On the Topological Hochschild Homology of bu, I

Hopf algebra structure on topological Hochschild homology

“…It is well known that HF p and HZ (p) admit unique structures as commutative S -algebras, and that the p-complete Adams summand ℓ p ⊂ ku p admits at least one such structure [MS93,§9]. In fact each of ℓ, ku, ℓ p and ku p admits a unique commutative S -algebra structure [BaRi].…”

“…See [May77, VIII.3.2] and [MS93,§9]. Similarly, κ : j → ko becomes a map of commutative S -algebras after 2-adic completion, since there is a discrete model KO(F 3 ) 2 for ko 2 , and κ can be identified with the natural map…”

“…The first case is the analogue for p = 2 of the discussion in [MS93,[5][6][7], where McClure and Staffeldt compute the mod p homotopy groups π * (THH (ℓ); Z/p) of the Adams summand ℓ ⊂ ku (p) for odd primes p.…”

“…The resulting skeleton filtration on THH (R) gives rise to the Bökstedt spectral sequence in homology E 2 * * (R) = HH * (H * (R; F p )) =⇒ H * (THH (R); F p ) , as we explain in Section 4. Compare [BM94,§2] and [MS93,3.1]. But the coproduct and conjugation maps are not induced by simplicial maps in this model, so some adjustment is needed in order for these structures to carry over to the spectral sequence.…”

“…In Section 8 we follow the strategy of [MS93] to compute the homotopy groups of THH (ku) ∧ M , where M = C 2 is the mod 2 Moore spectrum, and of THH (ko)∧Y , where Y = C 2 ∧C η is the 4-cell spectrum employed by Mahowald [Mah82]. The results appear in Theorems 8.13 and 8.14, respectively.…”