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From: Sen, Anindya
Sent: Tuesday, 05 June, 2007 11:40
To: Barth, Nils
Subject: RE: Algebraic topology saga
Am going trhough covering spaces,-- Hatcher presented
the proof of homotopy lifting so badly, I had to prove it myself to see
what he was talking about. But the homology stuff seems better so far.
Covering spaces are much clearer if you keep a few (geometric) examples
in mind, especially:
C^* -> C^* (z -> z^2)
R -> S^1 (t -> (cos t, sin t))
S^2 -> RP^2
[can also assign orientation combinatorially,
I think as ordering for vertices.
EG for 2-cell, I think you can order the vertices as 123,
and then the ordering on the sides are "delete that entry,
then read from there", so: 12, 23, 31]
What I don't see very clearly so far is, How exactly do
you assign signs to the boundary of a 3-cell?
With polygon you set up definition of clockwise &
anticlockwise, then just go around.
That's because the boundary of a polygon is parallelizable (it's a
circle), so you have not only an orientation, but a trivialization of
the tangent bundle on boundary.
The boundary of a 3-cell is a sphere, which is not parallelizable, so
you can't set up global trivialization:
you *just* get an orientation.
The idea is: the tangent bundle of the boundary + the normal bundle of
the boundary give the tangent bundle of the whole manifold (on the
boundary):
TM = TdM + N
So given an orientation on the interior and a choice of normal (and rule
for combining them), you get an orientation on the boundary.
AFAIK there are 4 conventions:
inward/outward normal first/last:
i.e., take a basis on the boundary, then put an inward/outward pointing
normal vector first/last and compare with orientation on the whole
space.
[NB: given a direct sum V = U + W, see how orientation
reversing/preserving *maps* act on orientation on V;
to get an orientation on V, you need orientation on U, and W, *and* the
order in which you add 'em.]
Concretely, view an orientation on 3-space as either left-handed or
right-handed.
Then "outward normal last" means that, from the outside of the 3-cell,
the orientation on each boundary 2-cell is counter-clockwise (for
right-handed orientation), or always clockwise (for left-handed
orientation).
[This is rather pretty: corkscrew on interior yields turn/loop on
boundary;
can visualize as "which direction will corkscrew be turning if it comes
out that way"]