Integral Invariants and Cohomology of BSpin(n)

D. J. Benson^{∗} Mathematical Institute 24–29 St. Giles Oxford OX1 3LB Great Britain

Jay A. Wood^{†} Dept. of Mathematics, Computer Science, and Statistics Purdue University Calumet Hammond, IN 46323, U.S.A. woodja@pucal.bitnet

1 Introduction

The purpose of this paper is to determine the integral cohomology H^{∗}(BSpin(n); ) of the classifying spaces of the spinor groups Spin(n). The mod two cohomology H^{∗}(BSpin(n); /2) was calculated by Quillen [11].

Let G be a compact Lie group with maximal torus ρ : T ↪ G, and let ρ_{∗} : BT → BG be the corresponding map of classifying spaces. Then a theorem of Borel [2] says that the restriction map

ρ^{∗} : H^{∗}(BG; ) → H^{∗}(BT; )

has as its kernel the ideal of torsion elements in H^{∗}(BG; ). The image lies in the invariants H^{∗}(BT; )^{W} of the Weyl group W = N_{G}(T)/T, and so ρ^{∗} passes to a well defined injective map

ρ̄^{∗} : H^{∗}(BG; )/torsion → H^{∗}(BT; )^{W} .

Thus the calculation of H^{∗}(BG; ) breaks into two parts: calculate the image of ρ̄^{∗}, and then calculate the torsion.

In the case of the special orthogonal groups G = SO(n), the map ρ̄^{∗} is an isomorphism, and Brown [5] and Feshbach [6] used this to give a complete calculation of H^{∗}(BSO(n); ). Their descriptions also apply to the stable special orthogonal group G = SO. Turning to the spinor groups, Thomas [12] calculated H^{∗}(BSpin; ) (and H^{∗}(BSpin; /2)) in the stable case. For the finite-dimensional spinor groups G = Spin(n), Feshbach [7] showed that ρ̄^{∗} is not an isomorphism for n = 11, 12. The second author ([13], Corollary 4.2) showed that ρ̄^{∗} is not an isomorphism for n = 13, but is again an isomorphism for 14 ≤ n ≤ 18. We shall see that for n ≥ 6, ρ̄^{∗} is an isomorphism if and only if n is not congruent to 3, 4 or 5 modulo 8, confirming a speculation made by the second author in Section 4 of [13]. In any case, we shall calculate the exact image, and use this to determine the integral cohomology of BSpin(n).

∗Current address: Department of Mathematics, University of Georgia, Athens GA 30602, USA.

djb@sloth.math.uga.edu

†Partially supported by NSA grant MDA904-91-H-0029, Purdue Research Foundation International Travel Grant, and Purdue University Calumet Scholarly Research Awards.

Here is a brief summary of the paper. After covering some preliminary material and establishing notation in Section 2, our first goal is to calculate the integral Weyl group invariants H^{∗}(BT̃; )^{W}, for T̃ a maximal torus in Spin(n). To that end, we first produce, in Section 3, a sequence q_{1}, q_{2}, … of new integral invariants which arise because certain combinations of Pontrjagin classes become divisible by 2 when inflated to BT̃. In Section 4 we produce additional integral invariants by taking orbit products over certain subgroups of W and observing that the elements so obtained are actually invariant under all of W. Having produced a number of integral invariants, we proceed to show that these new integral invariants, together with the known integral invariants inherited from BSO(n), account for all the integral invariants. The argument, which is the main technical result in the paper and which encompasses Sections 5, 6, and 7, goes as follows. Because the Weyl group W is a split extension of its normal subgroup of sign changes by its subgroup of permutations, the integral invariants of W are obtained by first calculating the invariants under sign changes, and then finding invariants under permutations. The calculation of the invariants under sign changes can be accomplished inductively, with the detailed arguments appearing in Section 5 for the case where n is odd and in Section 6 for n even. (The underlying idea is actually quite simple. In the notation of Theorem 5.1, η_{j} + ε_{j+1}(η_{j}) can be handled by q_{j} + lower order terms, while η_{j} · ε_{j+1}(η_{j}) = η_{j+1}.) Once the invariants under sign changes are known, the integral invariants under all of W are quickly determined in Section 7.

With H^{∗}(BT̃; )^{W} now understood, our second goal is to calculate the image of

ρ̄^{∗} : H^{∗}(BSpin(n); )/torsion → H^{∗}(BT̃; )^{W} .

Once again, there are certain things one can say a priori. The Pontrjagin classes p_{i} ∈ H^{∗}(BT̃; )^{W} are in the image of ρ̄^{∗} because they are just inflated from BSO(n). The integral invariants of Section 4 (or their squares) occur as Euler classes of spin representations of Spin(n), and hence are in the image of ρ̄^{∗}. This is proved in Section 8. In order to show that the q_{i}’s of Section 3 are in the image of ρ̄^{∗}, we make use of the fact that the reduction modulo 2 of H^{∗}(BSpin(n); )/torsion is precisely the Bockstein cohomology of H^{∗}(BSpin(n); /2). The structure of the Bockstein cohomology is reviewed in Section 9. We then argue inductively in Section 10 that, modulo lower order terms, the q_{i}’s are the images under ρ̄^{∗} of integral lifts of generators in the Bockstein cohomology. We reconcile the Euler class information and the Bockstein cohomology information to calculate the image of ρ̄^{∗} in H^{∗}(BT̃; )^{W}. The only circumstances when ρ̄^{∗} fails to be surjective are when n ≥ 6 and is congruent to 3, 4 or 5 modulo 8.

Finally, in Section 11, we describe the integral cohomology H^{∗}(BSpin(n); ) as a pullback. Although we give precise generators for H^{∗}(BSpin(n); )/torsion and the general form of their relations (see Proposition 3.3), we do not give explicit generators and relations for H^{∗}(BSpin(n); ) itself.

2 Notation and preliminaries

In the next few sections, which form the core of the paper, we calculate H^{∗}(BT̃; )^{W}, the invariants of the Weyl group on the integral cohomology ring of the classifying space of a maximal torus T̃ in Spin(n). This computation, to our knowledge, is one of the few known calculations of integral invariants of a finite group beyond the cases of the Weyl groups of the unitary, orthogonal, and symplectic groups.

The argument has two parts. First we produce a number of integral invariants via a recursive algorithm. Then we show that these invariants, together with known invariants coming from the maximal torus in SO(n), account for all integral invariants.

We review some standard notation and theorems. References for this material include Borel [2], Bröcker and tom Dieck [4], and Husemoller [8].

We write SO(n) for the compact connected Lie group consisting of the n×n real matrices of determinant one which are orthogonal with respect to a positive definite quadratic form. We write T for the maximal torus SO(2)^{r} ⊆ SO(n), of dimension r = [n/2]. Then we have

H^{∗}(BT; ) = [x_{1}, …, x_{r}],

with deg(x_{i}) = 2. We write c_{i} for the ith elementary symmetric function in x_{1}, …, x_{r}, and p_{i} for the ith elementary symmetric function in x_{1}^{2}, …, x_{r}^{2}, so that deg(c_{i}) = 2i while deg(p_{i}) = 4i. We use the convention that c_{0} = p_{0} = 1 to save notation. We also write e for c_{r} = x_{1} ⋯ x_{r} of degree 2r. These elements may be thought of as Chern classes, Pontrjagin classes, and the Euler class respectively, in a suitable sense. The relationship between the p_{i} and the c_{i} is given by

p_{i} = c_{i}^{2} − 2c_{i−1}c_{i+1} + 2c_{i−2}c_{i+2} − ⋯ + (−1)^{i}2c_{2i}.

See, for example, Milnor and Stasheff [10], p. 177.

For n ≥ 3 we have π_{1} SO(n) ≅ /2, so that the simply connected double cover Spin(n) is the universal covering group

1 → /2 → Spin(n) ⟶^{π} SO(n) → 1.

Let T̃ = π^{−1}(T) be the inverse image in Spin(n) of T; T̃ is a maximal torus in Spin(n). Then

H^{∗}(BT̃; ) = [x_{1}, …, x_{r}, A],

where A = c_{1}/2 = \frac{1}{2}(x_{1}+⋯+x_{r}); see, for example, [4], pp. 174–175. In this notation, the induced map π^{∗} : H^{∗}(BT; ) → H^{∗}(BT̃; ) is the obvious inclusion. Note that we use square brackets to denote the ring generated by some elements, even when the ring is not polynomial in these generators. The above ring is in fact a polynomial ring over , in A and any subset of r−1 of the x_{i}, but we write it in the above form to make transparent the action of the Weyl group W = N_{SO(n)}(T)/T = N_{Spin(n)}(T̃)/T̃.

If n = 2r + 1 is odd, W is of type B_{r}, and is generated by the sign changes ε_{i} : x_{i} ↦ −x_{i}, x_{j} ↦ x_{j} (j ≠ i) and permutations on the x_{i}, so that it is a split extension

1 → ( /2)^{r} → W → Σ_{r} → 1.

A sign change ε_{i} will send A to A − x_{i}, while the permutations fix A. It is easy to see that in this case

H^{∗}(BT; )^{W} ≅ [p_{1}, …, p_{r}].

If n = 2r is even, W is of type D_{r}, and is generated by the even sign changes ε_{i}ε_{j} and permutations on the x_{i}, so that it is a split extension

1 → ( /2)^{r−1} → W → Σ_{r} → 1.

In this case we have

H^{∗}(BT; )^{W} ≅ [p_{1}, …, p_{r−1}, e].

3 Construction of invariants

Recall that A = c_{1}/2 = \frac{1}{2}(x_{1} + ⋯ + x_{r}), and that we use square brackets to denote the ring generated by some elements, even when the ring is not polynomial in these generators.

Lemma 3.1 We have [x_{1}, …, x_{r}, A]^{Σ_{r}} = [A, c_{2}, …, c_{r}].

Proof The left hand side contains the right hand side, and is contained in the field of fractions of the right hand side. Since [x_{1}, …, x_{r}, A] is integral over the right hand side, so are the invariants. Moreover, the right hand side is integrally closed, so we have equality. □

Lemma 3.2 We have H^{∗}(BT̃; )^{W} = H^{∗}(BT; [\frac{1}{2}])^{W} ∩ [A, c_{1}, …, c_{r}].

Proof We have

H^{∗}(BT̃; [\frac{1}{2}]) = H^{∗}(BT; [\frac{1}{2}]),

and

H^{∗}(BT̃; )^{W} = H^{∗}(BT̃; [\frac{1}{2}])^{W} ∩ H^{∗}(BT̃; )^{Σ_{r}}.

Now apply the previous lemma. □

Proposition 3.3 There are elements q_{i} ∈ H^{∗}(BT̃; )^{W} (i ≥ 1) satisfying (i) deg(q_{i}) = 2^{i+1}, (ii) 2q_{1} = p_{1}, (iii) q_{i} − 2A^{2^{i}} ∈ [x_{1}, …, x_{r}](1, A, …, A^{2^{i}−1}), the module over [x_{1}, …, x_{r}] gen-

erated by 1, A, …, A^{2^{i}−1}, (iv) 2q_{i+1} − q_{i}^{2} ∈ [p_{2}, …, p_{r}].

Proof Let c_{i} be the ith elementary symmetric function of x_{1}, …, x_{r} (and c_{i} = 0 for i > r). Then (see Section 2) we have

p_{1} = c_{1}^{2} − 2c_{2} = 4A^{2} − 2c_{2}

and for i > 1,

p_{i} = c_{i}^{2} − 2c_{i−1}c_{i+1} + ⋯ ± 4Ac_{2i−1} ∓ 2c_{2i}.

We inductively define polynomial functions f_{i}(c) of c_{2}, …, c_{2^{i}}, and g_{i} = g_{i}(A, c) of A, c_{2}, …, c_{2^{i}} for i ≥ 1 by f_{1}(c) = c_{2} and

f_{i}(p) = f_{i}(c)^{2} + 4Ag_{i} + 2f_{i+1}(c).

Here, f_{i}(p) denotes the polynomial in the p_{j} obtained by replacing each c_{j} by p_{j} in f_{i}(c). This inductive definition is possible because of the formula above for p_{i}, which ensures that all terms in f_{i}(p) − f_{i}(c)^{2} have even coefficients, and all terms involving A have coefficients divisible by four.

Next, we inductively define functions h_{i} = h_{i}(A, c) for i ≥ 1 by h_{1} = A and

h_{i+1} = Ah_{i}^{2} − h_{i}f_{i}(c) − g_{i}.

Thus, for example, we have

f_{1}(p) = p_{2} = c_{2}^{2} − 4Ac_{3} + 2c_{4},

and so g_{1} = −c_{3}, f_{2}(c) = c_{4}, and h_{2} = A^{3} − Ac_{2} + c_{3}.

We now define q_{i} = 2Ah_{i} − f_{i}(c) for i ≥ 1. Thus for example q_{1} = 2A^{2} − c_{2} so that 2q_{1} = p_{1}, and q_{2} = 2A^{4} − 2A^{2}c_{2} + 2Ac_{3} − c_{4}. It follows that deg(q_{i}) = 2^{i+1} and that

q_{i} − 2A^{2^{i}} ∈ [x_{1}, …, x_{r}](1, A, …, A^{2^{i}−1}).

Observe that

2q_{i+1} = 4Ah_{i+1} − 2f_{i+1}(c)
= 4A(Ah_{i}^{2} − h_{i}f_{i}(c) − g_{i}) − 2f_{i+1}(c)
= (4A^{2}h_{i}^{2} − 4Ah_{i}f_{i}(c) + f_{i}(c)^{2})
− (f_{i}(c)^{2} + 4Ag_{i} + 2f_{i+1}(c))
= (2Ah_{i} − f_{i}(c))^{2} − f_{i}(p)
= q_{i}^{2} − f_{i}(p).

It follows that

2q_{i+1} − q_{i}^{2} = −f_{i}(p) ∈ [p_{2}, …, p_{r}],

and so we see inductively that q_{i} is in [\frac{1}{2}][p_{1}, …, p_{r}]. It is clear that q_{i} is also in [A, c_{2}, …, c_{r}], and so by Lemma 3.2 we have q_{i} ∈ H^{∗}(BT̃; )^{W}. □

4 Another invariant

We now describe a further integral invariant, whose form depends on the parity of n. If n = 2r + 1 is odd, and 0 ≤ j ≤ r, let E_{j} = ⟨ε_{1}, …, ε_{j}⟩ as a subgroup of W, so that E_{j} ≅ ( /2)^{j}, and E_{0} is the trivial group. Set

η_{j} = ∏_{g∈E_{j}} g(A) = ∏_{I⊆{1,2,...,j}}(A − ∑_{i∈I} x_{i}),

so that deg(η_{j}) = 2^{j+1}, and η_{0} = A. Note that η_{j} is a product of the elements of an orbit of E_{j}, and is hence E_{j}-invariant.

If n = 2r is even, and 1 ≤ j ≤ r, let F_{j} = ⟨ε_{1}ε_{2}, …, ε_{1}ε_{j}⟩ as a subgroup of W, so that F_{j} ≅ ( /2)^{j−1}, and F_{1} is the trivial group. Set

μ_{j} = ∏_{g∈F_{j}} g(A) = ∏_{{I⊆\{1,2,...,j\} \atop |I| even}}(A − ∑_{i∈I} x_{i}),

so that deg(μ_{j}) = 2^{j}, and μ_{1} = A. Note that μ_{j} is a product of elements of an orbit of F_{j}, and is hence F_{j}-invariant.

Proposition 4.1 (i) If n = 2r + 1 ≥ 7, so that W has type B_{r}, then η_{r−1} is an invariant of degree 2^{r} in H^{∗}(BT̃; )^{W}. We have η_{r−1}^{2} = η_{r}.

(ii) If n = 2r ≥ 8 is divisible by 4, so that W has type D_{r} with r even, then μ_{r−1} is an invariant of degree 2^{r−1} in H^{∗}(BT̃; )^{W}. We have μ_{r−1}^{2} = μ_{r}.

(iii) If n = 2r ≥ 6 is congruent to 2 modulo 4, so that W has type D_{r} with r odd, then μ_{r} is an invariant of degree 2^{r} in H^{∗}(BT̃; )^{W}.

Proof (i) We first check invariance under the sign change ε_{i} for i < r. This permutes the factors in the product defining η_{r−1} by swapping each term in which x_{i} has been subtracted with the corresponding term in which it has not. Therefore the product is invariant. Similarly, any permutation of the variables x_{1}, …, x_{r−1} permutes the terms in η_{r−1}, and so it leaves the product invariant.

Next, we observe that

A − ∑_{i∈I} x_{i} = −(A − ∑_{i∈{1,...,r}\I} x_{i}).

There are 2^{r−2} terms in which x_{1} has been subtracted. Since n ≥ 7, we have r − 2 ≥ 1, and so 2^{r−2} is even. So negating these terms does not alter the total sign of η_{r−1}, which shows that

η_{r−1} = ∏_{I⊆{2,...,r}}(A − ∑_{i∈I} x_{i}).

It follows that η_{r−1} is invariant under the sign changes and permutations on the variables x_{2}, …, x_{r}, and hence under the whole Weyl group. By negating the 2^{r−1} terms in η_{r} which involve x_{r}, it also follows that η_{r} = η_{r−1}^{2}.

The proofs of (ii) and (iii) are similar. □

5 Invariants of sign changes: n odd

We now come to the heart of the matter, namely determining the integral invariants of the subgroup of W generated by the sign changes. The integral invariants of the whole of W will be discussed in Section 7. We begin with the case where n is odd.

Theorem 5.1 Suppose that n = 2r + 1 is odd. Then for 1 ≤ j ≤ r − 1, we have

H^{∗}(BT̃; )^{E_{j}} = [x_{1}^{2}, …, x_{j}^{2}, x_{j+1}, …, x_{r}, q_{1}, …, q_{j−1}, η_{j}].

Furthermore, provided n ≥ 7, for j = r we have

H^{∗}(BT̃; )^{E_{r}} = [x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}].

Proof We prove the first statement by induction on j. For this purpose, it is convenient to set q_{0} = 0 and adjoin it to the list of generators. Then the statement makes sense and is true for j = 0, since η_{0} = A. So suppose that 0 ≤ j ≤ r − 1 and that the theorem is true for j. Then

H^{∗}(BT̃; )^{E_{j+1}} = (H^{∗}(BT̃; )^{E_{j}})^{⟨ε_{j+1}⟩}.

The element ε_{j+1} fixes all the generators except x_{j+1} and η_{j}. Set

R = [x_{1}^{2}, …, x_{j+1}^{2}, x_{j+2}, …, x_{r}, q_{1}, …, q_{j}, η_{j+1}] ⊆ H^{∗}(BT̃; )^{E_{j+1}}.

Examining the coefficients of A^{2^{j}} in η_{j} and q_{j} (see Proposition 3.3), we see that

η_{j} + ε_{j+1}(η_{j}) − q_{j} ∈ H^{∗}(BT̃; )^{E_{j}} ∩ [x_{1}, …, x_{r}](1, A, …, A^{2^{j}−1})
⊆ [x_{1}^{2}, …, x_{j}^{2}, x_{j+1}, …, x_{r}, q_{1}, …, q_{j−1}].

It is also ε_{j+1}-invariant and hence in R. Similarly, η_{j} −ε_{j+1}(η_{j}) is in the same subring and is negated by ε_{j+1}, and is hence in Rx_{j+1}. So we write η_{j} − ε_{j+1}(η_{j}) = x_{j+1}η′_{j} with η′_{j} ∈ R.

Adding, we see that

2η_{j} = (η_{j} + ε_{j+1}(η_{j})) + (η_{j} − ε_{j+1}(η_{j})) ∈ R + Rx_{j+1}, (∗)

while

η_{j}^{2} = η_{j}(ε_{j+1}(η_{j}) + x_{j+1}η′_{j}) = η_{j+1} + (x_{j+1}η′_{j})η_{j}.

So every element of H^{∗}(BT̃; )^{E_{j}} may be written (non-uniquely) as

f = (f_{1} + f_{2}x_{j+1}) + (f_{3} + f_{4}x_{j+1})η_{j}

with f_{1}, …, f_{4} ∈ R. Using (∗), we may change f_{3} + f_{4}x_{j+1} by any multiple of two, at the expense of changing f_{1} and f_{2}.

The image of this element under ε_{j+1} is

ε_{j+1}(f) = (f_{1} − f_{2}x_{j+1}) + (f_{3} − f_{4}x_{j+1})(η_{j} − x_{j+1}η′_{j}),

which differs from the original expression by

f − ε_{j+1}(f) = x_{j+1}[2f_{2} + 2f_{4}(η_{j} − x_{j+1}η′_{j}) + (f_{3} + f_{4}x_{j+1})η′_{j}].

So if f is invariant under the action of ε_{j+1}, then (f_{3} + f_{4}x_{j+1})η′_{j} must be divisible by two in [x_{1}, …, x_{r}, A].

Now provided j +1 < r, so that x_{1}, …, x_{j+1}, A are algebraically independent, we can argue as follows. We have η′_{j} ∈ [x_{1}, …, x_{j+1}, A] and η′_{j} is congruent to −x_{j+1}^{2^{j}−1} modulo the ideal (x_{1}, …, x_{j}, A), so that it is not divisible by two in [x_{1}, …, x_{r}, A]. So if (f_{3} + f_{4}x_{j+1})η′_{j} is divisible by two in [x_{1}, …, x_{r}, A] then by uniqueness of factorization in this ring, it is f_{3} +f_{4}x_{j+1} which is divisible by two, say f_{3} +f_{4}x_{j+1} = 2f′ with f′ ∈ [x_{1}, …, x_{r}, A]. Since 2f′ ∈ ⟨R, x_{j+1}⟩ ⊆ H^{∗}(BT̃; )^{E_{j}}, we have f′ ∈ H^{∗}(BT̃; )^{E_{j}}, and so we may write it in the form

f′ = (f′_{1} + f′_{2}x_{j+1}) + (f′_{3} + f′_{4}x_{j+1})η_{j}.

Substituting back, we see that the original element f can be expressed in the form

f = (f′′_{1} + f′′_{2}x_{j+1}) + 2(f′′_{3} + f′′_{4}x_{j+1})η_{j}.

By adjusting f′′_{1} and f′′_{2} appropriately, we may set f′′_{3} = f′′_{4} = 0. Since f is invariant under ε_{j+1}, we have f′′_{2} = 0 and so f = f′′_{1} ∈ R as required.

In the remaining case, where j + 1 = r, by Proposition 4.1 the element η_{j} is already invariant under ε_{j+1}, and so we are done. □

6 Invariants of sign changes: n even

Theorem 6.1 Suppose that n = 2r is even. Then for 2 ≤ j ≤ r − 1,

H^{∗}(BT̃; )^{F_{j}} = [x_{1}^{2}, …, x_{j}^{2}, x_{j+1}, …, x_{r}, x_{1} ⋯ x_{j}, q_{1}, …, q_{j−2}, μ_{j}].

Furthermore, provided n ≥ 6, for j = r we have

H^{∗}(BT̃; )^{F_{r}} = \{ \begin{matrix}[x_{1}^{2}, …, x_{r}^{2}, e, q_{1}, …, q_{r−2}, μ_{r}], & r odd, \\ [x_{1}^{2}, …, x_{r}^{2}, e, q_{1}, …, q_{r−3}, μ_{r−1}], & r even. \end{matrix}

Proof We prove the first statement (and the first case of the second statement) by induction on j. As in the proof of Theorem 5.1, we set q_{0} = 0, so that the statement makes sense and is true for j = 1, since μ_{1} = A. So suppose that 1 ≤ j ≤ r − 1 and that the statement is true for j. Then

H^{∗}(BT̃; )^{F_{j+1}} = (H^{∗}(BT̃; )^{F_{j}})^{⟨ε_{1}ε_{j+1}⟩}.

The element ε_{1}ε_{j+1} fixes all generators except x_{1} ⋯ x_{j}, x_{j+1} and μ_{j}. Set

R = [x_{1}^{2}, …, x_{j+1}^{2}, x_{j+2}, …, x_{r}, x_{1} ⋯ x_{j+1}, q_{1}, …, q_{j−1}, μ_{j+1}] ⊆ H^{∗}(BT̃; )^{F_{j+1}}.

Examining the coefficients of A^{2^{j−1}} in μ_{j} and q_{j−1}, we see that

μ_{j} + ε_{1}ε_{j+1}(μ_{j}) − q_{j−1} ∈ H^{∗}(BT̃; )^{F_{j}} ∩ [x_{1}, …, x_{r}](1, A, …, A^{2^{j−1}−1})
⊆ [x_{1}^{2}, …, x_{j}^{2}, x_{j+1}, …, x_{r}, x_{1} ⋯ x_{j}, q_{1}, …, q_{j−2}].

Every element of this ring is of the form

f = f_{1} + f_{2}x_{j+1} + f_{3}x_{1} ⋯ x_{j}

with f_{1}, f_{2}, f_{3} ∈ R (though note that q_{j−1} and μ_{j+1} are in R but not in this ring). The image of such an element under ε_{1}ε_{j+1} is

ε_{1}ε_{j+1}(f) = f_{1} − f_{2}x_{j+1} − f_{3}x_{1} ⋯ x_{j},

which differs from the original expression by

f − ε_{1}ε_{j+1}(f) = 2(f_{2}x_{j+1} + f_{3}x_{1} ⋯ x_{j}).

Thus if f is invariant under the action of ε_{1}ε_{j+1} then f = f_{1} is in R. In particular, this shows that

μ_{j} + ε_{1}ε_{j+1}(μ_{j}) − q_{j−1} ∈ R.

Similarly, μ_{j} − ε_{1}ε_{j+1}(μ_{j}) is in this same subring and is negated by ε_{1}ε_{j+1}, and is hence in Rx_{j+1} + Rx_{1} ⋯ x_{j}. So we write

μ_{j} − ε_{1}ε_{j+1}(μ_{j}) = x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j}

with μ′_{j} and μ′′_{j} in R. Adding, we see that

2μ_{j} = (μ_{j} + ε_{1}ε_{j+1}(μ_{j})) + (μ_{j} − ε_{1}ε_{j+1}(μ_{j})) ∈ R + Rx_{j+1} + Rx_{1} ⋯ x_{j}, (∗∗)

while

μ_{j}^{2} = μ_{j}(ε_{1}ε_{j+1}(μ_{j}) + x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j}) = μ_{j+1} + (x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j})μ_{j}.

So every element of H^{∗}(BT̃; )^{F_{j}} may be written (non-uniquely) as

f = (f_{1} + f_{2}x_{j+1} + f_{3}x_{1} ⋯ x_{j}) + (f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j})μ_{j}

with f_{1}, …, f_{6} ∈ R. Using (∗∗), we may change the second half of this expression by any multiple of two at the expense of changing the first half.

The image of this element under ε_{1}ε_{j+1} is

(f_{1} − f_{2}x_{j+1} − f_{3}x_{1} ⋯ x_{j}) + (f_{4} − f_{5}x_{j+1} − f_{6}x_{1} ⋯ x_{j})(μ_{j} − x_{j+1}μ′_{j} − x_{1} ⋯ x_{j}μ′′_{j})

which differs from the original expression by

f − ε_{1}ε_{j+1}(f) = x_{j+1}[2f_{2} + 2f_{5}(μ_{j} − x_{j+1}μ′_{j} − x_{1} ⋯ x_{j}μ′′_{j})
+ (f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j})μ′_{j}]
+ x_{1} ⋯ x_{j}[2f_{3} + 2f_{6}(μ_{j} − x_{j+1}μ′_{j} − x_{1} ⋯ x_{j}μ′′_{j})
+ (f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j})μ′′_{j}].

So if f is invariant under the action of ε_{1}ε_{j+1}, then

(f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j})(x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j})

is divisible by two in [x_{1}, …, x_{r}, A].

Now provided j + 1 < r, so that x_{1}, …, x_{j+1}, A are algebraically independent, we can argue as follows. We have μ′_{j}, μ′′_{j} ∈ [x_{1}, …, x_{j+1}, A] and

x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j} = μ_{j} − ε_{1}ε_{j+1}(μ_{j})

is congruent to −x_{j+1}^{2^{j−1}} modulo the ideal (x_{1}, …, x_{j}, A), so that it is not divisible by two in [x_{1}, …, x_{r}, A]. It follows that f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j} is divisible by two in [x_{1}, …, x_{r}, A], say

f_{4} + f_{5}x_{j+1} + f_{6}x_{1} ⋯ x_{j} = 2f′.

Since 2f′ ∈ H^{∗}(BT̃; )^{F_{j}}, we have f′ ∈ H^{∗}(BT̃; )^{F_{j}} and so we may write it in the form

f′ = (f′_{1} + f′_{2}x_{j+1} + f′_{3}x_{1} ⋯ x_{j}) + (f′_{4} + f′_{5}x_{j+1} + f′_{6}x_{1} ⋯ x_{j})μ_{j}.

Substituting back, we obtain an expression of the form

f = (f′′_{1} + f′′_{2}x_{j+1} + f′′_{3}x_{1} ⋯ x_{j}) + 2(f′′_{4} + f′′_{5}x_{j+1} + f′′_{6}x_{1} ⋯ x_{j})μ_{j}.

By adjusting f′′_{1}, f′′_{2}, f′′_{3} suitably, we may set f′′_{4} = f′′_{5} = f′′_{6} = 0. Since f is invariant under ε_{1}ε_{j+1}, we have 2(f′′_{2}x_{j+1} + f′′_{3}x_{1} ⋯ x_{j}) = 0 and so f = f′′_{1} ∈ R as required.

If j + 1 = r is odd, then x_{r} is congruent modulo the ideal (A) to x_{1} + ⋯ + x_{r−1}, which has an even number of terms. It follows that

x_{j+1}μ′_{j} + x_{1} ⋯ x_{j}μ′′_{j} = μ_{j} − ε_{1}ε_{j+1}(μ_{j})

is congruent modulo (A) to the product of the sums of the subsets of odd size of {x_{1}, …, x_{r−1}}, and is therefore not divisible by two in [x_{1}, …, x_{r}, A]. The rest of the argument proceeds as before.

Finally, if j + 1 = r is even, then by Proposition 4.1 the element μ_{j} is already invariant under ε_{1}ε_{j+1}, and so we are done. □

7 Integral invariants of the Weyl group

We have now done most of the hard work involved in finding the invariants of the Weyl group.

Theorem 7.1 (i) If n = 2r + 1 ≥ 7 is odd, then

H^{∗}(BT̃; )^{W} = [p_{1}, …, p_{r}, q_{1}, …, q_{r−2}, η_{r−1}].

(ii) If n = 2r ≥ 6 is even, then

H^{∗}(BT̃; )^{W} = \{ \begin{matrix}[p_{1}, …, p_{r−1}, e, q_{1}, …, q_{r−2}, μ_{r}], & r odd, \\ [p_{1}, …, p_{r−1}, e, q_{1}, …, q_{r−3}, μ_{r−1}], & r even. \end{matrix}

Proof (i) Using Theorem 5.1, we have

H^{∗}(BT̃; )^{W} = (H^{∗}(BT̃; )^{E_{r}})^{Σ_{r}} = ( [x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}])^{Σ_{r}}.

We have a short exact sequence

0 → [x_{1}^{2}, …, x_{r}^{2}] → [x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}]
→ [q_{1}, …, q_{r−2}, η_{r−1}]/( [q_{1}, …, q_{r−2}, η_{r−1}] ∩ [x_{1}^{2}, …, x_{r}^{2}]) → 0

and hence an exact sequence

0 → [x_{1}^{2}, …, x_{r}^{2}]^{Σ_{r}} → [x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}]^{Σ_{r}}
→ [q_{1}, …, q_{r−2}, η_{r−1}]/( [q_{1}, …, q_{r−2}, η_{r−1}] ∩ [x_{1}^{2}, …, x_{r}^{2}]).

The last map in this sequence is surjective since q_{1}, …, q_{r−2}, η_{r−1} are invariant, and so [x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}]^{Σ_{r}} is generated by [x_{1}^{2}, …, x_{r}^{2}]^{Σ_{r}} = [p_{1}, …, p_{r}] and q_{1}, …, q_{r−2}, η_{r−1}.

The proof of (ii) is similar. □

Corollary 7.2 (i) If n = 2r + 1 ≥ 7 is odd, then

2η_{r−1} − q_{r−1} ∈ [p_{2}, …, p_{r}, q_{1}, …, q_{r−2}].

(ii) If n = 2r ≥ 8 is divisible by 4, then

2μ_{r−1} − q_{r−2} ∈ [p_{2}, …, p_{r−1}, e, q_{1}, …, q_{r−3}].

Proof (i) By Proposition 3.3, q_{r−1} ∈ H^{∗}(BT̃; )^{W}. Then, by considering the highest powers of A that appear, we have

2η_{r−1} − q_{r−1} ∈ H^{∗}(BT̃; )^{W} ∩ [x_{1}, …, x_{r}](1, A, …, A^{2^{r−1}−1})
⊆ [p_{1}, …, p_{r}, q_{1}, …, q_{r−2}].

Since p_{1} = 2q_{1}, p_{1} is redundant.

The proof of (ii) is similar. □

Remark 7.3 In all the results of Sections 4, 5, 6, and 7, there have been restrictions upon the values of n for which the results apply. This is an appropriate place to summarize what happens for small values of n.

The restrictions are necessary because the proof that η_{r−1} (respectively μ_{r−1}) is an invariant (Proposition 4.1) breaks down if r < 3. Of course, η_{r} (respectively μ_{r}) is still an invariant. It is easy to compute that

H^{∗}(BT̃; )^{W} = \{ \begin{matrix}[e, μ_{1}] = [μ_{1}], & n = 2, \\ [p_{1}, q_{1}, η_{1}] = [η_{1}], & n = 3, \\ [p_{1}, e, μ_{2}] = [e, μ_{2}], & n = 4, \\ [p_{1}, p_{2}, q_{1}, η_{2}] = [q_{1}, η_{2}], & n = 5. \end{matrix}

8 Euler classes of spin representations

The purpose of this section is to prove that the elements η_{r−1}, μ_{r}, μ_{r−1} which appear in the description of H^{∗}(BT̃; )^{W} arise as the Euler classes of vector bundles associated to spin representations of Spin(n). Most of the material in this section is well-known. The reader is referred to [1], [4], Section VI.6, or [8], Chapters 11 and 13.

Given a representation ρ : Spin(n) → GL(V), V a complex vector space, denote by e(ρ) and c_{i}(ρ) the Euler class and Chern classes in integral cohomology of BSpin(n). Similarly, if V is a real vector space, denote by e(ρ), p_{i}(ρ) the Euler class and Pontrjagin classes in integral cohomology, and by w_{i}(ρ) the Stiefel–Whitney classes in cohomology modulo two. In this section, we shall calculate the Euler class e(ρ) when ρ is a spin representation.

For the sake of completeness in the Theorems below, we define one more invariant along the lines of η and μ of Section 4. For 2 ≤ j ≤ r, set

ν_{j} = ∏_{{I⊆\{1,2,...,j\} \atop |I| odd}}(A − ∑_{i∈I} x_{i}).

Note that μ_{j} is obtained if one requires |I| even instead. The following Proposition has a proof similar to that of Proposition 4.1.

Proposition 8.1 (i) If n = 2r ≥ 8 is divisible by 4, then ν_{r−1} is an invariant of degree 2^{r−1} in H^{∗}(BT̃; )^{W}, and ν_{r} = ν_{r−1}^{2}.

(ii) If n = 2r ≥ 6 is congruent to 2 modulo 4, then ν_{r} is an invariant of degree 2^{r} in H^{∗}(BT̃; )^{W}, and ν_{r} = μ_{r}. □

We first summarize information on the complex spin representations. Recall that ρ^{∗} maps H^{∗}(BSpin(n); ) → H^{∗}(BT̃; ), with image in H^{∗}(BT̃; )^{W}.

Theorem 8.2 The dimensions and Euler classes of the complex spin representations of Spin(n) appear in Table 1.

Table 1: Complex spin representations

n  Δ dim_{}Δ ρ^{∗}e(Δ)
2r + 1 Δ_{}2^{r}  η_{r}
2r Δ_{}^{+} 2^{r−1}  μ_{r}
Δ_{}^{−} 2^{r−1}  ν_{r}

Proof All but the last column is standard information. The characters of the complex spin representations Δ are known ([8], Proposition 9.4, p. 186), so that the Chern classes of Δ can be obtained from work of Borel and Hirzebruch ([3], Theorem 10.3, p. 491, or [8], Theorem 11.4, p. 282). Since the Euler class of a complex vector bundle equals the top Chern class, the result follows by inspection. □

Next we summarize information concerning the real spin representations.

Theorem 8.3 The types, dimensions, relationships to complex spin representations, and Euler classes of the real spin representations of Spin(n) appear in Table 2.

Table 2: Real spin representations

n r = [n/2]  Δ  h =log_{2} dim_{}Δ End(Δ) Δ_{}|_{}ρ^{∗}e(Δ)
8k + 1  4k  Δ_{}4k  2Δ_{}±η_{r−1}
8k + 2 4k + 1  Δ_{}4k + 1  Δ_{}, Δ_{}μ_{r} = ν_{r}
8k + 3 4k + 1  Δ_{}4k + 2  Δ_{}η_{r}
8k + 4 4k + 2 Δ_{}^{+}, Δ_{}^{−}  4k + 2  Δ_{}^{+}, Δ_{}^{−}  μ_{r}, ν_{r}
8k + 5 4k + 2  Δ_{}4k + 3  Δ_{}η_{r}
8k + 6 4k + 3  Δ_{}4k + 3  Δ_{}, Δ_{}μ_{r} = ν_{r}
8k + 7 4k + 3  Δ_{}4k + 3  2Δ_{}±η_{r−1}
8k + 8 4k + 4 Δ_{}^{+}, Δ_{}^{−}  4k + 3  2Δ_{}^{+}, 2Δ_{}^{−} ±μ_{r−1}, ±ν_{r−1}

Proof Once again, most of this information is standard. The dimension dim_{}Δ of a real spin representation Δ is often written as 2^{h} (for example, [11], Table 6.2), where the numbers h = h(n) have the values given in Table 2. The integers 2^{h} are called the Radon–Hurwitz numbers.

To understand the Euler classes, we make use of Theorem 8.2, remembering that the Euler class of a complex representation ρ depends only upon the underlying real representation, denoted ρ|_{}. If n ≡ 1, 7 mod 8, then Δ_{}= ⊗_{}Δ_{}, so that, as real representations, Δ_{}|_{}= Δ_{}⊕ Δ_{}= 2Δ_{}. Then ρ^{∗}e(Δ_{})^{2} = ρ^{∗}e(2Δ_{}) = ρ^{∗}e(Δ_{}) = η_{r}. But η_{r} = η_{r−1}^{2}, so that ρ^{∗}e(Δ_{})^{2} = η_{r−1}^{2}. Since H^{∗}(BT̃; ) is an integral domain, we have ρ^{∗}e(Δ_{}) = ±η_{r−1}. A similar proof holds for n congruent to 0 modulo 8.

In the remaining cases, n congruent to 2, 3, 4, 5, 6 modulo 8, the real spin representation is obtained from the corresponding complex spin representation by restriction of scalars. Thus e(Δ_{}) = e(Δ_{}), and the result follows from Theorem 8.2. □

Recall that ρ̄^{∗} : H^{∗}(BSpin(n); )/torsion → H^{∗}(BT̃; )^{W}.

Corollary 8.4 (i) If n = 2r + 1 is odd, then η_{r} is in the image of ρ̄^{∗}. If n ≡ 1, 7 mod 8, then η_{r−1} is in the image of ρ̄^{∗}.

(ii) If n = 2r is even, then μ_{r} is in the image of ρ̄^{∗}. If n ≡ 0 mod 8, then μ_{r−1} is in the image of ρ̄^{∗}. □

9 Bockstein cohomology

We now calculate the Bockstein cohomology, namely the kernel modulo the image of the action of the Bockstein operation β = Sq^{1}, on H^{∗}(BSpin(n); /2), following Kono [9] and Wood [13]. This is in preparation for the calculation of the image of

ρ̄^{∗} : H^{∗}(BSpin(n); )/torsion → H^{∗}(BT̃; )^{W} .

First, recall from Quillen [11] that

H^{∗}(BSpin(n); /2) = H^{∗}(BSO(n); /2)/J_{h} ⊗ /2[w_{2^{h}}(Δ)]
= /2[w_{2}, …, w_{n}]/(r_{1}, …, r_{h}) ⊗ /2[w_{2^{h}}(Δ)].

Here, w_{2}, …, w_{n} are the inflations of the Stiefel–Whitney classes from BSO(n), and Δ is a real spin representation of Spin(n). The elements r_{j} ∈ H^{2^{j−1}+1}(BSO(n); /2) are defined by r_{1} = w_{2}, and r_{j+1} = Sq^{2^{j−1}}r_{j} for j ≥ 1; they form a regular sequence in H^{∗}(BSO(n); /2). The values of h = h(n) are given in Table 2. The action of the Bockstein operation is given by β(w_{2j}) = w_{2j+1}, β(w_{2j+1}) = 0, β(r_{1}) = r_{2}, β(r_{2}) = 0, β(r_{j}) = r_{j−1}^{2} (j ≥ 3). Thus for j ≥ 2, the ideal J_{j} generated by r_{1}, …, r_{j} in H^{∗}(BSO(n); /2) is invariant under the action of the Bockstein, so that we obtain a well defined Bockstein operation on R_{j} = H^{∗}(BSO(n); /2)/J_{j}.

We have R_{2} = /2[w_{4}, …, w_{n}]. Decomposing this as a tensor product of polynomial rings in two and one variables, and applying the Künneth theorem for the action of β, we see that the Bockstein cohomology of this ring is

/2[w_{4}^{2}, …, w_{2r}^{2}]

if n = 2r + 1 is odd, and

/2[w_{4}^{2}, …, w_{2r−2}^{2}, w_{2r}]

if n = 2r is even. In both cases, this ring is concentrated in even degrees.

Lemma 9.1 There are elements s_{j} ∈ H^{2^{j−1}}(BSO(n); /2), 2 ≤ j ≤ h, satisfying s_{2} = r_{1}, and

r′_{j} = r_{j} + ∑_{i=2}^{j−1} s_{i}^{2^{j−i}−1}r′_{i}

for 3 ≤ j ≤ h, where r′_{j} = β(s_{j}).

The Bockstein cohomology of R_{j} (2 ≤ j ≤ h) is concentrated in even degrees. As a module over H^{∗}(R_{2}, β), H^{∗}(R_{j}, β) is free of rank 2^{j−2} on the square-free monomials in s_{3}, …, s_{j}. Each s_{i}^{2} (3 ≤ i ≤ h) lies in R_{2}.

Proof This is proved by induction on j. The existence of the elements s_{j} follows by defining r′_{j} by the given formula, checking that β(r′_{j}) = 0, and noting that the Bockstein homomorphism is exact in odd degrees.

To prove the second part, we use the exact sequence of modules for β:

0 → Σ^{2^{j}+1}R_{j} ⟶^{×r′_{j+1}} R_{j} → R_{j+1} → 0

(where Σ^{2^{j}+1} denotes a shift in degree by 2^{j} +1). The left hand map is multiplication by r′_{j+1}, which is a map of modules for β because β(r′_{j+1}) = 0. Arguing inductively, we see that the long exact sequence in Bockstein cohomology of this short exact sequence breaks into short exact sequences

0 → H^{2m}(R_{j}; β) → H^{2m}(R_{j+1}; β) → H^{2m−2^{j}}(R_{j}; β) → 0,

and that the Bockstein cohomology of R_{j} is concentrated in even degrees. Thus H^{∗}(R_{j+1}; β) is free as a module over H^{∗}(R_{j}, β), on the two generators 1 and s_{j+1}. □

Now suppose that n ≥ 3, so that h ≥ 2. Then β(w_{2^{h}}(Δ)) = 0, and so applying the Künneth theorem for the action of β, we see that

H^{∗}(H^{∗}(BSpin(n); /2); β) ≅ H^{∗}(R_{h}; β) ⊗ /2[w_{2^{h}}(Δ)].

We denote this ring by H_{β}.

Since H_{β} is concentrated in even degrees, it follows immediately that there are no higher Bocksteins, and so the 2-torsion in H^{∗}(BSpin(n); ) is killed by multiplication by two. Thus

H_{β} ≅ /2 ⊗_{}(H^{∗}(BSpin(n); )/torsion).

10 Integral cohomology modulo torsion

We calculate H^{∗}(BSpin(n); )/torsion by using the maps

H^{∗}(BSpin(n); )/torsion ⟶^{ρ̄^{∗}} H^{∗}(BT̃; )^{W}
↓
H_{β}

It is clear by inflating from BSO(n) that the elements p_{i} ∈ H^{∗}(BT̃; )^{W} are in the image of ρ̄^{∗}. Namely, they are the images of the Pontrjagin classes with the same name.

Lemma 10.1 The Euler class e(Δ) reduces modulo two to w_{2^{h}}(Δ), and its image under ρ̄^{∗} is given in Table 2.

The elements q_{j} ∈ H^{∗}(BT̃; )^{W} (1 ≤ j ≤ h−2) described in Proposition 3.3 are of the form ρ̄^{∗}(q′_{j}), where q′_{j} is an element of H^{∗}(BSpin(n); ) whose reduction modulo two is s_{j+2} plus terms only involving w_{4}^{2}, …, w_{2r}^{2}, s_{3}, …, s_{j+1} (n odd), respectively w_{4}^{2}, …, w_{2r−2}^{2}, w_{2r}, s_{3}, …, s_{j+1} (n even).

Proof The element p_{1} ∈ H^{∗}(BSO(n); ) reduces modulo two to the element w_{2}^{2} ∈ H^{∗}(BSO(n); /2), which then inflates to zero in H^{∗}(BSpin(n); /2). So p_{1} is divisible by two in H^{∗}(BSpin(n); ), say p_{1} = 2q′_{1}. Then ρ̄^{∗}(2q′_{1}) = 2q_{1}, and since H^{∗}(BT̃; )^{W} is torsion free, it follows that ρ̄^{∗}(q′_{1}) = q_{1}. Since q_{1} is not divisible by two, neither is q′_{1}, and so its reduction modulo two is non-zero, and hence equal to s_{3}.

The reductions modulo two of p_{2}, …, p_{r} (n odd), respectively p_{2}, …, p_{r−1}, e (n even) generate the subring H^{∗}(R_{2}, β) of H_{β}. Working by induction on j, these, together with the reductions modulo two of the elements q′_{1}, …, q′_{j−1}, generate the subring H^{∗}(R_{j+1}, β). Choose an element S_{j+2} ∈ H^{∗}(BSpin(n); ) with image s_{j+2} ∈ H_{β}. If we suppose that j+2 < h in the cases where n is congruent to 3, 4 or 5 modulo 8, then q_{j} is not divisible by two in H^{∗}(BT̃; Z)^{W}. So for degree reasons, ρ̄^{∗}(S_{j+2}) can be expressed as some integer multiple of q_{j} plus a polynomial in p_{2}, …, p_{r} (respectively p_{2}, …, p_{r−1}, e) and q_{1}, …, q_{j−1}. Since 2q_{j} − q_{j−1}^{2} ∈ [p_{2}, …, p_{r}], the coefficient of q_{j} may be taken to be zero or one. The coefficient is not zero, since otherwise s_{j+2} is a redundant generator in H_{β}. Thus by adjusting S_{j+2} by a suitable polynomial in the p’s, e (if n is even) and q′_{1}, …, q′_{j−1}, we obtain the required element q′_{j}.

Finally, in case n is congruent to 3, 4 or 5 modulo 8, and j + 2 = h, we argue as follows. The element ρ̄^{∗}(S_{h}) can be expressed as some integer multiple of η_{r−1} (n congruent to 3 or 5 modulo 8), respectively μ_{r−1} (n congruent to 4 modulo 8) plus a polynomial in p_{2}, …, p_{r} (respectively p_{2}, …, p_{r−1}, e) and q_{1}, …, q_{h−3}. Since Proposition 3.3 and Corollary 7.2 imply that 4η_{r−1}−q_{h−3}^{2} ∈ [p_{2}, …, p_{r}, q_{1}, …, q_{h−3}] (respectively 4μ_{r−1} − q_{h−3}^{2} ∈ [p_{2}, …, p_{r−1}, e, q_{1}, …, q_{h−3}]), the coefficient of η_{r−1} (respectively μ_{r−1}) may be taken to be zero, one, two or three. If this coefficient is odd, then squaring and reducing modulo two, it would follow that s_{h}^{2} involves w_{2^{h}}(Δ). This doesn’t happen, so the coefficient is even. If it were zero, s_{h} would be a redundant generator, so in fact the coefficient is two. Since 2η_{r−1} (respectively 2μ_{r−1}) is equal to q_{h−2} plus a polynomial in the p’s, e (if n is even) and q_{1}, …, q_{h−3}, by Corollary 7.2, we may now complete the argument as before. □

Theorem 10.2 Suppose that n ≥ 6. Then the following hold.

(i) The map

ρ̄^{∗} : H^{∗}(BSpin(n); )/torsion → H^{∗}(BT̃; )^{W}

is an isomorphism if and only if n is not congruent to 3, 4 or 5 modulo 8. (ii) If n is congruent to 3 or 5 modulo 8, then ρ̄^{∗} is injective, with image

[p_{1}, …, p_{r}, q_{1}, …, q_{r−2}, 2η_{r−1}, η_{r−1}^{2}]
= [p_{1}, …, p_{r}, q_{1}, …, q_{r−2}, q_{r−1}, η_{r}].

(iii) If n is congruent to 4 modulo 8, then ρ̄^{∗} is injective, with image

[p_{1}, …, p_{r−1}, e, q_{1}, …, q_{r−3}, 2μ_{r−1}, μ_{r−1}^{2}]
= [p_{1}, …, p_{r−1}, e, q_{1}, …, q_{r−3}, q_{r−2}, μ_{r}].

Proof Injectivity of ρ̄^{∗} is a general theorem of Borel [2]. In Lemma 10.1, explicit elements were produced to show that the image is at least as large as is claimed. The subring of H^{∗}(BSpin(n); )/torsion generated by these elements already surjects onto H_{β}, so the image cannot be any larger. □

Remark 10.3 Using Remark 7.3, it is also easy to see that

ρ̄^{∗} : H^{∗}(BSpin(n); )/torsion → H^{∗}(BT̃; )^{W}

is an isomorphism for n < 6.

Corollary 10.4 For n ≥ 6, the ring H_{β} has nilpotent elements if and only if n is congruent to 3, 4 or 5 modulo 8.

Proof A theorem of Feshbach [7] states that in general for a compact Lie group, the map ρ̄^{∗} is surjective if and only if there are no nilpotent elements in the E_{∞} page of the Bockstein spectral sequence.

When n is congruent to 3 or 5 modulo 8, the element 2η_{r−1} reduces modulo 2 to a nilpotent element in H_{β}. When n is congruent to 4 modulo 8, the element 2μ_{r−1} so reduces. □

11 Piecing together the integral cohomology

We calculated H^{∗}(BSpin(n); )/torsion in the last section. We saw that the torsion is killed by multiplication by two, and so we have a pullback diagram

H^{∗}(BSpin(n); ) → H^{∗}(BSpin(n); )/torsion
↓  ↓
Ker β  →  H_{β}

Thus we have the following.

Theorem 11.1 H^{∗}(BSpin(n); ) is isomorphic to the subring of

Ker β ⊕ H^{∗}(BSpin(n); )/torsion

consisting of pairs of elements with the same image in H_{β}. □

We have not set about the rather daunting task of using this description to give explicit generators and relations.

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