My main research interests are in algebraic topology and group theory, and their interactions with algebra, geometry and combinatorics. In early work with X. Liu [18, 19], we studied the stable homotopy theory of spheres using the May spectral sequence and the Adams spectral sequence. In a paper with X. Liu and W. Li [17], we described differentials in a certain spectral sequence converging to ...

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...

We describe Bott towers as sequences of toric manifolds M, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and com...

When A is a commutative local ring with residue field k, the derived tensor product −⊗ L A k : D(A) → (k −Mod) lifts to a functor taking values in a category of modules over the ‘Tate cohomology’ RHom∗A(k, k), which is the universal enveloping algebra of a certain Lie algebra. Under reasonable conditions this lift satisfies a spectral sequence of Adams (or Bockstein) type. In a suitable categor...

Let M1 n−1 denote the cokernel of the localization map BP∗/I → v−1 n−1BP∗/I, where I denotes the ideal of BP∗ generated by vi’s for 0 ≤ i ≤ n − 2. The chromatic Ext0(M1 n−1) on Γ(m + 1), which we denote Ex0(m, n), is isomorphic to the 0-th line of the E2-term of the Adams-Novikov spectral sequence for computing the homotopy groups of a spectrum, whose BP∗-homology is M1 n−1 ⊗BP∗ BP∗[t1, t2, . ....

Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP -homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and whilst η : S1,1 → S0,0 induces zero on BP -homology, it is nonnilpotent. Recent work with Haynes Miller has led to a calculation of ηπ∗...

Introduction. In this paper, we derive formulæ in Brown-Peterson homology at the prime 2 related to the family of elements φn ∈ MSp8n−3 of N. Ray, whose central rôle in the structure of MSp has been highlighted by recent work of V. Vershinin and other Russian topologists. In effect, we give explicit “chromatic” representatives for these elements, which were known to be detected in KO and mod 2 ...