COMPUTATIONS IN THE MOD 2 LAZARD RING

MARKUS ROST

                         preliminary version

Contents

Introduction 1 1. Basic definitions 1 2. A preliminary computation 3 3. Operations defined by power series 4 4. Steenrod squares 5 5. A contracting property of Sq^{0} 7 6. The logarithm 9 References 10

Introduction

This text contains purely algebraic considerations in the mod 2 Lazard ring L. We define operations

Sq^{k}: L → L.

and prove a vanishing property for them (Theorem 2). We apply this to obtain the canonical logarithm of the universal mod 2 Lazard formal group law. This approach does not involve the usual game with binomial coefficients.

This text is preliminary in a manifold sense: We only define the logarithm, but do not describe it in more detail. Missing are also the Landweber-Novikov operations. The geometric analogies of the material are only mentioned partially in some side remarks. No attempt has been made on the mod p-analogies.

1. Basic definitions

Let R be a F_{2}-algebra. By a mod 2 formal group law we understand a power series

F (x, y) ∈ R[[x, y]]

such that

(1)  F (x, 0) = F (0, x) = x,
(2)  F (x, F (y, z)) = F (F (x, y), z),
(3)  F (y, x) = F (x, y),
(4)  F (x, x) = 0.

Date: March 10, 2000.

These equations are understood in the rings R[[x]], R[[x, y, z]], etc.

It follows from (1) that

F (x, y) = x + y + ∑_{i,j≥1} a_{i,j}x^{i}y^{j}

with a_{i,j} ∈ R. Sometimes we write for simplicity

F (x, y) = ∑_{i,j} a_{i,j}x^{i}y^{j},

thereby understanding i, j ≥ 0, i + j ≥ 1, a_{1,0} = a_{0,1} = 1, and a_{i,0} = a_{0,j} = 0 for i, j > 1.

Let

L̃ = F_{2}[u_{i,j}]

be the polynomial ring in the variables u_{i,j}, i, j ≥ 1 and let

I ⊂ L̃

be the smallest ideal such that

F_{univ}(x, y) = x + y + ∑_{i,j≥1} u_{i,j}x^{i}y^{j}

becomes a mod 2 formal group law over the quotient

L = L̃/I.

The ring L is called the mod 2 Lazard ring and F_{univ}, considered as element of L[[x, y]], is called the universal mod 2 formal group law.

For any mod 2 formal group law

F (x, y) = x + y + ∑_{i,j≥1} a_{i,j}x^{i}y^{j} ∈ R[[x, y]]

there exist a unique ring homomorphism

ρ_{F} : L → R

with

ρ_{F} (u_{i,j}) = a_{i,j}.

All formal group laws considered in this text are mod 2 formal group laws. We will call them simply “formal group laws” and L will be called “the Lazard ring”.

We consider L̃ as a Z-graded ring with

deg u_{i,j} = 1 − i − j.

For a Z-graded ring R we extend the grading to any power series ring R[[x_{1}, …, x_{r}]] by associating the degree 1 to each of variables x_{i}.

This way F_{univ}(x, y) ∈ L̃[[x, y]] is homogeneous of degree 1. Also, the equations (1)–(4) are homogeneous. It follows that the ideal I is a homogeneous ideal. Therefore the Lazard ring carries a Z-grading

L = ⨁_{k≤0} L^{k}

with

u_{i,j} ∈ L^{1−i−j}.

Remark. The cobordism ring of a space X can be described as

N^{∗}(X) = ⨁_{k∈Z} N^{k}(X),

N^{k}(X) = [X_{+}, S^{k} ∧ MO].

One has N^{k}(X) = 0 for k > dim X. For nonempty X the groups N^{k}(X) are nonzero in non-positive degrees. The Lazard ring L is isomorphic to the unoriented cobordism ring N^{∗} = N^{∗}(point).

Once in a while we will refer to these facts for some explanations. We will certainly not make use of them, because this text is supposed as a preparation to establish the isomorphism L = N^{∗}.

The negative grading on L introduced above coincides with the natural grading on N^{∗}. For k > 0 one has L^{k} = N^{k} = 0 and for k ≥ 0 the group L^{−k} = N^{−k} is the group of bordism classes of k-dimensional smooth compact manifolds.

2. A preliminary computation

Let F (x, y) be a (mod 2) formal group law over R. We consider the (continuous) homomorphism over R[[t]]

τ: R[[t, x]] → R[[t, x]],
x ↦ F (t, x).

Note that τ is an involution:

τ^{2}(x) = F (t, F (t, x)) = F (F (t, t), x) = F (0, x) = x.

Lemma 1. Let f ∈ R[[t, x]] with τ(f) = f. Then there exist a unique element g ∈ R[[t, u]] such that

f(t, x) = g(t, xF (t, x)).

Proof. Let u = xτ(x) = xF (t, x). Then

u = x^{2} + tα

for some α ∈ R[[t, x]]. Using standard arguments for power series rings, it follows that

R[[t, x]] = R[[t, u]] ⊕ xR[[t, u]].

Note that τ(u) = u, so every element of the subring R[[t, u]] is τ-invariant. Write f as

f(t, x) = g(t, u) + xh(t, u).

Then τ(f) = f implies

F (t, x)h(t, u) = xh(t, u).

However

F (t, x) − x = t + xt ∑_{i,j≥1} a_{i,j}x^{i−1}t^{j−1}

is not a zero divisor. Thus h = 0.

3. Operations defined by power series

Let

F (x, y) = ∑_{i,j} a_{i,j}x^{i}y^{j}

be a (mod 2) formal group law over R and let

q(x) = ∑_{i≥0} c_{i}x^{i+1} ∈ R[[x]]

be a power series with c_{0} invertible in R. Then

R[[x]] → R[[x]],
x ↦ q(x)

defines an automorphism of R[[x]] over R. In particular there exist the inverse power series

q^{−1}(x) ∈ R[[x]]

with

q(q^{−1}(x)) = q^{−1}(q(x)) = x.

We put

F_{q}(x, y) = q(F (q^{−1}(x), q^{−1}(y))) ∈ R[[x, y]].

This is a formal group law, obtained from the formal group law F by means of the coordinate change x ↦ q(x). Therefore there exist a ring homomorphism

θ_{q} = ρ_{F_{q}}: L → R

such that

q(F (x, y)) = ∑_{i,j} θ_{q}(u_{i,j})q(x)^{i}q(y)^{j}. (5)

We extend θ_{q} to the continuous homomorphism

θ̄_{q}: L[[x_{1}, …, x_{r}]] → R[[x_{1}, …, x_{r}]],
x_{i} ↦ q(x_{i}).

Then

θ̄_{q}(F_{univ}(x_{k}, x_{l})) = q(F (x_{k}, x_{l})).

by (5).

This means for instance that the following diagram is commutative:

\begin{matrix}L[[z]] & ⟶^{θ̄_{q}} & R[[z]] \\ ^{α}↓ & & ^{α}↓ \\ L[[x, y]] & ⟶^{θ̄_{q}} & R[[x, y]]. \end{matrix} (6)

Here the maps α are the identity on L resp. R and map z to F (x, y). The homomorphisms θ̄_{q} are understood as above: they extend θ_{q}: L → R by u ↦ q(u) for u = x, y, z.

Remark. These considerations are the formal group analogies of a construction in (unoriented) cobordism theory, see [1]. Given the power series q(x), one may define for topological spaces X a natural transformation

Θ_{q}: N^{∗}(X) → N^{∗}(X) ⊗_{L} R.

Here the tensor product is understood via the isomorphism L → N^{∗}, given by the canonical formal group law in N^{∗}(P^{∞} × P^{∞}) = N^{∗}[[x, y]], and via ρ_{F} : L → R, given by the formal group law F.

The homomorphisms θ̄_{q}: L[[x_{1}, …, x_{r}]] → R[[x_{1}, …, x_{r}]] are the formal analogies of the operations Θ_{q} for X = P^{∞} × ⋯ × P^{∞}.

The commutative diagram (6) corresponds to the functoriality of Θ_{q} with respect to the map P^{∞} ×P^{∞} → P^{∞}, the sum map for the Eilenberg-MacLane space P^{∞} = H(Z/2, 1).

The operations considered in the next section are the formal analogies of the Steenrod squares in cobordism theory.

4. Steenrod squares

Now let

R = L[[t]][t^{−1}]

be the ring of Laurent series over L, let

F (x, y) = F_{univ}(x, y) = ∑_{i,j} u_{i,j}x^{i}y^{j} ∈ R[[x, y]]

be the universal formal group law considered as formal group law over R (so that ρ_{F} : L → R is the inclusion), and let

q(x) = \frac{xF (x, t)}{t} ∈ L[[t, x]][t^{−1}] ⊂ R[[x]].

Then

q(x) = x + \frac{x^{2}}{t} + x^{2} ∑_{i,j≥1} u_{i,j}x^{i−1}t^{j−1},

and x → q(x) defines an invertible endomorphism of R[[x]] over R. Thus we may apply the construction of the previous section and get a ring homomorphism θ_{q}: L → R. We write Sq = θ_{q}. Thus Sq is the ring homomorphism

Sq: L → L[[t]][t^{−1}]

such that

q(∑_{i,j} u_{i,j}x^{i}y^{j}) = ∑_{i,j} Sq(u_{i,j})q(x)^{i}q(y)^{j}. (7)

Note that q is homogeneous of degree 1, and therefore Sq is homogeneous of degree 0 (with respect to the Z-gradings given by the grading on L and by deg x = deg t = 1). We write

Sq = ∑_{k∈Z} t^{−k} Sq^{k}

with additive homomorphisms

Sq^{k}: L → L.

Then Sq^{k} is homogeneous of degree k, that is

Sq^{k}(L^{n}) ⊂ L^{n+k}.

The maps Sq^{k} are called Steenrod squares. In the following we establish the properties to be expected from operations with this name.

The Cartan formula

Sq^{k}(αβ) = ∑_{h+l=k} Sq^{h}(α) Sq^{l}(β)

follows from the multiplicativity of the total Steenrod square Sq.

Since L is concentrated in non-positive degrees, it follows that

Sq^{k}(α) = 0 for α ∈ L^{n}, k > −n.

The next theorem sharpens this a priori vanishing property.

Theorem 2. Let α ∈ L^{n}. Then

(8)  Sq^{n}(α) = α^{2},
(9)  Sq^{k}(α) = 0 for k > n.

Proof. By (7) we have

\frac{F (x, y)F (F (x, y), t)}{t} = ∑_{i,j} Sq(u_{i,j}) (\frac{xF (x, t)}{t})^{i} (\frac{yF (y, t)}{t})^{j} ,

or

F (x, y)F (F (x, y), t) = ∑_{i,j} \frac{Sq(u_{i,j})}{t^{i+j−1}}(xF (x, t))^{i}(yF (y, t))^{j}.

The left hand side is an element of L[[t, x, y]] and is invariant under the involutions given by

τ_{x}: t ↦ t, x ↦ F (t, x), y ↦ y,
τ_{y}: t ↦ t, x ↦ x,  y ↦ F (t, y).

Indeed, we have

τ_{x}(F (x, y)F (F (x, y), t)) = F (F (t, x), y)F (F (F (t, x), y), t))
= F (F (x, y), t)F (x, y).

Similarly for τ_{y}.

The involutions τ_{x}, τ_{y} commute, so we may apply Lemma 1 to them successively and find that

F (x, y)F (F (x, y), t) = ∑_{i,j} Q_{i,j}(t)u_{x}^{i}u_{y}^{j} (10)

with u_{x} = xF (t, x), u_{y} = yF (t, y), and Q_{i,j} ∈ L[[t]]. Comparing coefficients we get

\frac{Sq(u_{i,j})}{t^{i+j−1}} = Q_{i,j}(t).

This proves (9) for α = u_{i,j}.

Moreover, setting t = 0 in (10), we get

F (x, y)^{2} = ∑_{i,j} Q_{i,j}(0)x^{2i}y^{2j}.

Hence Q_{i,j}(0) = u_{i,j}^{2}. This proves (8) for α = u_{i,j}.

Since L is generated by the u_{i,j}, the claims follow from the Cartan formula.

Remark. The vanishing properties

Sq^{k}(α) = 0 for α ∈ L^{n}, −n ≥ k > n. (11)

have the following geometric analogue:

Proposition 3. Let M be a compact n-manifold. The pair (P(T M), L(T M)) consisting of the projective tangent bundle of M and its tautological line bundle is bordant.

Proof. By the strict blow up of a manifold Y in a submanifold Z we understand the manifold X → Y obtained from Y by “cutting along Z”, that is by replacing Z by the sphere bundle S(N) of the normal bundle of Z in Y. If Y has no boundary, then X is a manifold with boundary S(N). The usual (real) blow up of Y in Z is the quotient of the strict blow up by the involution v ↦ −v on S(N). See [6, p. 56–57] for details.

Now let X → M ×M be the strict blow up in the diagonal. The switch involution on M × M lifts to an involution σ on X. This involution is fixed point free. The double cover X → X/σ has as boundary the double cover S(T M) → P(T M). The latter is the sphere bundle of the line bundle L(T M) which therefore extends to a line bundle on X/σ.

One may represent the pair (P(T M), L(T M)) by a map f: P(T M) → P^{∞}. Then the proposition means that f is bordant.

Since dim P(T M) = 2n − 1, we may represent the pair by a map f: P(T M) → P^{2n−1}. The bordism group of P^{r} injects into the bordism group of P^{∞}. Therefore the map f will be bordant also as map to P^{2n−1}. Thus we get a relation

0 = [f] ∈ N^{0}(P^{2n−1}) = ⨁_{i=0}^{2n−1} N^{i}.

The 2n relations of (11) are the formal analogies of the vanishing of the 2n components of [f].

All the Wu-relations among Stiefel-Whitney numbers are encoded in these relations. For instance the 0-th component of [f] is just w_{n}(−T M)[M] ∈ F_{2}. The formal analogue of this is Sq^{n}(α) = 0 for α ∈ L^{−n}.

These relations appear in some form in the approaches to the cobordism ring by Quillen [7], [1] and Buonchristiano and Hacon [2], [3], [4], [5].

5. A contracting property of Sq^{0}

For our power series

q(x) = \frac{xF (x, t)}{t}

we consider now the associated homomorphisms θ̄_{q} (see section 3), which we denote by Sq as well. Thus we have ring homomorphisms

Sq: L[[x_{1}, …, x_{r}]] → L[[t]][t^{−1}][[x_{1}, …, x_{r}]]

extending Sq on L and with Sq(x_{i}) = q(x_{i}).

Note that if we restrict to the polynomial rings, we get homomorphisms

Sq: L[x_{1}, …, x_{r}] → L[[t, x_{1}, …, x_{r}]][t^{−1}].

Again we write

Sq = ∑_{k∈Z} t^{−k} Sq^{k}

with additive homomorphisms

Sq^{k}: L[[x_{1}, …, x_{r}]] → L[[x_{1}, …, x_{r}]]

of degree k.

Remark. These operations are the formal versions the Steenrod squaring operations on N^{∗}(P^{∞} × ⋯ × P^{∞}).

Let

I = ⟨x_{1}, …, x_{r}⟩L[[x_{1}, …, x_{r}]] ⊂ L[[x_{1}, …, x_{r}]]

be the ideal generated by the x_{i}.

We write

L[[x_{1}, …, x_{r}]] = ⨁_{k∈Z} U^{k}

with U^{k} the homogeneous component of degree k.

Proposition 4. One has

Sq^{k}(I^{n+k} ∩ U^{l}) ⊂ I^{2n+l−k} ∩ U^{k+l}.

We will need and prove this only in the following special case:

Proposition 5. For n ≥ 0 one has

Sq^{0}(I^{n+1} ∩ U^{1}) ⊂ I^{2n+1} ∩ U^{1}.

Proof. Since Sq^{0} is homogeneous of degree 0, it suffices to show: Let n ≥ 0, a_{−n} ∈ L^{−n}, let further p = (p_{1}, …, p_{r}) be a multi-index with ∑ p_{s} = n + 1, and let α = a_{−n}x^{p}. Then

Sq^{0}(α) ∈ I^{2n+1}.

We have

Sq(α) = Sq(a_{−n}) ∏_{s}(\frac{x_{s}F (x_{s}, t)}{t})^{p_{s}}

= ∑_{k≤−n} t^{−k} Sq^{k}(a_{−n})t^{−(n+1)} ∏_{s}(x_{s}F (x_{s}, t))^{p_{s}}.

Here we have used Theorem 2 to get the upper bound for the index k. Multiplying both sides by t we get

t Sq(α) = ∑_{h≥0} t^{h} Sq^{−n−h}(a_{−n}) ∏_{s}(x_{s}F (x_{s}, t))^{p_{s}} (12)

with h = −k − n.

On the other hand we have

(13) t Sq(α) = Sq^{1}(α) + t Sq^{0}(α) + t^{2} Sq^{−1}(α) + · · · .

COMPUTATIONS IN THE MOD 2 LAZARD RING 9

Since

F (x, t) = x + t + xtP (x, t)

for some P ∈ L[[t, x]], we have

∏_{s} F (x_{s}, t)^{p_{s}} ∈ I^{n+1} + tI^{n} + t^{2}L[[t, x_{1}, …, x_{r}]].

Therefore, by (12),

t Sq(α) ∈ I^{2n+2} + tI^{2n+1} + t^{2}L[[t, x_{1}, …, x_{r}]]

and (13) yields

Sq^{1}(α) + t Sq^{0}(α) ∈ I^{2n+2} + tI^{2n+1}.

This proves the claim.

6. The logarithm

We apply this to the case r = 1 and write z = x_{1}. So let I = zL[[z]] and

L[[z]] = ⨁_{k∈Z} U^{k}.

Then

U^{1} = zF_{2} ⊕ z^{2}L^{−1} ⊕ z^{3}L^{−2} ⊕ · · · .

Proposition 6. Sq^{0}(z) − z ∈ I^{2} ∩ U^{1}, (14) and for m ≥ 0 one has (Sq^{0})^{m}(I^{2} ∩ U^{1}) ⊂ I^{2^{m}+1}. (15) Proof. We have

Sq(z) = \frac{zF (z, t)}{t} = \frac{z^{2}}{t} + z + ∑_{i,j≥1} u_{i,j}z^{i+1}t^{j−1}

and therefore

Sq^{0}(z) = z + z^{2} ∑_{i≥0} u_{i+1,1}z^{i}.

This proves (14). Claim (15) follows from Proposition 5. Corollary 7. For any ℓ_{0} ∈ U^{1}, the series of elements

ℓ_{m} = (Sq^{0})^{m}(ℓ_{0}) ∈ U^{1}

is convergent in the I-adic topology on U^{1}. The limit

ℓ_{∞} = lim_{m→∞} ℓ_{m}

depends alone on the class of ℓ_{0} in U^{1}/(I^{2} ∩ U^{1}).

Since U^{1}/(I^{2} ∩ U^{1}) = F_{2}, there is only one nontrivial such limit. It is obtained for ℓ_{0}(z) = z and given by

ℓ = lim_{m→∞}(Sq^{0})^{m}(z).

It is called the canonical logarithm of the universal formal group law. In the following we show that ℓ is indeed a logarithm.

Obviously we have

Sq^{0}(ℓ) = ℓ. (16)

The commutative diagram (6) yields the commutative diagram

\begin{matrix}L[[z]] & ⟶^{Sq^{0}} & R[[z]] \\ ^{α}↓ & & ^{α}↓ \\ L[[x, y]] & ⟶^{Sq^{0}} & R[[x, y]]. \end{matrix} (17)

with α(z) = F (x, y). Let β_{x}: L[[x]] → L[[x, y]], β_{y}: L[[y]] → L[[x, y]] be the inclusions. Further let

I = ⟨x, y⟩L[[x, y]].

Lemma 8. Let ℓ ∈ L[[z]] with deg ℓ = 1 and let ℓ′ = Sq^{0}(ℓ).

Let further n ≥ 0 and suppose that

α(ℓ) = β_{x}(ℓ) + β_{y}(ℓ) mod I^{n+1}.

Then

α(ℓ′) = β_{x}(ℓ′) + β_{y}(ℓ′) mod I^{2n+1}.

Proof. This follows from Proposition 5 and the fact that α, β_{x}, β_{y} are compatible with Sq^{0}.

Corollary 9. ℓ(F (ℓ^{−1}(x), ℓ^{−1}(y)) = x + y.

This clear from the Lemma and (16).

Remark. For the moment we have completely omitted a further description of the logarithm. It is desirable to give the formal analogy of the description of the coefficients of the logarithm in [8].

References

[1] T. Bröcker and T. tom Dieck, Kobordismentheorie, Springer-Verlag, Berlin, 1970, Lecture Notes in Mathematics, Vol. 178.

[2] S. Buoncristiano and D. Hacon, An elementary geometric proof of two theorems of Thom, Topology 20 (1981), no. 1, 97–99.

[3] , The geometry of Chern numbers, Ann. of Math. (2) 118 (1983), no. 1, 1–7.

[4] , A geometrical approach to characteristic numbers, Geometry today (Rome, 1984), Birkhäuser Boston, Boston, MA, 1985, Edited and with a preface by E. Arbarello, C. Procesi and E. Strickland, pp. 21–38.

[5] , On the formal groups approach to cobordism, Proc. Amer. Math. Soc. 97 (1986), no. 3, 559–564.

[6] K. Jänich, On the classification of O(n)-manifolds, Math. Ann. 176 (1968), 53–76.

[7] D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971).

[8] K. Shibata, A note on the formal group law of unoriented cobordism theory, Osaka J. Math. 10 (1973), 33–42.

School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

E-mail address: rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516