Wilson Spaces, Snaith Constructions, and Elliptic Orientations

Abstract

We construct a canonical family of even periodic E∞-ring spectra, with exactly one member of the family for every prime p and chromatic height n. At height 1 our construction is due to Snaith, who built complex K-theory from CP ∞ . At height 2 we replace CP ∞ with a p-local retract of BU 6 , producing a new theory that orients elliptic, but not generic, height 2 Morava E-theories.In general our construction exhibits a kind of redshift, whereby BP n − 1 is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K(n)-localization of our height n ring to work of Peterson and Westerland building E hSG ± n from K(Z, n + 1).Theorem 1.5. Suppose that E k, C is an elliptic Morava E-theory associated to a supersingular elliptic curve C over a perfect field k of characteristic 2. Then there is a natural homotopy equivalence between(1) The space of E ∞ -ring maps MSU → E k, C , and(2) The subspace of E ∞ -ring maps R → E k, C that respect the Weil pairing on C.The subspace in (2) is the union of a collection of path components in the space of E ∞ -ring homomorphisms R → E k, C . We clarify what it means to respect the Weil pairing in the more precisely stated Definition 7.5 and Theorem 7.6.Remark 1.6. There is geometric interest in E ∞ -ring maps out of MSU, since such maps represent highly structured invariants of manifolds up to bordism.Remark 1.7. Morava E-theories that are not elliptic often fail to receive even homotopy commutative ring maps from R (cf. Example 6.14). In Corollary 6.27 we determine exactly which Morava E-theories receive such homotopy ring maps.1.1. The construction at a general prime and height.Convention 1.8. For the remainder of this paper, we fix a prime number p and work in the p-local category. That is to say, all spectra and simply connected spaces are implicitly p-localized.Convention 1.9. If E is a spectrum and n an integer, we sometimes use the notationConvention 1.10. For each integer h ≥ 0, we let ν(h) denote the integerDefinition 1.11. For each nonnegative integer h, let BP h denote the spectrum obtained from BP by quotienting out the Hazewinkel generators v h+1 , v h+2 , . . . ∈ π * (BP).We define W h = Ω ∞ Σ 2ν(h) BP h = BP h 2ν(h) . Steve Wilson proved [Wil75] that any choice of generators v h+1 , v h+2 , . . . yields the same space W h -we choose the Hazewinkel generators solely for concreteness (cf. Remark 2.6).Remark 1.12. Note that W 0 ≃ CP ∞ at all primes p, while W 1 ≃ BU 6 only for p = 2. At odd primes W 1 is a retract of BU 6 .

Stability of Loday constructions

“…We show that really smooth maps satisfy stability. Other examples where stability holds are Thom spectra as well as S → KU and other spectra of the form [5]. Using Galois descent we also obtain stability for S → KO.…”

“…satisfies étale descent, and therefore the composite In [5] Hood Chatham, Jeremy Hahn, and Allen Yuan construct interesting examples of E ∞ -ring spectra. For a prime p they consider the infinite loop space…”

“…, which has remarkable features [5,Theorem 1.13]: R h has torsion-free homotopy groups that vanish in odd degrees, it is Landweber exact, and its Morava-K…”

Algebraic theories of power operations

“…Example. In Chatham-Hahn-Yuan [CHY19], an interesting family of E ∞ ring spectra R h−1 at chromatic height h are constructed, and left open is the question of whether there exists an E ∞ map R h−1 → E, where E is a Lubin-Tate spectrum of height h. Combining [CHY19, Theorem 7.6] with the preceding, we learn that there are E ∞ maps R 1 → E whenever E is a Lubin-Tate spectrum of height 2 associated to a supersingular elliptic curve. 6.5.7.…”