X$, then you only wiggle a little bit in $Y$.  The yellow pages map is super not-continuous, because if you go to your alphabetical neighbor in $X$ you might have to cross all of town in $Y$ to find his address.  This graph is an example of a function which is continuous:

Puppy Growth Chart

THIS CHART TELLS YOU HOW BIG YOUR PUPPY IS GONNA GET http://goldendoodles.com/care/growth_chart.htm

The topic of this series of posts will be homeomorphisms of the torus: invertible, continuous maps from the donut to itself.  Next time we’ll talk about matrices (ooh goody) and how they have to do with these homeomorphisms, and then eventually we’ll classify all homeomorphisms by what kind of matrix they are.

 

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