Subject: two questions for the list
From: Allen Hatcher
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