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This result is an orphan of a project with Hopkins and Strickland. I
shall give a description of the elliptic homology of BO<8>, with proofs
for Morava K(1) and K(2). The proof uses pleasant algebro-geometric
formulations of results about bicommutative Hopf algebras over finite
We will show that the Mitchell-Richter filtration of
the space of loops on complex Stiefel manifolds stably
splits. We obtain this result as a special case of a more
general splitting theorem. Another special case is
H. Miller's stable splitting of Stiefel manifolds.
The proof uses Weiss' orthogonal calculus, and it is
very much inspired by Goodwillie's old argument for
a different, but related, general splitting result, which
used his own calculus of homotopy functors. We will use
this opportunity to discuss the (close, but perhaps not
entirely obvious) relationship between Goodwillie's
homotopy calculus and Weiss' orthogonal calculus.
The existence of a homotopy idempotent functor whose
equivalences are precisely the equivalences of a given cohomology
theory has remained unproved since Bousfield settled the same problem
for homology theories in the seventies. This problem is closely
related with the question of whether every continuous, homotopy
idempotent functor can be obtained by inverting a single map in the
sense of Dror Farjoun. We prove that both problems have an
affirmative solution assuming the validity of a suitable
large-cardinal axiom from set theory, namely Vopenka's principle.
This is joint work with Dirk Scevenels and Jeff Smith.
This is a join work with W.G. Dwyer and M. Intermont.
An ordinary CW complex is built by gluing cells in possibly
One can however be more efficient. Stover proved that any space
can be constructed out of spheres in only one step: any space
can be expressed as the realization of a simplicial space whose
values are wedges of spheres. In our work we investigate what
is so special about the spheres. For a space A we ask: when any
A-CW complex can be expressed as a realization of a simplicial
space whose values are wedges of suspensions of A? It turns out
that if this happens, then A has to be Bousfield equivalent to
a Moore space. Thus the higher Morava-Hopkins-Smith type of A is
responsible for the analogue of the Stover's result to be false.
We introduce algebraic methods of studying how complicated
Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. In this paper, we apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.
There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8).
Eigenvectors of psi^2 must also be eigenvectors of psi^k for
any k. The matrix of these eigenvectors is the key to the
analysis. Its determinant is closely related to the homotopy
decomposition of E8 localized at each prime. By taking careful
combinations of eigenvectors, we obtain a set of generators of
K(E8) on which we have a nice formula for all Adams
operations. Bousfield's theorem (and considerable Maple
computation) allows us to obtain from this the v1-periodic