I’m back! Having a busy month moving and settling in @Stanford, but this hasn’t totally kept me away from making cool pictures :) Here’s a depiction of an Anosov mapping class on a torus 1/npic.twitter.com/kOFV9vd5ov
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This identifies the set of ways to put clothes on our torus with the linear maps of the plane preserving the integer lattice: that is SL(2,Z)! (If we additionally allowed the torus to wear its clothes inside out, we would get twice as many)
Each of these is called a Mapping Class. This set also inherits the group structure of SL(2,Z) (maybe harder to see given the clothing analogy, but remember we are really talking about self-homeomorphisms of the torus, and those can be composed!). This is the Mapping Class Group.
Matrices in SL(2,Z) come in three types: finite order, diagonalizable, and having a generalized 1-eigenspace. Does this distinction mean anything for mapping classes? Yes! (brb - more after I teach)
Finite order are somewhat boring (like putting your shirt on backwards, if you take it off and do it again you’re back to a normal front-facing shirt) also, if you look for finite order integer matrices you’ll quickly realize there aren’t many anyway!
The main group of mapping classes for the torus correspond to the diagonalizable matrices (this is everything with trace greater than 2 in absolute value)- which are called Anosov (yay! We have finally returned to the picture!) we will talk about the remainder in another thread.
These diagonalizable mapping classes have two eigen-directions: one stretching and one compressing (because determinant is 1) so this corresponds to unzipping the sweater into a square, stretching into a parallelogram, twisting it back around the tours and zipping back up.
Here’s what that looks like for the mapping class in the original picture (corresponding to the matrix {{2,1},{1,1}}) (image from Wikipedia)pic.twitter.com/X3BYoO5Fyf
Now a little treat for those of you who have stuck with me so far. Why am I thinking about clothes on a donut? To make cool 3-manifolds to do raytracing in of course! After a little more explanation (sorry not sorry
) I promise to show you some 3D virtual reality renders.
Recall to build a closed surface, we can start with a polygon and pair up the sides particular ways. We can do the same for 3 manifolds! Let’s start here with a cubepic.twitter.com/HLpIKb5w8o
To get started, let’s pair up opposing vertical faces and identify them. This takes each horizontal plane and turns it into a torus - so the result looks kind of like a thickened toruspic.twitter.com/yjTcvbLdgq
To get a closed 3 manifold we somehow need to still glue together the remaining surfaces - that is we need to specify a map from the inner torus to the outer! If we perturb this map a little bit it won’t affect the topology of the result, so we really need a Mapping Class.
As a crazy fact - the type of mapping class we choose determines the geometry that this 3 manifold can have! If we pick an Anosov mapping class, the resulting 3 manifold has Sol geometry.pic.twitter.com/TEsdo95IdT
Here’s a view from inside the resulting manifold, where we are rendering the edges of the fundamental domain by raytracing along geodesics: this is what we would actually see if we were placed inside the identified cube given the Sol metricpic.twitter.com/Tp5u0Ritgx
Some color coding may help here. Imagine standing in this manifold and looking down - we see the torus coming from the bottom of the cube (Orange). If we look up we see the torus coming from the top (Blue).pic.twitter.com/iqJzncHnzc
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