MAHOWALD’S INFINITE FAMILY

John Rognes

February 10th 2006

1. Adams filtrations 1 and 2

Let p = 2 and write H_{∗}(X) for H_{∗}(X; ℤ/2), H for Hℤ/2, etc.

The E_{2}-term of the Adams spectral sequence

E_{2}^{s,t} = Ext_{A}^{s,t}(ℤ/2, ℤ/2) ⟹ π_{t−2}(S)_{2}^{∧}

can be computed as the homology of the normalized cobar complex

ℤ/2 ⟶^{d^{0}} Ā_{∗} ⟶^{d^{1}} Ā_{∗} ⊗ Ā_{∗} ⟶^{d^{2}} Ā_{∗} ⊗ Ā_{∗} ⊗ Ā_{∗} → …

where

A_{∗} = H_{∗}(H) = ℤ/2[ξ_{k} | k ≥ 1]

is the dual Steenrod algebra with |ξ_{k}| = 2^{k} − 1, Ā_{∗} = coker(η: ℤ/2 → A_{∗}). We write [a_{1}| … |a_{s}] for a_{1} ⊗ ⋯ ⊗ a_{s} ∈ Ā_{∗}^{⊗s}.

Here d^{0} = 0, so the Adams 0-line is just

E_{2}^{0,∗} = Ext_{A}^{0,∗}(ℤ/2, ℤ/2) = ℤ/2{1}

concentrated in bidegree (s, t) = (0, 0). Here 1 detects the unit generator ι ∈ π_{0}(S).

Next, d^{1} is induced by the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗}, dual to the composition in A,

ψ(ξ_{k}) = ∑_{i+j} ξ_{i}^{2^{j}} ⊗ ξ_{j} .

Here ξ_{0} = 1. So ker(d^{1}) consists of the primitives in A_{∗}, with ψ(x) = x ⊗ 1 + 1 ⊗ x, which are spanned by the powers ξ_{1}^{2^{i}} for i ≥ 0. These are dual to the indecomposables Sq^{2^{i}} in A. Let h_{i} = [ξ_{1}^{2^{i}}] ∈ Ext_{A}^{1,2^{i}}(ℤ/2, ℤ/2). The Adams 1-line

E_{2}^{1,∗} = Ext_{A}^{1,∗}(ℤ/2, ℤ/2) = ℤ/2{h_{i} | i ≥ 0}

is concentrated in bidegrees (s, t) = (1, 2^{i}). Here h_{0}, h_{1}, h_{2} and h_{3} detect 2ι, η, ν and σ in π_{∗}(S).

The map d^{2} is induced by the difference (= sum) of ψ⊗id and id⊗ψ: A_{∗} ⊗A_{∗} → A_{∗} ⊗ A_{∗} ⊗ A_{∗}. Adams shows that the products

h_{i}h_{j} = [ξ_{1}^{2^{i}}|ξ_{1}^{2^{j}}]

Typeset by AMS-TEX

generate the kernel of d^{2}. Furthermore, the coproduct

ψ(ξ_{2}^{2^{i}}) = ξ_{2}^{2^{i}} ⊗ 1 + ξ_{1}^{2^{i+1}} ⊗ ξ_{1}^{2^{i}} + 1 ⊗ ξ_{2}^{2^{i}}

implies the coboundary d^{1}([ξ_{2}^{2^{i}}]) = [ξ_{1}^{2^{i+1}}|ξ_{1}^{2^{i}}] = h_{i+1}h_{i} for i ≥ 0. Thus

h_{i}h_{i+1} = 0

in Ext_{A}. This is the only relation, so

E_{2}^{2,∗} = Ext_{A}^{2,∗}(ℤ/2, ℤ/2) = ℤ/2{h_{i}h_{j} | 0 ≤ i ≤ j, j ≠ i + 1}

is the Adams 2-line.

The products h_{i}h_{j}h_{k} do not generate all of the Adams 3-line Ext_{A}^{3,∗}(ℤ/2, ℤ/2). For example there is an indecomposable class c_{0} = ⟨h_{1}, h_{0}, h_{2}^{2}⟩ ∈ E_{2}^{3,11}, which detects ϵ = ⟨η, 2ι, ν^{2}⟩ ∈ π_{8}(S). But Adams showed that the only relations between the product classes are

h_{i}^{2}h_{i+2} = h_{i+1}^{3}

and

h_{i}h_{i+2}^{2} = 0

for i ≥ 0. Compare with the relations 4ν = η^{3} and 2ν^{2} = 0 in π_{∗}(S), for i = 0. The comparison is not as perfect in higher degrees, since η^{2}σ = ν^{3} + ηϵ, but still ησ^{2} = 0. Wang showed that the classes c_{i} = ⟨h_{i+1}, h_{i}, h_{i+2}^{2}⟩ generate the rest of the Adams 3-line.

Adams proved that there is a spectral sequence differential

d_{2}(h_{i}) = h_{0}h_{i−1}^{2}

which is nonzero in Ext_{A}^{3,∗}(ℤ/2, ℤ/2) for all i ≥ 4, so there are no further infinite cycles on the 1-line.

Mahowald and Tangora went on to prove that the differential

d_{2}(h_{i}h_{j}) = h_{0}h_{i−1}^{2}h_{j} + h_{0}h_{i}h_{j−1}^{2}

is nonzero in Ext_{A}^{4,∗}(ℤ/2, ℤ/2) for most cases of 0 ≤ i < j, which together with a few more known differentials (like d_{4}(h_{3}h_{5}) = h_{0}x, and later corrections, like d_{3}(h_{2}h_{5}) = h_{0}p) leaves only the two infinite families

h_{i}^{2} ∈ E_{3}^{2,2^{i+1}}

for i ≥ 0 and

h_{1}h_{j} ∈ E_{3}^{2,2^{j}+2}

for j ≠ 0, 2, and the finitely many cases

h_{0}h_{2}, h_{0}h_{3}, h_{2}h_{4}

in the Adams 2-line at the E_{3}-term. The latter three are infinite cycles, represented in π_{∗}(S) by 2ν, 2σ and ν_{∗}, respectively, in Toda’s notation.

It is known that h_{i}^{2} survives to E_{∞} for 0 ≤ i ≤ 5, represented by 4ι, η^{2}, ν^{2}, σ^{2}, θ_{4} and θ_{5}, while for i ≥ 6 this is the subject of the open Kervaire invariant problem. ((References for θ_{4} = ⟨σ, 2σ, σ, 2σ⟩, θ_{5}?))

References: Adams (Comm. Math. Helv., 1958), Mahowald–Tangora (Topology, 1967), Mahowald–Tangora (Bol. Soc. Math. Mex., 1967), Barratt–Mahowald– Tangora (Topology, 1970), Ravenel’s green book (now mauve).

2. Mahowald’s η_{j}-family

Theorem (Mahowald). For each j ≥ 3 there are classes η_{j} ∈ π_{2^{j}}(S) represented by h_{1}h_{j} ≠ 0 in the Adams spectral sequence.

Let k = 2^{j−3}, so that 8k = 2^{j}.

Sketch of proof. Mahowald constructs a finite CW spectrum T (k) and stable maps

\xymatrix{S^{8k} \ar[r]^{g_{j}} \ar@{-->}[drr]_{h_{j}} & T (k) \ar[rr]^{f_{j}} &  & ℝP^{∞} \ar[r]^{t} & S^{0} \\  &  & S^{1} \ar[ul]_{i} \ar[ur] \ar@{-->}[urr]_{h_{1}}}

satisfying the following three claims:

(1) H_{1}(i) and H_{1}(f_{j}) are isomorphisms, so f_{j} ∘ i represents the bottom 1-cell in ℝP^{∞},

(2) Sq^{2^{j}} acts nontrivially on H^{1} of the mapping cone C_{g_{j}} = T (k) ∪ e^{8k+1} of g_{j}, so g_{j} is represented by i_{∗}(h_{j}) in Ext_{A}^{1,1}(H^{∗}T (k), ℤ/2), and

(3) t is the ℤ/2-transfer map, so Sq^{2} acts nontrivially on H^{0} of the mapping cone C_{t} = S^{0} ∪ CℝP^{∞}.

Let η_{j} = t ∘ f_{j} ∘ g_{j}: S^{2^{j}} → S^{0}. Then η_{j} is represented by the Yoneda product

[t ∘ f_{j}] · [g_{j}] = [t ∘ f_{j}] · i_{∗}(h_{j}) = [t ∘ f_{j} ∘ i] · h_{j}

and since t maps the 1-cell in ℝP^{∞} to S^{0} by η, represented by h_{1}, this equals h_{1}h_{j}. □

Here S^{8k} has homology concentrated in degree 8k, T (k) has homology concentrated in degrees 1 ≤ ∗ ≤ k, ℝP^{∞} has homology in degrees ∗ ≥ 1 and S^{0} has homology concentrated in degree 0.

We shall discuss claim (1) in section 7, and claim (2) in section 6. Claim (3) is standard, so this will complete the proof.

References: Mahowald (Topology, 1977), Hunter–Kuhn (Math. Proc. Camb. Phil. Soc., 1999).

3. Brown–Gitler spectra

For n ≥ 0, T (n) will be Spanier–Whitehead n-dual to the Brown–Gitler spectrum B(n), T (n) = Σ^{n}DB(n). In the original paper, B(n) is denoted B([n/2]). The following two properties characterize the (connective, 2-complete) spectrum B(n) up to homotopy equivalence.

(1) H^{∗}(B(n)) = M(n) := A/A{χSq^{i} | 2i > n}, and

(2) the natural map B(n)_{n}(Z) → H_{n}(Z) is onto for each space Z.

Considering the admissible monomial basis for A, and conjugating, easily gives:

Lemma. H^{∗}(B(n)) = M(n) is concentrated in degrees 0 ≤ ∗ ≤ n − α(n), where α(n) is the number of 1’s in the binary expansion of n. Dually, H^{∗}(T (n)) = J(n) is concentrated in degrees α(n) ≤ ∗ ≤ n.

For n = k = 2^{j−3}, α(n) = 1 and i: S^{1} → T (k) is the bottom cell. The top class of M(k), in degree k − 1, is generated by χ(Sq^{k/2} … Sq^{2}Sq^{1}) on the bottom class 1, in degree 0, so k-dually Sq^{k/2} … Sq^{2}Sq^{1} acts isomorphically on J(k) from the bottom class in degree 1, to the top class in degree k.

The Steenrod operation Sq^{k/2} ⋯ Sq^{2}Sq^{1} acts isomorphically from H^{1} to H^{k} for both T (k) and ℝP^{∞}, since H^{∗}(ℝP^{∞}) = ℤ/2[x] with Sq^{a}(x^{b}) = \binom{b}{a}x^{a+b}, so f_{j} induces an isomorphism on H_{1} if and only if it induces an isomorphism on H_{k}.

References: Brown–Gitler (Topology, 1973), Brown–Peterson (Trans. A.M.S., 1978).

4. The May–Milgram model

Let C_{2}(n) be the space of n little 2-cubes = squares. It is the n-th space in the little squares operad, with associated monad functor also denoted C_{2}. For any based space (Y, y_{0}) (a CW complex, say) there is a natural map

C_{2}(Y) := ∐_{n≥0} C_{2}(n) ×_{Σ_{n}} Y^{n}/(∼) ⟶ Ω^{2}Σ^{2}Y

which is a homotopy equivalence for path-connected Y.

Here, to a diagram c of n little disjoint squares in I^{2}, and an n-tuple (y_{1}, …, y_{n}) of points in Y, a partial map I^{2} → I^{2} × Y is defined to take the i-th little square, for 1 ≤ i ≤ n to I^{2} × {y_{i}}. Continuing by a map I^{2} × Y → Σ^{2}Y that collapses ∂I^{2} × Y ∪ I^{2} × {y_{0}} to ∗, the partial map can be extended to all of I^{2} by taking what lies outside of the little squares to ∗. Then ∂I^{2} goes to ∗, so we have a map S^{2} → Σ^{2}Y, or a point in Ω^{2}Σ^{2}Y.

The left hand side, C_{2}(Y), is naturally filtered by only allowing n to range up to some fixed number, and the filtration has subquotients

F_{n−1}C_{2}(Y) ⊂ F_{n}C_{2}(Y) → C_{2}(n)_{+} ∧_{Σ_{n}} Y^{∧n} .

Each of these cofiber sequences is stably split, by Snaith, so

Σ^{∞}Ω^{2}Σ^{2}Y ≃ ⋁_{n≥0} Σ^{∞}C_{2}(n)_{+} ∧_{Σ_{n}} Y^{∧n} .

Mahowald and Brown–Peterson prove:

Proposition. For Y an odd sphere, C_{2}(n)_{+} ∧_{Σ^{n}} Y^{∧n} is a suspension of a Brown– Gitler spectrum B(n).

Taking Y = S^{7}, then

C_{2}(n)_{+} ∧_{Σ^{n}} S^{7n} ≃ Σ^{7n}B(n)

is stably a direct summand of Ω^{2}S^{9}. So to produce stable maps from Σ^{7n}B(n), it suffices to produce a map from Ω^{2}S^{9}.

Remark (Arnold, Brown). There is an equivariant homotopy equivalence C_{2}(n) ≃ F (ℝ^{2}, n), the configuration space of n distinct points (p_{1}, p_{2}, …, p_{n}) in ℝ^{2} = ℂ. The symmetric group Σ_{n} acts freely on F (ℝ^{2}, n) by permuting the points, as well as on ℝ^{2n} = ℂ^{n} by permuting the complex coordinates (z_{1}, z_{2}, …, z_{n}). The associated ℝ^{2n}-bundle with total space F (ℝ^{2}, n) ×_{Σ_{n}} ℝ^{2n} does in fact admit a trivialization, given by

(p_{1}, p_{2}, …, p_{n}; z_{1}, z_{2}, …, z_{n}) ⟼ (p_{1}, p_{2}, …, p_{n}; ∑_{i} z_{i}, ∑_{i} p_{i}z_{i}, …, ∑_{i} p_{i}^{n−1}z_{i})

(using complex multiplications; all sums are over 1 ≤ i ≤ n). Over the orbit of (p_{1}, p_{2}, …, p_{n}) the (Vandermonde) determinant of this linear map is ± ∏_{a<b}(p_{a} − p_{b}) ≠ 0. So the Thom complex C_{2}(n)_{+} ∧_{Σ_{n}} (S^{2ℓ+1})^{∧n} is Σ^{2ℓn}(C_{2}(n)_{+} ∧_{Σ_{n}} S^{n}).

References: Cohen–Mahowald–Milgram (Proc. Symp. Pure Math., 32(2), 1978), May (The geometry of iterated loop spaces, 1972), Milgram (Annals of Math., 1966), Snaith (J. London Math. Soc., 1974),

5. Thom complexes of bundles over suspensions

Let (X, x_{0}) be a based space, g: X → O the clutching map of a stable vector bundle γ: ΣX → BO. Let

Jg: Σ^{∞}X → S

be the stable map Σ^{∞}X → Σ^{∞}X_{+} (taking x to x−x_{0}), followed by the left adjoint of

X_{+} ⟶^{g_{+}} O_{+} ⟶^{J} Ω^{∞}(S) = Q(S^{0}) .

Lemma. The Thom spectrum T h(γ ↓ ΣX) = (ΣX)^{γ} is homotopy equivalent to the mapping cone C_{Jg} of the map Jg: Σ^{∞}X → S.

Proof. Write g as a colimit of maps g(n): X(n) → O(n). The Thom complex of γ restricted to ΣX(n) is the n-th space in the Thom spectrum of γ. It equals the pushout in

\xymatrix{Σ_{+}^{n}(X) \ar[r]^{^{\overline{Jg}^{+}}} \ar[d] & S^{n} \ar[d] \\ Σ_{+}^{n}(CX) \ar[r] & T h(γ)_{n}}

where \overline{Jg}_{+} is left adjoint to Jg_{+} : X(n)_{+} → O(n)_{+} → Ω^{n}S^{n}. This follows by collapsing one of the two cones in ΣX to a point. Stably, the S^{n}’s on the left hand side corresponding to the base point x_{0} can be split off, leaving Σ^{n}X and CΣ^{n}X, respectively. □

Corollary. Under the Thom isomorphism H^{∗}(ΣX) ≅ H^{∗}(C_{Jg}), the i-th Stiefel– Whitney class w_{i}(γ) = γ^{∗}(w_{i}) corresponds to Sq^{i}(1).

References: Mahowald (Topology, 1977).

6. Construction of g_{j}

Let Y = S^{7}, X = Ω^{2}Σ^{2}Y = Ω^{2}S^{9}.

Let g: Ω^{2}S^{9} → O be Ω^{2} of a map generating π_{9}(BBO) = π_{8}(BO) ≅ ℤ. The adjoint clutching map γ fits in a commutative diagram

\xymatrix{S^{8} \ar[r]^{η} & ΩS^{9} \ar[r]^{Bg} & BO \\  & ΣX = ΣΩ^{2}S^{9} \ar[u]^{ϵ} \ar[ur]_{γ}}

where η and ϵ are the unit and counit for the loop–suspension adjunction.

Lemma. Bg ∘ η generates π_{8}(BO) and (Bg ∘ η)^{∗}(w_{8}) ≠ 0.

Proof. The first claim is by definition, and thus Bg ∘ η factors through BString = BO⟨8⟩ → BO, and S^{8} → BString induces an isomorphism on π_{8} and H^{8}, by the Hurewicz theorem. The claim is then that w_{8} in H^{8}(BO) restricts to the generator of H^{8}(BString). To check this, start with H^{∗}(BSO) = ℤ/2[w_{i} | i ≥ 2] and use the Serre spectral sequence for K(ℤ/2, 1) → BSpin → BSO (pulled back from w_{2}) to compute

H^{∗}(BSpin) = ℤ/2[w_{i} | i ≥ 4]/(w_{5}, Sq^{4}w_{5}, …) ,

and then use the Serre spectral sequence for K(ℤ, 3) → BString → BSpin to compute H^{∗}(BString) = {1, w_{8}, … }. □

Lemma. (Bg)^{∗}(w_{8ℓ}) ≠ 0 for all ℓ ≥ 1.

Proof. Map S^{8} × ⋯ × S^{8} → ΩS^{9} (ℓ copies of S^{8}) by the loop sum of ℓ copies of η. The H-map Bg takes loop sum in ΩS^{9} to Whitey sum in BO, so by the Cartan formula (Bg)^{∗}(w_{8ℓ}) pulls back to the cross product in H^{∗}(S^{8} × ⋯ × S^{8}) of the generators in the ℓ copies of H^{8}(S^{8}), which is nonzero. □

Lemma. γ^{∗}(w_{2^{j}}) ≠ 0 for all j ≥ 3.

Proof. The Serre spectral sequence for the loop–path fibration over ΩS^{9} shows that ϵ^{∗}: H^{∗}(ΩS^{9}) → H^{∗}(ΣΩ^{2}S^{9}) is injective in degree 2^{j} for all j ≥ 3. □

Proof of claim (2). By the corollary and the last lemma, Sq^{2^{j}} acts non-trivially from degree 0 in the cohomology of the mapping cone of Jg.

Now restrict Jg to the kth stable summand (k = 2^{j−3}, j ≥ 3)

Σ^{7k}B(k) ≃ C_{2}(k)_{+} ∧_{Σ_{k}} S^{7k} ↪ Σ^{∞}Ω^{2}S^{9} ⟶^{Jg} S .

The mapping cone of the composite map Σ^{7k}B(k) → S has cohomology ℤ/2 ⊕ Σ^{7k+1}M(k), concentrated in degrees 0, 7k + 1, …, 8k, and one can check that Sq^{2^{j}} = Sq^{8k} acts non-trivially.

Take Spanier–Whitehead 8k-duals, to get the desired map

g_{j}: S^{8k} → Σ^{8k}DΣ^{7k}B(k) = T (k) .

Then Sq^{2^{j}} also acts non-trivially in its mapping cone C_{g_{j}}, from degree 1.

The inclusion map i: S^{1} → T (k) on the bottom cell is 8k-dual to the pinch map Σ^{7k}B(k) → S^{8k−1} to the top cell. Thus g_{j} ≠ 0 is represented in the image under

ℤ/2{h_{j}} = Ext_{A}^{1,1}(Σ^{1}ℤ/2, Σ^{8k}ℤ/2) ⟶^{i_{∗}} Ext_{A}^{1,1}(H^{∗}(T (k)); Σ^{8k}ℤ/2) ,

i.e., by i_{∗}(h_{j}). □

References: Mahowald (Topology, 1977).

7. Construction of f_{j}

The Adams spectral sequence edge homomorphism can be continued

B(n)_{n}(Z) ≅ [T (n), Z] → Hom_{A}(H^{∗}(Z), H^{∗}(T (n)) = J(n))
→ Hom_{ℤ/2}(H^{n}(Z), H^{n}(T (n)) = ℤ/2) ≅ H_{n}(Z) ,

taking f first to the A-module homomorphism H^{∗}(f) and then to the ℤ/2-module homomorphism H^{n}(f). By the (second) Brown–Gitler property, the composite map is onto for all spaces Z.

Remark. In fact, J(n) is the injective hull of Σ^{n}ℤ/2 in the category U of unstable A-modules, so the second arrow is an isomorphism whenever H^{∗}(Z) is an unstable A-module, which it is for spaces Z.

Proof of claim (1). Take n = k = 2^{j−3}, Z = ℝP^{∞} and let

f_{j}: T (k) → ℝP^{∞}

be a map that goes to the generator a_{k} ∈ H_{k}(ℝP^{∞}) under the composite map above. So H^{k}(f_{j}) is an isomorphism. By naturality of the Steenrod operations, we have a commutative square

\xymatrix{H^{k}(T (k)) & H^{k}(ℝP^{∞}) \ar[l]_{H^{k}(f_{j})}^{≅} \\ H^{1}(T (k)) \ar[u]^{Sq^{k/2}···Sq^{2}Sq^{1}}_{≅} & H^{1}(ℝP^{∞}) . \ar[l]^{H^{1}(f_{j})} \ar[u]_{Sq^{k/2}···Sq^{2}Sq^{1}}^{≅}}

Here Sq^{k/2} ⋯ Sq^{2}Sq^{1} is an isomorphism from H^{1} in both H^{∗}(T (k)) and H^{∗}(ℝP^{∞}), so also H^{1}(f_{j}) is an isomorphism. □

References: Schwartz (Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, 1994).

8. Brown–Gitler’s approach to B(n)

9. Mahowald’s approach to B(n)

10. Goerss–Lannes–Morel’s approach to B(n)