UC Berkeley

We explore the generalization of cellular decomposition in chromatically localized stable categories suggested by Picard--graded homotopy groups. In particular, for K(d) a Morava K-theory, we show that the Eilenberg-Mac Lane space K(Z, d+1) has a K(d)-local cellular decomposition tightly analogous to the usual decomposition of infinite-dimensional complex projective space (alias K(Z, 2)) into affine complex cells. Additionally, we identify these generalized cells in terms of classical invariants --- i.e., we show that their associated line bundles over the Lubin-Tate stack are tensor powers of the determinant bundle. (In particular, these methods give the first choice--free construction of the determinantal sphere S[det].) Finally, we investigate the bottom attaching map in this exotic cellular decomposition, and we justify the sense in which it selects a particular K(d)-local homotopy class

Susp^{-1} S[det]^2 --> S[det]

generalizing the classical four "Hopf invariant 1" classes h = (2, eta, nu, sigma) in the cofiber sequences

Susp^{-1} S^(2n) --h-> P_k^1 --> P_k^2

associated to the four normed real division algebras k = (R, C, H, O) with real dimension n = (1, 2, 4, 8). We also include a lengthy introduction to the subject of chromatic homotopy theory, outlining all of the tools relevant to the statements of our original results.