Space Filling
In 1890 Peano showed that there existed a 1-dimensional curve (with endpoints) filling the square [0,1]x[0,1]. One can similarly fill any hypercube with a curve. This works because the cardinality of the continuum is a fixed number, no matter what the dimension.
In the 1990s, Thurston studied 3-dimensional geometry and found some other beautiful space-filling curves, in particular curves filling a sphere. The example shown on page 3 of the paper comes from a fibration of a 3-manifold over a circle with fibre the punctured torus, where the puncture is bounded by a figure8 knot. Such manifolds have tori as boundaries, namely a neighbourhood of the knot. Geometry in 3 dimensions is very rich. For example, the set of volumes of finite volume hyperbolic manifolds is of cardinality omega^omega. Here is a helpful outline of notes about Thurston's work.
This is nice, because in three dimensions we can still picture what is going on. What about higher dimensions? The fact is that higher dimensions are actually a lot simpler than dimension three. It was only recently that the Poincare conjecture was proven for dimension 3, whereas it has been known for dimensions > 4 since the early 1960s (Smale) and for dimension 4 since 1982 (Freedman).
The importance of knot theory in understanding 3-manifolds begs the question: is there a categorical way to understand this complexity? Knots, after all, come from diagrams describing the structure of braided tensor categories. Ribbons appear with the addition of extra structure. Computations involving moving knots and ribbons about are much, much simpler than more traditional analogues.
In the 1990s, Thurston studied 3-dimensional geometry and found some other beautiful space-filling curves, in particular curves filling a sphere. The example shown on page 3 of the paper comes from a fibration of a 3-manifold over a circle with fibre the punctured torus, where the puncture is bounded by a figure8 knot. Such manifolds have tori as boundaries, namely a neighbourhood of the knot. Geometry in 3 dimensions is very rich. For example, the set of volumes of finite volume hyperbolic manifolds is of cardinality omega^omega. Here is a helpful outline of notes about Thurston's work.
This is nice, because in three dimensions we can still picture what is going on. What about higher dimensions? The fact is that higher dimensions are actually a lot simpler than dimension three. It was only recently that the Poincare conjecture was proven for dimension 3, whereas it has been known for dimensions > 4 since the early 1960s (Smale) and for dimension 4 since 1982 (Freedman).
The importance of knot theory in understanding 3-manifolds begs the question: is there a categorical way to understand this complexity? Knots, after all, come from diagrams describing the structure of braided tensor categories. Ribbons appear with the addition of extra structure. Computations involving moving knots and ribbons about are much, much simpler than more traditional analogues.
posted by Kea | 1:07 PM
2 Comments:
There's an interesting passage (pp. 14-15) in Atiyah's Mathematics in the 20th Century, Bulletin of the London Mathematical Society 34(1), 1-15, 2002, where he remarks that the information about 3-manifolds from Vaughan Jones type work with quantum groups and knots is almost orthogonal to Thurston's program. He then mysteriously hints at a bridge between the approaches.
There's an interesting passage (pp. 14-15) in Atiyah's Mathematics in the 20th century, where he suggests that the information about 3-manifolds from Vaughan Jones like considerations - knots and quantum groups - is almost orthogonal to Thurston's program. He then hints at a bridge between them.
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