THE HOPKINS–MILLER THEOREM

John Rognes

March 24th 1999

Lecture 16. The Hopkins–Miller theorem

16.1. Operads.

An operad C is a sequence of spaces C(n) for each integer n ≥ 0, equipped with (i) a right action of Σ_{n} on C(n), (ii) a chosen unit element 1 ∈ C(1), and (iii) for each k ≥ 0 and each sequence (i_{1}, …, i_{k}) of non-negative integers, a map

γ: C(k) × C(i_{1}) × ⋯ × C(i_{k}) → C(n)

where n = i_{1} + … i_{k}. These are subject to equivariance, unitality and associativity axioms, which we omit.

A map of operads C → D is a sequence of maps C(n) → D(n), compatible with the chosen structures. A product of operads C × D is defined levelwise, with nth space C(n) × D(n), etc.

We think of the points in C(n) as abstract operations taking n inputs and yielding one output. The Σ_{n}-action describes the result of reordering the inputs, the unit 1 describes the identity operation, and the map γ defines composition of operations.

Let U be a universe, i.e., a countably infinite dimensional inner product space. The linear isometries operad L = L_{U} has nth space L(n) equal to the space of linear isometries U^{n} → U. Here U^{n} denotes the direct sum of n copies of U. Such linear isometries are necessarily injective, but need not be surjective. The group Σ_{n} acts by permuting the summands in the source U^{n}. The unit 1 is the identity map of U, and the composition law γ is given by direct sum followed by composition of linear isometries. Each space L(n) is contractible.

The associative operad A has nth space A(n) = Σ_{n}, with the usual right Σ_{n}- action. The unit 1 is the identity in Σ_{1}, and γ concatenates the permutations in Σ_{i_{1}} × ⋯ × Σ_{i_{k}} to give an element in Σ_{n}, and then shuffles {1, …, n} divided into k blocks with i_{1}, …, i_{k} elements, respectively, as specified by the permutation in Σ_{k}.

An A_{∞} operad C is an operad for which there is a map C → A to the associative operad which is a level-wise homotopy equivalence, i.e., the maps C(n) → A(n) = Σ_{n} are homotopy equivalences for all n. The name means homotopy Associative to all orders.

An E_{∞} operad C is an operad for which each space C(n) is (non-equivariantly) contractible. The name means homotopy Everything (associative and commutative) to all orders.

The linear isometries operad L is an E_{∞} operad. Its product with the associative operad A is an A_{∞} operad, with nth space L(n) × Σ_{n}.

Typeset by AMS-TEX

16.2. Operads acting on spaces.

Let C be an operad and X a space. An action of C on X is given by a sequence of maps

ξ: C(n) × X^{n} → X

for all n ≥ 0, which are suitably compatible with the Σ_{n}-action, the unit element, and the composition law. We think of ξ as specifying how points in C(n) act as operations on X taking n inputs and yielding one output. A space with an action by C is also known as a C-algebra.

A map X → Y of C-algebras is required to commute with the structure maps ξ. When C is an A_{∞} operad, a C-algebra X is called an A_{∞} space. Then X

admits a pairing X × X → X, which is homotopy associative and satisfies all higher associativity conditions, in the sense of Stasheff. Likewise with E_{∞} operads and E_{∞} spaces, which admits homotopy associative and commutative pairings, including all higher coherence conditions.

Examples of A_{∞} spaces include all loop spaces, and all (associative) topological monoids. Examples of E_{∞} spaces include all infinite loop spaces, and all commutative topological monoids.

The forgetful functor from C-algebras to spaces has a left adjoint, taking a space X to the space

C(X) := ∐_{n≥0} C(n) ×_{Σ_{n}} X^{n} .

Then C(X) is acted upon by C, using the composition maps γ to define the structure maps ξ. We call C(X) the free C-algebra on X.

16.3. Smash products of spectra.

Let U be a universe. A prespectrum E over U is a collection of based spaces E_{V} for each finite dimensional subspace V ⊂ U, and based structure maps Σ^{W −V} E_{V} → E_{W} for all pairs V ⊆ W of finite dimensional subspaces of U. Here W − V denotes the orthogonal complement of V in W, S^{W −V} is its one-point compactification, and Σ^{W −V} E_{V} = E_{V} ∧ S^{W −V}. A map of prespectra E → F is a collection of based maps E_{V} → F_{V} for all finite dimensional subspaces V ⊂ U, which commute with the given structure maps. Let PU denote the category of prespectra over U.

A spectrum E over U is a prespectrum such that the adjoint structure maps E_{V} → Ω^{W −V} E_{W} are homeomorphisms for all pairs V ⊆ W. Here Ω^{W −V} E_{W} = F (S^{W −V} , E_{W}) is the based mapping space. Let SU be the full subcategory of spectra in PU.

The forgetful functor ℓ: SU → PU admits a left adjoint, namely the spectrification functor L: PU → SU which converts a prespectrum into a spectrum. When E is a prespectrum such that each adjoint structure map E_{V} → Ω^{W −V} E_{W} is an embedding, then its spectrification L(E) has V th space

L(E)_{V} := colim_{V ⊆W} Ω^{W −V} E_{W}

where the colimit runs over the finite dimensional subspaces W ⊂ U containing V. (In general the spectrification functor is given by a transfinite repetition of this procedure, and is typically not homotopy invariant.)

Given universes U and U′ and prespectra E and E′ over U and U′ respectively, the external smash product E ∧ E′ ∈ P(U ⊕ U′) is the prespectrum whose value at a finite dimensional subspace W ⊂ U ⊕ U′ is

Σ^{W −V ⊕V′}E_{V} ∧ E′_{V′}

where V ⊂ U and V′ ⊂ U′ are maximal subspaces such that V ⊕ V′ ⊆ W. For spectra E ∈ SU and E′ ∈ SU′, we define the spectrum E ∧ E′ ∈ S(U ⊕ U′) to be L(ℓ(E) ∧ ℓ(E′)).

For a linear isometry f: U → U′ there is a push-forward change of universe functor f_{∗}: PU → PU′ taking a prespectrum E over U to the prespectrum f_{∗}(E) whose value at a finite dimensional subspace W′ ⊂ U′ is

Σ^{W′−f(V)}E_{V}

where V ⊂ U is maximal such that f(V) ⊆ W. For a spectrum E ∈ SU let f_{∗}(E) ∈ SU′ be L(f_{∗}(ℓ(E))).

Iterating, we can form the n-fold external smash product E^{∧n} ∈ SU^{n}, and its push-forward f_{∗}(E^{∧n}) ∈ SU, for any given spectrum E ∈ SU and linear isometry f: U^{n} → U.

Let C be an operad over the linear isometries operad, i.e., one equipped with a map C → L. Then each point in C(n) determines a linear isometry U^{n} → U, for which push-forward of the n-fold external smash product is a spectrum over U. These constructions may be continuously glued together for all points in C(n), yielding a functorial construction of a spectrum

C(n) ⋊ E^{∧n} ∈ SU

called the twisted half-smash product.

16.4. A_{∞} ring spectra.

Let U be a universe, let C be an operad over L = L_{U}, and let E be a spectrum over U. An action of C on E is a sequence of spectrum maps

ξ: C(n) ⋊ E^{∧n} → E

in SU, subject to suitable compatibility conditions. We also call E a C-algebra spectrum.

If C is an A_{∞} operad, we say that E is an A_{∞} ring spectrum. Then a point in C(2) determines a map

E ∧ E := f_{∗}(E ∧ E) → E

in SU, which is the multiplication making E a (homotopy associative) ring spectrum. In fact the multiplication then satisfies all higher associativity conditions, as in the space level case. If C is an E_{∞} operad, then we say that E is an E_{∞} ring spectrum, which in particular is a (homotopy associative and) commutative ring spectrum.

A map E → F of A_{∞} ring spectra is required to commute with the structure maps ξ. We denote the category of A_{∞} ring spectra and A_{∞} ring spectrum maps by A_{∞}.

The category of A_{∞} ring spectra is equivalent to the category of 𝕊-algebras in the sense of Elmendorf, Kriz, Mandell and May, i.e., the category of monoids in their symmetric monoidal category of 𝕊-modules. The category of E_{∞} ring spectra is equivalent to the subcategory of commutative 𝕊-algebras, i.e., the category of commutative monoids in 𝕊-modules.

Examples of A_{∞} ring spectra include the Thom spectra induced by loop maps X → BG, the algebraic K-theory spectra of monoidal categories, and the Morava K-theories K(n).

Examples of E_{∞} ring spectra include the Thom spectra induced by infinite loop maps X → BG, the algebraic K-theory spectra of symmetric monoidal categories, and the sphere spectrum S.

16.5. The theorem.

Let FGL be the category of pairs (k, Γ), where k is a perfect field of characteristic p > 0, and Γ is a (commutative, one-dimensional) formal group law over k, of height n < ∞. The maps (k_{1}, Γ_{1}) → (k_{2}, Γ_{2}) are pairs (i, f) where i: k_{2} → k_{1} is a field homomorphism and f: Γ_{1} → i_{∗}(Γ_{2}) is an isomorphism of formal group laws.

Recall that with these notations there is a universal deformation of (k, Γ) to a formal group law F = F (k, Γ) defined over a complete local ring E(k, Γ) ≅ 𝕎k[[u_{1}, …, u_{n−1}]] with residue field k, such that π_{∗}(F) = Γ. Here 𝕎k denotes the Witt ring of k, and π: E(k, Γ) → k is the residue class homomorphism.

Theorem (M. Hopkins and H. Miller). There is a functor (k, Γ) ↦ E_{(k,Γ)} from FGL^{op} to the category A_{∞} of A_{∞} ring spectra, such that:

(1) E_{(k,Γ)} is a commutative ring spectrum.

(2) π_{i}(E_{(k,Γ)}) = 0 for i odd and π_{2}(E_{(k,Γ)}) contains a unit. Thus E_{(k,Γ)} is complex orientable.

(3) The corresponding formal group law over π_{0}(E_{(k,Γ)}) is isomorphic to the universal deformation of (k, Γ).

When k = 𝔽_{p^{n}} and Γ = F_{n} is the unique p-typical formal group law defined over 𝔽_{p}, with p-series [p]_{F_{n}}(x) = x^{p^{n}} (the Honda formal group law), then E_{(k,Γ)} is briefly called E_{n}. It homotopy is

π_{∗}(E_{n}) ≅ 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u, u^{−1}]

where deg(u_{i}) = 0 and deg(u) = 2.

The theorem then asserts that E_{n} is an A_{∞} ring spectrum, and that it admits an action through A_{∞} ring spectrum maps by the group of automorphisms of (𝔽_{p^{n}}, F_{n}) in the category FGL.

Let S_{n} be the Morava stabilizer group, i.e., the group of strict automorphisms over 𝔽_{p^{n}} of the formal group law F_{n}. This is a pro-p-group, with a corresponding totally disconnected topology. Then contained in the group of automorphisms of (𝔽_{p^{n}}, F_{n}) in FGL we have the subgroup of maps (i, f) where f is a strict isomorphism, and i is a Galois automorphism of 𝔽_{p^{n}} over the field of definition 𝔽_{p} of F_{n}. Thus there is a semi-direct extension of groups

S_{n} → S_{n} ⋊ C_{n} → C_{n}

acting on E_{n} in A_{∞}.

The Morava change of rings theorem can be interpreted as saying that the homotopy fixed points for this action of S_{n} ⋊ C_{n} on E_{n} is homotopy equivalent to the Bousfield K(n)-localization L_{K(n)}S of the sphere spectrum.

When S_{n} ⋊ C_{n} has a maximal finite subgroup (up to conjugacy) G_{n}, we let

EO_{n} = E_{n}^{hG_{n}}

be the homotopy fixed points for this action.

For n = 1, E_{1} = KU is p-completed complex K-theory. For p odd, EO_{1} is the p-completed Adams summand L ⊂ KU, while for p = 2, EO_{1} = KO is 2-completed real K-theory.

For n = 2 and p = 2, the maximal finite subgroup of S_{2} is the binary tetrahedral group G_{24} = Q_{8} ⋊ C_{3} previously considered. Its extension by C_{2} is a group G_{48} = Q_{8} ⋊ Σ_{3}, which up to conjugacy is the maximal finite subgroup of S_{2} ⋊ C_{2}. We are interested in EO_{2} as an approximation to L_{K(2)}S.

16.6. The construction.

The category of spectra SU is a topological category, i.e., it is enriched in based spaces. For spectra E and F, the space of maps SU(E, F) is a subspace of the product space

∏_{V} F (E_{V} , F_{V})

where V runs over the finite dimensional subspaces of U. The category of spectra SU admits the structure of a topological closed model category. This means there are subcategories of cofibrations, weak equivalences (i.e, π_{∗}-isomorphisms) and fibrations satisfying the closed model category axioms of Quillen, in a manner compatible with the topological enrichment. (The notion of a simplicial closed model category is perhaps a little more familiar.)

Specifically, for a cofibration i: A → X and a fibration p: E → B, the map

f = i^{#} × p_{#} : SU(X, E) ⟶ SU(A, E) ×_{SU(A,B)} SU(X, B)

is a Serre fibration. If i or p is a weak equivalence, then so is f.

Likewise the category A_{∞} of A_{∞} ring spectra is a topological category. For A_{∞} ring spectra E and F, the space of A_{∞} ring maps A_{∞}(E, F) is a subspace of SU(E, F). To be precise, we should have a specific A_{∞} operad C over L_{U} in mind, and consider the category of C-algebra spectra as our model for A_{∞}.

According to Mike Hopkins, the category of C-algebra spectra admits the structure of a topological closed model category. Also according to Hopkins, the associated homotopy category of C-algebra spectra is independent up to equivalence of the choice of A_{∞} operad C.

There is also a model category of simplicial operads. A cofibrant A_{∞} operad C has the property that the underlying spectrum of any cofibrant C-algebra spectrum E is also cofibrant as a spectrum.

Let A_{∞}^{LT} be the full subcategory of A_{∞} with objects the A_{∞} ring spectra E such that:

(1) E is cofibrant as an A_{∞} ring spectrum, i.e., the unit map S → E is a cofibration in the model category structure.

(2) E is a commutative ring spectrum.

(3) π_{i}(E) = 0 for i odd, and π_{2}(E) contains a unit.

(4) π_{0}(E) is a complete local ring with maximal ideal 𝔪, such that the residue field π_{0}(E)/𝔪 is a perfect field of positive characteristic.

(5) The degree 0 formal group law F over π_{0}(E), which exists because E is complex orientable by (3), is a universal deformation of its reduction mod 𝔪.

Thus there is a functor π: A_{∞}^{LT} → FGL^{op} taking E to (π_{0}(E)/𝔪, F). This is the functor we would like to invert.

Theorem. Let C be a cofibrant A_{∞} operad over L. The topological functor

π: A_{∞}^{LT} → FGL^{op}

is fully faithful, i.e., induces a weak equivalence on mapping spaces

A_{∞}(E, F) → FGL(π(F), π(E))

for all objects of A_{∞}^{LT}. Here FGL(π(F), π(E)) is viewed as a discrete space.

Theorem. Let C be a cofibrant A_{∞} operad over L. For each formal group law (k, Γ) in FGL the Morava ring spectrum E_{(k,Γ)} admits an A_{∞} structure (as a C-algebra spectrum) compatible with its usual ring structure.

Hence the functor π: A_{∞}^{LT} → FGL^{op} is a weak equivalence of topological categories. Fix small models for FGL and for A_{∞}^{LT}. Then perform the homotopy right Kan extension of the topological inclusion functor i: A_{∞}^{LT} → A_{∞} along the weak equivalence π: A_{∞}^{LT} → FGL^{op}. The result is the desired functor ρ: FGL^{op} → A_{∞}.

The homotopy right Kan extension ρ takes a formal group law (k, Γ) to the homotopy limit of the functor from the comma category with objects (E, α) with E in A_{∞}^{LT} and α a map in FGL^{op} from (k, Γ) to π(E), to A_{∞}, whose value at (E, α) is E. (Did I get the directions right ?)

Since π is a weak equivalence of topological categories, ρ has the desired properties. In particular if (k, Γ) = π(E) then ρ(k, Γ) is weakly equivalent to E in A_{∞}.