Spheres and Hopf Rings

J. Michael Boardman

September 1999

Let E be any of the usual ring spectra, such as H(𝔽_{p}), MU, BP, KU, K(n), or P (n). It defines the multiplicative cohomology theory E^{∗}(−) and homology theory E_{∗}(−). Write \underline{E}_{k} for the k-th space of its Ω-spectrum; it represents the ungraded cohomology theory E^{k}(−). We have the standard Hopf ring elements e, e_{k} and b_{k}, whose definitions we recall below.

Theorem 1 We have the Hopf ring relation

e^{∘} e = −b_{1} in E_{∗}(\underline{E}_{2}).  (2)

More generally,

e_{k} ^{∘} e_{m} = (−1)^{km}e_{k+m} in E_{∗}(\underline{E}_{k+m})  (3)

for any k > 0 and m > 0, and hence (by induction) e_{k} = (−1)^{k(k−1)/2}e^{^{∘}k}.

The purpose of this note is to explain why equation (2) differs in sign from the original formula e^{∘} e = b_{1} given in Ravenel–Wilson [2]. The point is that the thoroughly algebraic machinery of Boardman–Johnson–Wilson [1] leaves no negotiating room for signs. To establish equation (3), we must be precise about what all the elements and maps are. We work throughout in the homotopy category of unbased spaces.

Representing cohomology We represent E-cohomology in the following sense. For each k, we fix a universal class ι_{k} ∈ E^{k}(\underline{E}_{k}). Then given any space X and a cohomology class x ∈ E^{k}(X), there is a map f: X → \underline{E}_{k}, unique up to homotopy, that induces f^{∗}ι_{k} = x; we say that f represents x. We use this correspondence throughout to identify cohomology classes with (homotopy classes of) maps.

In particular, the multiplication map ϕ: \underline{E}_{k} × \underline{E}_{m} → \underline{E}_{k+m} represents the cross product ι_{k} × ι_{m} ∈ E^{k+m}(\underline{E}_{k}×\underline{E}_{m}); that is,

ϕ^{∗}ι_{k+m} = ι_{k} × ι_{m}.  (4)

More generally, if also y ∈ E^{m}(Y) is represented by g: Y → \underline{E}_{m}, the cross product x × y is represented by

ϕ^{∘}(f ×g): X × Y ⟶ \underline{E}_{k} × \underline{E}_{m} ⟶ \underline{E}_{k+m},

since

(ϕ^{∘}(f ×g))^{∗}ι_{k+m} = (f ×g)^{∗}ϕ^{∗}ι_{k+m} = (f ×g)^{∗}(ι_{k}×ι_{m}) = f^{∗}ι_{k} × g^{∗}ι_{m} = x × y.

Spheres For each k > 0, E^{∗}(S^{k}) is an exterior algebra on one generator u_{k} ∈ E^{k}(S^{k}), which we fix. Then the formula [1, (13.8)]

r(u_{k}) = ⟨r, 1_{k}⟩1_{S} + ⟨r, e_{k}⟩u_{k}  in E^{∗}(S^{k})  (5)

defines the Hopf ring element e_{k} ∈ E_{k}(\underline{E}_{k}), where r is any unstable operation defined on E^{k}(−) and 1_{S} denotes the unit element of E^{∗}(S^{k}). In particular, e = e_{1}. We note that e_{k} is independent of the choice of u_{k}, since for any rational number λ ∈ E^{0}, in particular λ = −1, the Cartan formula [1, (10.36)] gives

r(λu_{k}) = ⟨r, [λ]^{∘}1_{k}⟩1_{S} + ⟨r, [λ]^{∘}e_{k}⟩u_{k} = ⟨r, 1_{k}⟩1_{S} + ⟨r, e_{k}⟩λu_{k},

as [λ]^{∘} e_{k} = λe_{k} by [1, Prop. 13.7(g)].

Alternatively, we can construct e_{k} from the homology point of view. We represent u_{k} by the map f_{k}: S^{k} → \underline{E}_{k}, so that f_{k}^{∗}ι_{k} = u_{k}. Let z_{k} ∈ E_{k}(S^{k}) be the element dual to u_{k}, which is defined by ⟨u_{k}, z_{k}⟩ = 1 and ⟨1_{S}, z_{k}⟩ = 0. Then by [1, Prop. 10.5],

(f_{k})_{∗}z_{k} = ⟨1, z_{k}⟩1_{k} + (−1)^{k(k+k)}⟨u_{k}, z_{k}⟩e_{k} = e_{k} in E_{∗}(\underline{E}_{k}),

with no sign. For k = 1, this agrees with the Ravenel–Wilson definition [2] of e = e_{1}.

Complex orientation We choose a complex orientation x ∈ E^{2}(ℂP^{∞}), so that E^{∗}(ℂP^{∞}) = E^{∗}[[x]], the ring of formal power series in x. Then the formula [1, (15.2)]

r(x) = ⟨r, b_{0}⟩1 + ⟨r, b_{1}⟩x + ⟨r, b_{2}⟩x^{2} + …  in E^{∗}(ℂP^{∞})  (6)

defines the Hopf ring elements b_{k} ∈ E_{2k}(\underline{E}_{2}) for all k ≥ 0, where r is any unstable operation defined on E^{2}(−). By definition, the inclusion j: S^{2} ≅ ℂP^{1} ⊂ ℂP^{∞} sends x to a generator of E^{∗}(S^{2}), which we may assume is u_{2} exactly. Then by naturality from equation (6),

r(u_{2}) = r(j^{∗}x) = ⟨r, b_{0}⟩1_{S} + ⟨r, b_{1}⟩u_{2} in E^{∗}(S^{2}).

Comparison with equation (5) shows that b_{0} = 1_{2} and b_{1} = e_{2}, as in [1, Prop. 15.3(a)].

From the homology point of view, let β_{k} ∈ E_{2k}(ℂP^{∞}) be dual to x^{k}, so that ⟨x^{m}, β_{k}⟩ is 1 for m = k and zero otherwise. Let h: ℂP^{∞} → \underline{E}_{2} be the map that represents x. Then again by [1, Prop. 10.5], h_{∗}β_{k} = b_{k} for all k, as in the Ravenel– Wilson definition [2] of b_{k}.

First proof of Theorem Having chosen orientations for our spheres, we define the map q: S^{k} × S^{m} → S^{k+m} to induce

q^{∗}u_{k+m} = u_{k} × u_{m}.  (7)

The Cartan formula [1, (10.39)] gives

r(u_{k}×u_{m}) = ⟨r, 1_{k+m}⟩1_{S} ×1_{S} + (−1)^{km}⟨r, e_{k}^{∘}e_{m}⟩u_{k} × u_{m} in E^{∗}(S^{k}×S^{m}),

with other terms that drop out. If we apply q to equation (5), we obtain

r(u_{k}×u_{m}) = ⟨r, 1_{k+m}⟩1_{S} ×1_{S} + ⟨r, e_{k+m}⟩u_{k}×u_{m}.

Comparison of these two formulae yields equation (3), and hence (2).

Second proof of Theorem We adopt the homology point of view. First,

e_{k} ^{∘} e_{m} = ϕ_{∗}((f_{k})_{∗}z_{k} × (f_{m})_{∗}z_{m}) = ϕ_{∗}(f_{k}×f_{m})_{∗}(z_{k}×z_{m}).

Next, we note that the diagram

[figure: S^{k} × S^{m}; \underline{E}_{k} × \underline{E}_{m}; S^{k+m}; \underline{E}_{k+m}; f_{k}×f_{m}; q; ϕ; f_{k+m}]  (8)

commutes (up to homotopy), because

(f_{k}×f_{m})^{∗}ϕ^{∗}ι_{k+m} = (f_{k}×f_{m})^{∗}(ι_{k}×ι_{m}) = f_{k}^{∗}ι_{k} × f_{m}^{∗}ι_{m} = u_{k} × u_{m}

and

q^{∗}f_{k+m}^{∗}ι_{k+m} = q^{∗}u_{k+m} = u_{k} × u_{m}.

So e_{k} ^{∘} e_{m} = (f_{k+m})_{∗}q_{∗}(z_{k}×z_{m}).

To compute q_{∗}(z_{k}×z_{m}), we evaluate

⟨u_{k+m}, q_{∗}(z_{k} × z_{m})⟩ = ⟨q^{∗}u_{k+m}, z_{k} × z_{m}⟩
= ⟨u_{k} × u_{m}, z_{k} × z_{m}⟩
= (−1)^{km}⟨u_{k}, z_{k}⟩⟨u_{m}, z_{m}⟩ = (−1)^{km}.

The conclusion is that

q_{∗}(z_{k} × z_{m}) = (−1)^{km}z_{k+m}.  (9)

Thus e_{k} ^{∘} e_{m} = (−1)^{km}(f_{k+m})_{∗}z_{k+m} = (−1)^{km}e_{k+m}, as required.

Remark If instead we arrange q to make q_{∗}(z_{k}×z_{m}) = z_{k+m} hold, the above calculation shows that q^{∗}u_{k+m} = (−1)^{km}u_{k} × u_{m}, and diagram (8) no longer commutes (if k and m are odd).

References

[1] J. M. Boardman, D. C. Johnson, W. S. Wilson, Chapter 15: Unstable operations in generalized cohomology, Handbook of Algebraic Topology (ed. I. M. James), Elsevier (Amsterdam, 1995), 687–828.

[2] D. C. Ravenel, W. S. Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Alg. 9 (1977), 241–280.

Department of Mathematics Johns Hopkins University Baltimore, MD 21218–2686

                                         E-mail: boardman@math.jhu.edu