UNSTABLE MODULES AND SULLIVAN’S CONJECTURE

John Rognes

May 1998

Steenrod operations. We consider mod p cohomology, and let q = 2p−2. Steenrod constructed natural transformations (operations)

Sq^{i}: H^{n}(X; 𝔽_{2}) ⟶ H^{n+i}(X; 𝔽_{2})

(Steenrod squares) for p = 2, and

P^{i}: H^{n}(X; 𝔽_{p}) ⟶ H^{n+qi}(X; 𝔽_{p})

(reduced powers) for odd primes p. Together with the mod p Bockstein operation β = β_{p} associated to the short exact sequence 0 → 𝔽_{p} → ℤ/p^{2} → 𝔽_{p} → 0, these generate all the stable cohomology operations in mod p cohomology, i.e., all those operations that commute with suspensions.

We have Sq^{0} = 1 and Sq^{1} = β when p = 2, while P^{0} = 1 for p odd.

The Steenrod algebra A. The mod 2 Steenrod algebra A = A(2) is the (associative unital) graded 𝔽_{2}-algebra generated by elements Sq^{i} for i > 0, modulo the two-sided ideal generated by the Adem relations

Sq^{a}Sq^{b} = ∑_{i=0}^{[a/2]} \binom{b − i − 1}{a − 2i}Sq^{a+b−i}Sq^{i}

for all a, b > 0 such that a < 2b. Here Sq^{0} is equal to the unit 1. The generator Sq^{i} has degree i.

For odd primes p, the mod p Steenrod algebra A = A(p) is the (associative unital) 𝔽_{p}-algebra generated by elements β and P^{i} for i > 0, modulo the two-sided ideal generated by β^{2} = 0 and the Adem relations

P^{a}P^{b} = ∑_{i=0}^{[a/p]}(−1)^{a+i}\binom{(p − 1)(b − i) − 1}{a − pi}P^{a+b−i}P^{i}

for all a, b > 0 such that a < pb, and

P^{a}βP^{b} = ∑_{i=0}^{[a/p]}(−1)^{a+i}\binom{(p − 1)(b − i)}{a − pi}βP^{a+b−i}P^{i}

+ ∑_{i=0}^{(a−1)/p}(−1)^{a+i−1}\binom{(p − 1)(b − i) − 1}{a − pi − 1}P^{a+b−i}βP^{i}

Typeset by AMS-TEX

for all a, b > 0 such that a ≤ pb. Here P^{0} is equal to the unit 1. The generator β (Bockstein) has degree 1, and the generator P^{i} has degree qi = 2i(p − 1).

The classes Sq^{2^{s}} for s ≥ 0 generate A(2) as an algebra, and the classes β and P^{p^{s}} for s ≥ 0 generate A(p) as an algebra.

Let p = 2. For a sequence of natural numbers I = (i_{1}, …, i_{n}) let

Sq^{I} = Sq^{i_{1}} … Sq^{i_{n}} ∈ A(2) .

The sequence I is admissible if i_{s} ≥ 2i_{s+1} for all s ≥ 1. The set of admissible monomials Sq^{I} form a basis for A(2) as a vector space.

Let p be odd. For a sequence of integers I = (ϵ_{0}, i_{1}, …, i_{n}, ϵ_{n}) where the ϵ_{s} are 0 or 1 and the i_{s} are positive, let

P^{I} = β^{ϵ_{0}}P^{i_{1}}β^{ϵ_{1}} … P^{i_{n}}β^{ϵ_{n}} ∈ A(p) .

The sequence I is admissible if i_{s} ≥ pi_{s+1} + ϵ_{s} for all s ≥ 1. The admissible monomials P^{I} form a basis for A(p) as a vector space.

The mod p cohomology H^{∗}(X; 𝔽_{p}) of any space or spectrum X is a graded left A(p)-module, with Sq^{I} (resp. P^{I}) acting by the composite of Steenrod’s operations with the same name. Hereafter we write H^{∗}(X) for H^{∗}(X; 𝔽_{p}) and A for A(p).

The category U of unstable A-modules. The mod p cohomology of a space X satisfies a further instability condition. For p = 2 it is Sq^{i}(x) = 0 when i > deg(x), for p odd it is β^{ϵ}P^{i}(x) = 0 when ϵ + 2i > deg(x).

Definition. A graded A-module M is unstable if it satisfies the instability condition above. Let the category U of unstable A-modules be the full subcategory of the category of graded A-modules, with objects the unstable modules.

The cohomology of a space is an unstable A-module. The cohomology of a spectrum is generally not unstable.

The Hopf algebra structure. The cohomology of a space X is a graded commutative unital 𝔽_{p}-algebra, with respect to the cup product xy = x ∪ y. The product and A-module structure are related: For any x, y ∈ H^{∗}(X) we have the Cartan formulas:

Sq^{k}(xy) = ∑_{i+j=k} Sq^{i}(x)Sq^{j}(y)

for p = 2 and

P^{k}(xy) = ∑_{i+j=k} P^{i}(x)P^{j}(y)
β(xy) = β(x)y + (−1)^{deg(x)}xβ(y)

for p odd. Milnor showed that the coproduct homomorphism Δ: A → A⊗A defined by

Δ(Sq^{k}) = ∑_{i+j=k} Sq^{i} ⊗ Sq^{j}

for p = 2 and

Δ(P^{k}) = ∑_{i+j=k} P^{i} ⊗ P^{j}
Δ(β) = β ⊗ 1 + 1 ⊗ β

makes A a Hopf algebra. This lets us define an A-module structure on the tensor product of two A-modules (over 𝔽_{p}), and the Cartan formulas assert that the cup product

H^{∗}(X) ⊗ H^{∗}(X) ⟶^{∪} H^{∗}(X)

is A-linear. Thus the cohomology H^{∗}(X) of any space is an A-algebra.

The category K of unstable A-algebras.

The mod p cohomology of a space satisfies a further instability condition. For p = 2 it is

Sq^{i}(x) = x^{2} for i = deg(x),

and for p odd it is

P^{i}(x) = x^{p} for 2i = deg(x).

Definition. A commutative unital graded A-algebra K is unstable if it satisfies the instability condition above. Let the category K of unstable A-algebras be the full subcategory of the category of graded A-algebras, with objects the unstable algebras.

The cohomology of a space is an unstable A-algebra.

A basic example. Let V = (ℤ/p)^{d} be an elementary abelian p-group, i.e., a finite dimensional 𝔽_{p}-vector space. The mod p cohomology of the classifying space BV is also the group cohomology of V, and denoted H^{∗}(V).

By the Künneth theorem, H^{∗}(V) ≅ H^{∗}(ℤ/p)^{⊗d}.

When p = 2, H^{∗}(ℤ/2) ≅ ℤ/2[x] is polynomial on a generator x of degree 1. The A-module structure is given by

Sq^{i}(x^{n}) = \binom{n}{i}x^{n+i} .

In general H^{∗}(V) ≅ S(V^{∗}) is the symmetric algebra on the dual of V in degree 1.

When p is odd, H^{∗}(ℤ/p) ≅ 𝔽_{p}[x, y]/(x^{2} = 0) is the tensor product of an exterior algebra on a generator x of degree 1 and a polynomial algebra on a generator y of degree 2. The A-module structure is given by β(x) = y and

P^{i}(y^{n}) = \binom{n}{i}y^{n+i(p−1)} .

In general H^{∗}(V) ≅ E(V^{∗}) ⊗ S(V^{∗}) is the exterior algebra on the dual of V in degree 1, tensored with the symmetric algebra on the dual of V in degree 2.

An A-algebra homomorphism from H^{∗}(V) is determined by its behavior in degree 1. Thus the map

Hom(V, W) ⟶ Hom_{K}(H^{∗}(W), H^{∗}(V))

taking f: V → W to f^{∗} is a bijection.

More generally a map of spaces f: X → Y induces a function

f^{∗}: Hom_{K}(H^{∗}(Y), H^{∗}(X)) .

When is this set in bijection with [X, Y] ?

Free objects F (n) in U. The category U is an abelian category. (The morphism sets are abelian groups, and morphisms have well-behaved kernels and cokernels.) We shall now see that it has enough projectives. We can therefore do classical homological algebra in U, i.e., define Ext_{U}^{s}(M, N).

Let M = (M^{n})_{n} be an unstable A-module. The functor M ↦ M^{n} from U to the category E of 𝔽_{p}-vector spaces, is co-representable. Hence there is, up to isomorphism, a unique unstable A-module F (n) such that there is a natural isomorphism

Hom_{U}(F (n), M) ≅ M^{n} .

Since M ↦ M^{n} is exact, it follows that F (n) is projective in the abelian category U.

The module F (n) can be explicitly constructed. For p = 2 let I = (i_{1}, …, i_{n}) be an admissible sequence. The excess of I is the sum

e(I) = ∑_{s}(i_{s} − 2i_{s+1}) = i_{1} − i_{2} − ⋯ − i_{n} .

Given n the sub-vector space of A generated by the Sq^{I} with e(I) > n is a left ideal. Similar definitions apply for odd p.

Proposition 1.6.2. The unstable A-module F (n) is the cyclic A-module on one generator ι_{n} in degree n, modulo the ideal of admissible monomials with e(I) > n. For p = 2 this is

F (n) = Σ^{n}A/𝔽_{2}{Sq^{I} | e(I) > n} .

The admissible monomials with e(I) ≤ n form a vector space basis for F (n). Thus F (n) is of finite type, i.e., finite dimensional in each degree.

For example, F (0) = 𝔽_{p}, while F (1) = 𝔽_{2}{x, x^{2}, x^{4}, x^{8}, … } for p = 2, and F (1) = 𝔽_{p}{x, y, y^{p}, y^{p^{2}}, … } for p odd.

The unstable module F (1)^{⊗n} has dimension 1 in degree n. Thus there is a nontrivial map F (n) → F (1)^{⊗n}. If p = 2 the image of ι_{n} in F (1)^{⊗n} is Σ_{n}-invariant. Hence there is a map of unstable A-modules

F (n) ⟶ (F (1)^{⊗n})^{Σ_{n}} .

This map is an isomorphism.

A representability lemma. To construct injective modules in U, we shall represent exact contravariant functors U → E.

Lemma 2.2.1. A (contravariant) functor R: U^{op} → E is representable if and only if it is right exact and takes direct sums to products.

An unstable representing module B(R) must satisfy

R(M) ≅ Hom_{U}(M, B(R))

for all M. With M = F (n) this says R(F (n)) ≅ Hom_{U}(F (n), B(R)) ≅ B(R)^{n}, so evaluating R on the F (n) determined the degree n part of the representing module B(R).

Brown–Gitler modules.

The functor H_{n}: U → E given by

H_{n}(M) = Hom_{E}(M^{n}, 𝔽_{p}) = M^{n∗}

is representable, by the lemma above. The notation is such that H_{n}(H^{∗}(X)) = H_{n}(X) for spaces X of finite type. This functor is right exact and takes direct sums to products.

Definition. The nth Brown–Gitler module J(n) is the representing unstable Amodule for the functor H_{n}. There is a natural isomorphism

H_{n}(M) ≅ Hom_{U}(M, J(n)) .

As H_{n} is also left exact, this module J(n) is in fact injective in the abelian category U. Any unstable A-module injects into a product of J(n)’s, so U has enough injectives.

We have J(n)^{m} ≅ Hom_{U}(F (m), J(n)) ≅ F (m)^{n∗}, which determines the groups J(n)^{m}. Since F (m) is (m − 1)-connected, it follows that J(n) is concentrated in degrees 0 ≤ ∗ ≤ n. Thus J(n) is finite.

For example, J(0) = 𝔽_{p} in degree 0 is injective in U.

Carlsson modules. Carlsson constructed certain unstable A-modules K(i) as sequential limits of the J(n)’s:

K(i) = lim_{s}(⋯ → J(2^{s}i) → J(2^{s−1}i) → ⋯ → J(i)).

These are also injective in U for general reasons.

Carlsson showed that for p = 2 the unstable A-module H^{∗}(ℤ/p) is a direct summand of K(1), and thus injective. Miller extended this to odd p, and Lannes and Zarati extended the injectivity assertion to H^{∗}(V) for general V = (ℤ/p)^{d}.

U-injectivity of H^{∗}(V) ⊗ J(n).

Theorem 3.1.1 (Carlsson, Miller, Lannes, Zarati). Let V be an elementary abelian p-group. Then the unstable A-module H^{∗}(V) ⊗ J(n) is injective in U for all n.

Lannes’ functor T_{V}. Let L be an unstable A-module of finite type. The following is a consequence of Freyd’s adjoint functor theorem.

Theorem 3.2.1 (Lannes). The functor M ↦ L ⊗ M from U to itself has a left adjoint denoted N ↦ (N : L)_{U}. There is a natural isomorphism

Hom_{U}((N : L)_{U}, M) ≅ Hom_{U}(N, L ⊗ M) .

We call N ↦ (N : L)_{U} the division by L functor.

Definition. Lannes’ functor T_{V} is the division by H^{∗}(V) functor:

T_{V} (N) = (N : H^{∗}(V))_{U} .

We write T = T_{𝔽_{p}}, so for V = (ℤ/p)^{d} we have T_{V} = T^{d} (the d-fold composition). There is an adjunction

Hom_{U}(T_{V} (N), M) ≅ Hom_{U}(N, H^{∗}(V) ⊗ M) .

6 JOHN ROGNES

Theorem 3.2.2. The functor T_{V} is exact. Proof. This is implied by the U-injectivity of H^{∗}(V) ⊗ J(n). □ First computations. Proposition 3.3.2. The functor T_{V} commutes with colimits.

This holds for any left adjoint. Proposition 3.3.2. The functor T_{V} commutes with suspensions. Sketch proof. The suspension functor Σ in U has a right adjoint, which commutes with tensor product with H^{∗}(V). □

An unstable A-module M is locally finite if each element x ∈ M is contained in a finite A-submodule. Thus M is a colimit of finite A-modules.

The split injection 𝔽_{p} → H^{∗}(V) induces a split surjection T_{V} (M) → M. Proposition 3.3.6. Let M be a locally finite unstable A-module. Then T_{V} (M) ≅ M. Proof. T_{V} (𝔽_{p}) ≅ 𝔽_{p} since H^{0}(V) ≅ 𝔽_{p}. By exactness T_{V} (M) ≅ M for any finite M, by induction over the dimension of M. By passage to colimits, the same holds for any locally finite M. □

For example, T_{V} (H^{∗}(X)) ≅ H^{∗}(X) for any finite dimensional CW-complex X. T_{V} and tensor products. Lannes’ functor T_{V} commutes with tensor products. Theorem 3.5.1 (Lannes). There is a natural isomorphism

T_{V} (M ⊗ N) ≅ T_{V} (M) ⊗ T_{V} (N)

for unstable A-modules M and N.

The map arises by a chain of adjunctions from the product H^{∗}(V) ⊗ H^{∗}(V) → H^{∗}(V). T_{V} and unstable algebras. If K is an unstable A-algebra, so is T_{V} (K). The product is given by

T_{V} (K) ⊗ T_{V} (K) ≅ T_{V} (K ⊗ K) ⟶ T_{V} (K)

using the isomorphism of 3.5.1 and the product K ⊗ K → K. Theorem 3.8.1 (Lannes). Let K, L be unstable A-algebras. The unstable Amodule T_{V} (K) is in a natural way an unstable A-algebra, and there is a natural isomorphism

Hom_{K}(T_{V} (K), L) ≅ Hom_{K}(K, H^{∗}(V) ⊗ L) .

In fact there exists a division functor K ↦ (K : H^{∗}(V))_{K} also in K, which equals T_{V}.

Thus T_{V} : K → K is also a left adjoint in the category of unstable A-algebras. Since T_{V} preserves injections of unstable A-algebras, viewed as unstable A-modules, it follows that H^{∗}(V) is categorically injective in K.

But K is not an abelian category (not even additive), so we cannot do ordinary homological algebra in K. Instead one uses simplicial resolutions and comonadderived functors.

There are then adjunctions

Ext_{K}^{s}(T_{V} (K), L) ≅ Ext_{K}^{s}(K, H^{∗}(V) ⊗ L)

and similar formulas for the derived functors of suitable groups of derivations, rather than homomorphisms.

Cosimplicial spaces. A cosimplicial space X^{•} is a (covariant) functor X: Δ → Spaces, where we by spaces typically mean simplicial sets. It is a sequence of spaces [q] ↦ X^{q}, together with coface and codegeneracy maps.

The cosimplicial space Δ^{•} is given by [q] ↦ Δ^{q}, with the usual coface and codegeneracy maps. The totalization Tot X^{•} of a cosimplicial space is the mapping space

Tot X^{•} = Map(Δ^{•}, X^{•}) .

Its p-simplices are the set of maps (= natural transformations) Δ^{p} × Δ^{•} → X^{•}.

Restricting a map from Δ^{•} to the s-skeleton (Δ^{q})^{(s)} ⊆ Δ^{q} in each codegree q, yields a map

Tot X^{•} ⟶ Tot_{s} X^{•}

to the sth partial totalization.

Bousfield–Kan R-completion. Let R be a commutative unital ring, Y a simplicial set (= space). Let R(Y) be the simplicial set which in degree q is the free R-module R{Y_{q}} on the set of q-simplices in Y. There is a monad (= triple) (R, μ, η), with product μ: R(R(Y)) → R(Y) and unit η: Y → R(Y). There is a cosimplicial space [q] ↦ R^{q+1}(Y) denoted R^{•}(Y), with coaugmentation η:

\xymatrix{Y \ar[r] & R(Y) \ar[r] \ar[r] & R(R(Y)) \ar[l] \ar[r] \ar[r] \ar[r] & … \ar[l] \ar[l]}

Then R_{∞}(Y) = Tot R^{•}(Y), and the coaugmentation defines the completion map Y → R_{∞}(Y).

Bousfield–Kan 𝔽_{p}-completion. The map Y → 𝔽_{p∞}(Y) is Bousfield localization with respect to H_{∗}(−; 𝔽_{p}) for virtually nilpotent spaces, i.e., connected spaces Y for which a subgroup of finite index in π_{1}(Y) acts nilpotently on π_{∗}(Y). This includes all connected spaces with finite fundamental group. We often write Y_{p}^{∧} = 𝔽_{p∞}(Y).

Bousfield–Kan homotopy spectral sequence. Let X^{•} be a fibrant and pointed cosimplicial space. Then [s] ↦ π_{t}(X^{s}) defines a cosimplicial group. Its cohomotopy π^{s}π_{t}(X^{•}) is the sth cohomology group of the associated cochain complex.

Get a pointed tower of principal fibrations

Tot X^{•} ⟶ … ⟶ Tot_{s} X^{•} ⟶ Tot_{s−1} X^{•} ⟶ … ⟶ Tot_{0} X^{•} .

The associated spectral sequence has E_{2}-term

E_{2}^{s,t} = π^{s}π_{t}(X^{•})

for t ≥ s ≥ 0 and converges (under mild conditions on X^{•}) to π_{t−s} Tot X^{•}.

Homotopy of Map(X, Y_{p}^{∧}). Let X and Y be connected spaces, with H^{∗}(X) and H^{∗}(Y) of finite type.

Proposition 8.4.1. Take the trivial map as base point in Map(X, 𝔽_{p∞}(Y)). Then

π^{s}π_{t}(Map(X, 𝔽_{p}^{•}(Y))) ≅ Ext_{K}^{s}(H^{∗}(Y), Σ^{t}(H^{∗}(X))^{+})

for t ≥ 1, s ≥ 0, and

π^{0}π_{0}(Map(X, 𝔽_{p}^{•}(Y))) ≅ Hom_{K}(H^{∗}(Y), H^{∗}(X)) .

The Ext-groups are formed in the sense of homotopical algebra in the (nonabelian) category K of unstable A-algebras, defined using resolutions generated by a comonad (= cotriple). This is the comonad generated by the forgetful functor K → U and its left adjoint U → K.

The p-adic completion 𝔽_{p∞}(Y) is the totalization of a cosimplicial space which is a generalized Eilenberg–Mac Lane space in each codegree. Hence its cohomology is accessible through the calculations of Cartan and Serre. This gives the E_{2}-term above.

Similar formulas apply for other choices of base point, replacing groups of homomorphisms with groups of derivations, and Ext-groups with the derived functors of the derivations.

Miller’s conjecture.

((Use the (s = t)-line in the spectral sequence. Lannes’ vanishing results.))

Theorem 8.1.1 (Lannes). Let Y be a connected nilpotent space such that H^{∗}(Y) is of finite type and π_{1}(Y) is finite. Then the natural map [f] ↦ f^{∗} induces a bijection

[BV, Y] ⟶^{≅} Hom_{K}(H^{∗}(Y), H^{∗}(V)) .

Remark. When Y is connected and finite dimensional, there is only the trivial unstable algebra homomorphism H^{∗}(Y) → H^{∗}(V). For (when p = 2) Sq_{0} : x ↦ x^{2} acts injectively on H^{∗}(V), and nilpotently on H^{∗}(Y) in positive degrees.

Miller’s theorem.

Theorem 8.6.1. Let Y be a connected nilpotent space such that H^{∗}(Y) is of finite type and π_{1}(Y) is finite. Then the canonical map H^{∗}(Y) → T H^{∗}(Y) is an isomorphism if and only if the space of based maps Map_{∗}(Bℤ/p, Y) is contractible.

Proof sketch. Evaluation at the base point of Bℤ/p defines a map Map(Bℤ/p, Y) → Y, and a map of spectral sequences with E_{2}-terms:

Ext_{K}^{s}(H^{∗}(Y), H^{∗}(ℤ/p) ⊗ H^{∗}(S^{t})) ⟶ Ext_{K}^{s}(H^{∗}(Y), H^{∗}(S^{t})) .

((Relate H^{∗}(ℤ/p) ⊗ H^{∗}(S^{t}) to Σ^{t}(H^{∗}(ℤ/p))^{+}.))

The left term is isomorphic to Ext_{K}^{s}(T H^{∗}(Y), H^{∗}(S^{t})) and the map is induced by the inclusion H^{∗}(Y) → T H^{∗}(Y). If this is an isomorphism the spectral sequences are isomorphic from the E_{2}-terms onwards. This proves that Map_{∗}(Bℤ/p, Y) is contractible after p-completion. An arithmetic square argument yields the integral result. □

Remark. Recall that H^{∗}(Y) → T H^{∗}(Y) is an isomorphism whenever H^{∗}(Y) is locally finite, e.g., when Y is finite dimensional.

The Sullivan conjecture. Let G be a group and X a G-space. The homotopy fixed point space X^{hG} = Map(EG, X)^{G} is the space of G-equivariant maps EG → X. There is a canonical map

γ: X^{G} ⟶ X^{hG}

taking a fixed point of X to the constant map from EG to it.

When the G-action is trivial, this is the “constant maps” section γ: X → Map(BG, X) in the fibration

Map_{∗}(BG, X) ⟶ Map(BG, X) ⟶^{ev} X .

Here ev evaluates a map at the base point.

The following was proven for G = ℤ/p and trivial G-action by Miller, then for a general ℤ/p-action by Carlsson, Lannes and Miller (independently), and for general p-groups by Dwyer and Zabrodsky.

Theorem 9.1.1 and 9.1.2. Let G be a finite p-group, and X a finite G-CW complex. Then the map

(X^{G})_{p}^{∧} ⟶^{γ} (X_{p}^{∧})^{hG}

is a homotopy equivalence.

The proof for G = ℤ/p is by reduction to Miller’s theorem 8.6.1. The proof for general p-groups is by an induction on the order of p, using the existence of a central C ≅ ℤ/p in any nontrivial p-group G.

Corollary. Let G be a finite p-group, and X a finite CW complex with trivial G-action. Then the based mapping space Map_{∗}(BG, X_{p}^{∧}) is contractible.

Remark. The finiteness hypothesis on X is necessary. Take for example X = K(ℤ/p, n). Then π_{0} Map_{∗}(BG, X_{p}^{∧}) ≅ H^{n}(BG) is typically nonzero.

The cohomology of Map(BV, Y_{p}^{∧}).

The evaluation map

BV × Map(BV, Y) → Y

induces a homomorphism

H^{∗}(Y) → H^{∗}(V) ⊗ H^{∗}(Map(BV, Y)),

adjoint to a homomorphism

T_{V} H^{∗}(Y) → H^{∗}(Map(BV, Y)) .

It admits a lift over H^{∗}(Map(BV, Y_{p}^{∧})).

Theorem 9.7.1 (Lannes). Let Y be a space such that H^{∗}(Y) and T_{V} H^{∗}(Y) are of finite type, and suppose T_{V} H^{∗}(Y) is trivial in degree 1. Then the natural map

T_{V} H^{∗}(Y) ⟶^{≅} H^{∗}(Map(BV, X_{p}^{∧}))

is an isomorphism.

This pretty much pins down the meaning of the functor T_{V}.