Subject: response to Hatcher
From: Daniel Isaksen
Date: Fri, 22 Dec 2006 15:51:16 -0500 (EST)
Daniel Biss, Dan Dugger, and I thought a bit about Allen Hatcher's
question about (O^n - 0)/~, where the equivalence relation is the one that
is generated by left multiplication. Here's what we came up with.
The space (O2 - 0)/~ is homeomorphic to a closed 2-disk. The basic idea
is this. Given a point (x,y) of O2, the ratio of the norms |x| and |y|
is invariant under left multiplication. So is the inner product .
It turns out that these two values completely detect the equivalence
relation.
The space (O3 - 0)/~ is contractible and contains an open dense set that
is homeomorphic to an open 5-disk. We could probably work out the exact
homeomorphism type of this space. The basic idea is the same. Given
(x,y,z) in O3, consider the ratios of the norms and also the three
pairwise inner products. These values detect equivalence classes.
We haven't completely finished with n = 4. We know that it has dimension
9 or 10. I would guess that it's a contractible space of dimension 10.
For n > 4, I would guess that it's a contractible space of dimension 10 +
8(n-4).
Dan Isaksen
isaksen@math.wayne.edu