'O Mister Hog,' cry the frenzied masses, surging like tidal jam, 'where have you been? What have you done? What great work of hoggishness has consumed thee?'
'O dear sweet readers,' I reply, 'o both of you, such wonders have I seen! I have crossed the Timeless Void that yawns sleepily beyond the outermost Sphere of the Fixed Stars, to reach the Realms of Light; and I took tea with Jesus Christ, Whose blood ransomed us from the cruel God of these Aeons. In the Realms of Light they drink only organic lapsang souchong, and they drink it very quickly, for the background radiation vapourises unshielded tea in half a microsecond flat. The crumpets are armoured with neutronium, and the butter is a quark-gluon plasma.
'And I asked the Saviour to share with me the wisdom of the Alien God, His Father.
'And Jesus said unto me, "I can suck my own dick."
'And I saw that this was Truth.'
The trip back from the Realms of Light took quite a long time, what with security and so forth. I was held up for two hours at the Sphere of Ialdabaoth, Archon of Saturn, when some lady started panicking over a mariachi band she claimed was using the lavatory 'swarthily' and menacing her with their moustaches. I got in a good bit of reading...
'O Mister Hog,' I hear you cry like tiny baby trumpets made of crab, 'what have you been reading?'
I hear this question; and I hear that this question is good.
'Several things!'
I've been reading John Milnor's book on characteristic classes, which is to do with the cohomology of vector bundles. Most of you probably aren't very interested in that. Fortunately for all of us, it's fun to try and picture certain vector bundles (and related spaces) in our heads, so let's all do that instead! You too, Jesus.
'Holy pigfuck,' exclaims the Christ. 'Hglaghlaghlaglug.'
Imagine you have some kind of topological space sitting around. Now imagine that, at every point in that space, you have a line, or a plane, or some higher-dimensional hyperplane, and that you can weave all these lines or planes together without overlapping to form a new space sort of sitting over your old one. That is essentially a vector bundle over your original space.
Imagine a circle. At every point on the circle, there is a tangent line. We'd like to assemble all of these tangent lines into a new object, the tangent bundle to the circle. We don't want any of the lines to overlap, though. It's easy to piece them together in four-dimensional space, but that's pretty impossible to really visualise. I think of it as living in three-dimensional space by giving all the lines a little twist. Imagine you tweak each line ever so slightly out of the plane of the circle, so that they never overlap; start from any one point on the circle, and twist the fibres more and more as you go until you go a quarter of the way around, when the tangent line is twisted vertically. Then you can keep twisting as you go until, halfway around, the tangent line is horizontal again. Now take this helical mess, and imagine yourself pulling on the first, horizontal tangent line until it becomes vertical. Since all the tangent lines fit together, all the other lines have to move, too; the effect sort of ripples along the circle, leaving all the tangent lines vertical in its wake. When the dust settles, you're left with a cylinder, living happily in three-dimensional space. You have just discovered that the circle is parallelisable.
It is not true that you can always twist the fibres of a vector bundle until they all become parallel. Imagine now the sphere. At every point on the surface of the sphere, we have a two-dimensional tangent plane; we can assemble all of these tangent planes to form the tangent bundle to the sphere. None of these tangent planes are supposed to overlap; I imagine them sort of fitting together like the petals of a rose, and again I use a sort of 'twistiness' to imagine their necessary extension into higher dimensions. Just as there was a circular hole in the centre of the tangent bundle to the circle, there is a spherical hole in the centre of this bundle. I try to picture myself both inside this cavity looking out at the bundle, and outside the bundle looking in. From the inside, it looks to me like a curvilinear zero-gravity Gothic cathedral made of glass, tinted, for some reason, with patches and streaks of blue. The planes seen at strange angles and edge-on seem to form an infinite web of buttresses, and everything tries to curve away. From the outside, it is more like a green glass carnation mating with a spiral galaxy, but with perfect spherical symmetry, and infinite in every direction. I can't twist this structure to arrange all the tangent planes in a nice parallel way without it breaking somewhere; wherever I start, the parallelising ripple always tears a hole at the antipodal point. This intuitive picture of mine actually is fairly accurate: the tangent bundle to the sphere is not parallelisable in the technical sense, but if you delete any one point from the sphere and its corresponding tangent plane, what's left is.
I'm not sure where the colours come from...
Projective space is deeply involved in the study of vector bundles. You can construct the projective plane as follows: take a sphere. Imagine taking a point on the sphere, and its antipodal point, and pulling them together to meet somewhere inside the sphere. Now do it with another pair of points, but make sure they meet somewhere else. Do this with every single point on the sphere, each point and its antipodal point meeting each other but meeting no other points. It's a weird, collapsed sphere that can't properly live in three dimensions, but I imagine it as looking a bit like a seashell, all curled up on itself. And pink. I've no idea why, but this projective plane is tinted a distinct pink. Like a salmon. Now imagine that, back on your original sphere, you attached to every point a line connecting the point to its antipodal point, and stretching off to infinity in both directions beyond the sphere. Picture what happens to all these lines as the sphere collapses into the projective plane; what pops out is called the canonical line bundle over the projective plane, and I still can't imagine what it might look like, but I'm working on it. I think it's rose red.
Posted by aloysius at July 31, 2004 02:16 PM | TrackBack |