Subject: two questions for the list
From: Allen Hatcher <hatcher@math.cornell.edu>
Date: Mon, 11 Dec 2006 11:02:57 -0500

Here are two questions for the discussion list related to octonionic  
projective spaces.

1.  It is well known that octonionic projective spaces OP^n exist  only 
for n = 1,2.  Associativity of multiplication is needed in order  for 
multiplication of nonzero vectors in O^n by nonzero scalars in O  to be an 
equivalence relation.  However, even if this isn't an  equivalence 
relation one can take the equivalence relation it  generates.  What is the 
resulting quotient space of O^n - {0} ?

2.  In the study of the exceptional Lie groups there arise  "projective 
planes" of the form (F tensor O)P2  for F = C, H, O.   These are 
Riemannian manifolds of dimensions 32, 64, and 128 whose  isometry groups 
are E_6, E_7, and E_8.  (See the article by John Baez  on the octonions in 
BAMS 39, pp.145-205.)  What is known about the  algebraic topology of 
these manifolds?  Is their homology known, or a  CW structure?

Allen Hatcher