From: Linus Kramer
Subject: pi_7(O) and octonions
Date: Tue, 09 Nov 1999 12:44:33 +0100
Newsgroups: sci.math.research
Keywords: generator of homotopy group of orthogonal group, spheres
John Baez asked if pi_7(O) is generated by
the (multiplication by) unit octonions.
View this as a question in KO-theory: the claim is
that H^8 generates the reduced real K-theory
\tilde KO(S^8) of the 8-sphere; the bundle
H^8 over S^8 is obtained by the standard glueing
process along the equator S^7, using the octonion
multiplication. So H^8 is the octonion Hopf bundle.
Its Thom space is the projective Cayley plane
OP^2. Using this and Hirzebruch's signature theorem,
one sees that the Pontrjagin class of H^8 is
p_8(H^8)=6x, for a generator x of the 8-dimensional
integral cohomology of S^8 [a reference for this
calulation is my paper 'The topology of smooth
projective planes', Arch. Math 63 (1994)].
We have a diagram
cplx ch
KO(S^8) ---> K(S^8) ---> H(S^8)
the left arrow is complexification, the second arrow
is the Chern character. In dimension 8, these maps form
an isomorphism. Now ch(cplx(H^8))=8+x (see the formula
in the last paragraph in Husemoller's "Fibre bundles",
the chapter on "Bott periodicity and integrality
theorems". The constant factor is unimportant, so the
answer is yes, pi_7(O) is generated by the map
S^7---> O which sends a unit octonion A to the
map l_A:x --> Ax in SO(8).
Linus Kramer