User:Typometer

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[edit] The Suspension of the 2-Torus as an Algebraic Variety

The 2-Torus
The 2-Torus

The equations \{x_1^2+x_2^2=1, x_3^2+x_4^2=1\} define a 2-torus in \mathbb{A}^4. Since it's compact, its projective closure X \subset \mathbb{P}^4 defined by the equations \{x_1^2+x_2^2=x_5^2, x_3^2+x_4^2=x_5^2\} is also a 2-torus. The equations defining X also define a variety Y \subset \mathbb{P}^5, which is related to ΣT2 as follows.

What I'd like to emphasize is:

  1. The foliation is singular.
  2. The foliation is twisted. (It restricts to a foliation of the Mőbius band away from the singular point.)


[edit] Collapsing the Boundary of the Mőbius Band to a point gives RP2

The Mőbius Band
The Mőbius Band

I claim that collapsing the boundary circle of the Mőbius band to a point gives RP2. There are several ways to see this. At first it might seem incredible that this doesn't lead to a singularity. (After all, collapsing the boundary of the cylinder does result in a singularity.) But bearing in mind that a neighborhood of this collapsed point is the cone on a single circle (as opposed to the cone on two circles), it becomes much more believable. Here's one way to see that the resulting manifold is in fact RP2. By identifying opposite points along the boundary of the Mőbius band, we can get a circle bundle, which is in fact the Klein bottle. This reduces the question to understanding what happens when you contract the "middle double circle" of the Klein bottle (the circle along which the two Mőbius bands making up RP2 are glued along) to a point. Depicting the Klein bottle as a square with edges identified in


[edit] To do

  • Spectral sequence calculations
  • Lecture summaries
  • What's going on with the grading of mod-2 IH groups? Compare with the Lagrangian IH groups of ΣT2?
  • Perverse sheaves
  • Cone questions
  • Manifolds and Modular Forms
  • [G&H] reading
  • Curvature
  • Representation theory
  • Scheme stuff


Typometer 09:24, 17 February 2007 (UTC)

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