HOMOTOPY METHODS IN ALGEBRAIC TOPOLOGY

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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 fields.

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 many steps. 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 A-CW complexes can be.

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 homotopy groups.

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