Lobachevskii Journal of Mathematics Vol. 5, 1999, 57–60 \begin{matrix}c \\ ◯ \end{matrix}Donald Yau

Donald Yau

COHOMOLOGY OF UNITARY AND SYMPLETIC GROUPS

(submitted by B.N.Shapukov)

Abstract. We compute the cohomology rings of U(n) and Sp(n) and of their Stiefel varieties by using the Serre spectral sequence. This approach is much simpler than the usual method, that of using the cell structures. The argument here also shows that the cohomology of U(n) is built from those of U(n − 1) and S^{2n−1} through a fiber bundle; a similar result holds for Sp(n).

1. Introduction

The cohomology rings of the unitary and sympletic groups and their Stiefel varieties are well-known. They are some exterior algebras with generators in odd dimensions. In particular, the cohomology ring of U(n) is an exterior algebra with generators in dimensions 1, 3, …, 2n − 1, and similarly the cohomology ring of Sp(n) is an exterior algebra with generators in dimensions 3, 7, …, 4n − 1. The cohomology ring of the Stiefel variety U(n)/U(n − k) of orthonormal k-frames in complex n-space can be regarded as a subring of H^{∗}(U(n)) with the first n − k generators deleted, and similarly for the sympletic Stiefel variety Sp(n)/Sp(n − k). The traditional way to compute these cohomology rings is to use their cell structures. While these cell structures are elegant, the traditional argument involved in computing these cohomology rings is not.

The aim of this paper is to do the computation in an elegant way, without using the cell structures of these groups. Our argument makes it clear that the cohomology ring of U(n) is constructed from lower dimensional unitary groups by means of a finite sequence of fibrations, and similarly for Sp(n). It allows us to understand how the cohomology rings of the various unitary (resp. sympletic) groups are related to each other.

Now we say a little bit about the organization of the paper. In Section 1 we compute the cohomology rings of SU(n) by using an inductive argument that involves the fibration SU(n − 1) → SU(n) → S^{2n−1}. Then we obtain the cohomology rings of U(n) as a corollary. We could have worked directly with U(n) without going through SU(n), but since the argument is the same, our approach gives us the cohomology rings of SU(n) as well as U(n) without any extra effort. In Section 2 we use the same method to obtain the cohomology rings of the Stiefel varieties, U(n)/U(n − k). Here we use the fibration S^{2n−2k+1} → U(n)/U(n − k) → U(n)/U(n − k + 1). In the final section we obtain the corresponding results for the sympletic cases.

I would like to thank Haynes Miller for asking the question that leads to this paper.

1991 Mathematical Subject Classification. Primary 55T10; Secondary 55R20.

The author is partially supported by the Liberty Mutual Pre-Doctoral Fellowship.

2. The cohomology of U(n) and SU(n)

In this section we compute the cohomology rings of the special unitary groups SU(n) for n ≥ 2. This in turn will tell us the cohomology rings of U(n) for n ≥ 1.

The following proof will make use of the sequence of fiber bundles:

\xymatrix{S^{3} = SU(2) \ar[r] & SU(3) \ar[d] \ar[r] & SU(4) \ar[d] \ar[r] & ⋯ \ar[r] & SU. \\  & S^{5} & S^{7}}

Each vertical map is the projection of a fiber bundle with the horizontal arrow immediately to the left of its tail the inclusion of the fiber into the total space.

Theorem 2.1. For each n ≥ 2 the integral cohomology ring H^{∗}(SU(n)) is isomorphic as a graded ring to the exterior algebra (over the integers) Λ_{ℤ}[x_{3}, …, x_{2n−1}], in which the degree of each generator x_{i} is i.

Proof. We proceed by induction on n. For n = 2 note that one can identify SU(2) with the sphere S^{3} in ℝ^{4}, so H^{∗}(SU(2)) ≅ H^{∗}(S^{3}) ≅ Λ_{ℤ}[x_{3}].

Now suppose n > 2 and the result holds for smaller n. There is a fiber bundle

\xymatrix{SU(n − 1) \ar[r] & SU(n) \ar[d]^{p} \\  & S^{2n−1}}

with bundle projection p. This is actually a fibration because the base space is paracompact and Hausdorff; see Spanier [S].The fiber SU(n − 1) is connected, and the base space S^{2n−1} is simply connected, since n > 2. Therefore there is a convergent Serre spectral sequence

\xymatrix{E_{2}^{p,q} = H^{p}(S^{2n−1}; H^{q}(SU(n − 1))) \ar@{=>}[r] & H^{p+q}(SU(n)).}

By induction hypothesis, the cohomology ring of SU(n − 1) is the exterior algebra Λ_{ℤ}[x_{3}, …, x_{2n−3}]. Since the base space of the above fiber bundle is the (2n − 1)−sphere, the E_{2} term of the Serre spectral sequence has only two nonzero columns, corresponding to p = 0, 2n − 1. Since the cohomology groups of the sphere S^{2n−1} are torsion-free, we have

E_{2}^{p,q} ≅ H^{p}(S^{2n−1}) ⊗ H^{q}(SU(n − 1)),

in which the tensor product is taken over ℤ. There is no nonzero differential until the E_{2n−1} term. In E_{2n−1} the generators are all killed by differentials because the differentials have degree (−2n + 2, 2n − 1). By the multiplicative property of differentials in the Serre spectral sequence, it follows that there is no nonzero differential in the E_{2n−1} term as well, so the spectral sequence collapses.

It now follows that the integral cohomology ring of SU(n) is torsion-free and has generators x_{3}, x_{5}, …, x_{2n−1}, in which the last generator comes from the base space S^{2n−1}. Each of these generators has odd dimensions, so the square of each one of them is zero because there is no torsion. Moreover, they anticommute, i.e. x_{i}x_{j} = −x_{j}x_{i} for all i, j = 3, 5, …, 2n − 1. Hence the integral cohomology ring of SU(n) is precisely the exterior algebra Λ_{ℤ}[x_{3}, x_{5}, …, x_{2n−1}]. This finishes the induction, and completes the proof of the theorem.

As an immediate corollary, we obtain the cohomology ring of U(n).

Corollary 2.2. For each n ≥ 1 the integral cohomology ring of the unitary group U(n) is isomorphic as a graded ring to the exterior algebra Λ_{ℤ}[x_{1}, x_{3}, …, x_{2n−1}], in which the degree of each generator x_{i} is i.

Proof. Since U(1) is just the circle S^{1}, its integral cohomology ring is Λ_{ℤ}[x_{1}]. For n > 1, note that one can identify U(n) with S^{1} × SU(n); see Hatcher [H]. Therefore we have

H^{∗}(U(n)) ≅ H^{∗}(S^{1} × SU(n))
≅ H^{∗}(S^{1}) ⊗ H^{∗}(SU(n))
≅ Λ_{ℤ}[x_{1}] ⊗ Λ_{ℤ}[x_{3}, …, x_{2n−1}]
≅ Λ_{ℤ}[x_{1}, x_{3}, …, x_{2n−1}].

We are indebted to Haynes Miller [M] for the following remark.

Remark 2.3. There is another space with the above integral cohomology ring, namely, S^{1} ×S^{3} ×⋯×S^{2n−1}. It is true that U(2) can be identified with S^{1} ×S^{3}, since SU(2) is just S^{3}. But it is not true that U(n) is a product of odd-dimensional spheres S^{1}×S^{3}×⋯×S^{2n−1} for any n ≥ 3. For S^{5} is not an H-space, while U(n) is a topological group, hence an Hspace. So if U(n) were a product S^{1} × ⋯ × S^{2n−1}, then S^{5} would be a retract of U(n), hence also an H-space, which we know is not the case.

3. Stiefel varieties

In this section we compute the cohomology rings of the Stiefel varieties by using the Serre spectral sequence again.

First we recall that the Stiefel variety W_{n,k} := U(n)/U(n − k) (0 ≤ k ≤ n) is the space of orthonormal k-frames in the complex n-space. It is clear that W_{n,0} = {pt.}, W_{n,1} = S^{2n−1}, and W_{n,n} = U(n). Thus we only need to consider the cases when k ≠ 0. The computation in this section will make use of the following sequence of fiber bundles:

\xymatrix{S^{1} \ar[d] & S^{3} \ar[d] & S^{5} \ar[d] &  & S^{2n−3} \ar[d] \\ W_{n,n} \ar[r] & W_{n,n−1} \ar[r] & W_{n,n−2} \ar[r] & ⋯ \ar[r] & W_{n,2} \ar[r] & W_{n,1} = S^{2n−1}}

Here each horizontal map is the projection of a fiber bundle, and each vertical map is the inclusion of the fiber into the total space.

Theorem 3.1. For n ≥ 1, 0 < k ≤ n the integral cohomology ring of the Stiefel variety of orthonormal k-frames in complex n-space, W_{n,k}, is isomorphic as a graded ring to the exterior algebra Λ_{ℤ}[x_{2(n−k+1)−1}, …, x_{2n−1}], in which the degree of each generator x_{i} is i.

Proof. The proof of this theorem is quite similar to that of 2.1. If n = 1 then k = 1, so

H^{∗}(W_{1,1}) ≅ H^{∗}(U(1)) ≅ H^{∗}(S^{1}) ≅ Λ_{ℤ}[x_{1}].

Now suppose n > 1. We do induction on k ≥ 1. For k = 1 we have

H^{∗}(W_{n,1}) ≅ H^{∗}(S^{2n−1}) ≅ Λ_{ℤ}[x_{2n−1}].

So suppose 1 < k < n. The Stiefel variety W_{n,k−1} is (2n − 2k + 2)-connected. This follows immediately from the identifications U(n)/U(n−1) = S^{2n−1}, and their long exact sequences in homotopy groups. In particular, W_{n,k−1} is simply connected, so the fiber bundle (which is actually a fibration)

\xymatrix{S^{2n−2k+1} \ar[r] & W_{n,k} \ar[d] \\  & W_{n,k−1}}

gives a convergent Serre spectral sequence

\xymatrix{E_{2}^{p,q} = H^{p}(W_{n,k−1}; H^{q}(S^{2n−2k+1})) \ar@{=>}[r] & H^{p+q}(W_{n,k}).}

Since H^{q}(S^{2n−2k+1}) is torsion-free, we have

E_{2}^{p,q} ≅ H^{p}(W_{n,k−1}) ⊗ H^{q}(S^{2n−2k+1}).

Since the fiber of the above bundle is the sphere S^{2n−2k+1}, there are only two nonzero rows in the E_{2} term, corresponding to q = 0, 2n − 2k + 1. The 0^{th} row is the cohomology of the base space W_{n,k−1}. By induction hypothesis, the integral cohomology ring of W_{n,k−1} is the exterior algebra Λ_{ℤ}[x_{2n−2k+3}, …, x_{2n−1}]. An argument very similar to that of 2.1 shows that the spectral sequence again collapses, so the integral cohomology ring of W_{n,k} is torsion-free and has generators x_{2n−2k+1} (coming from the fiber S^{2n−2k+1}), x_{2n−2k+3}, …, x_{2n−1}, whose dimensions are indicated by the indices. Since there is no torsion and since the dimensions of these generators are odd, the square of each one of them is zero, and they anticommute. Hence the integral cohomology ring of W_{n,k} is precisely the exterior alge

Remark 3.2. Putting k = n in the above theorem one obtains the integral cohomology ring of W_{n,n} = U(n) that agrees with 2.2.

4. Sp(n) and sympletic Stiefel varieties

In this final section we obtain the corresponding results for the sympletic groups Sp(n) and the sympletic Stiefel varieties Sp(n)/Sp(n − k), n ≥ 1, 1 ≤ k ≤ n. The corresponding result of 3.1 is:

Theorem 4.1. For each n ≥ 1, 1 ≤ k ≤ n the integral cohomology ring of the sympletic Stiefel variety Sp(n)/Sp(n − k) is isomorphic as a graded ring to the exterior algebra Λ_{ℤ}[x_{4(n−k+1)−1}, …, x_{4n−1}], in which the degree of each generator x_{i} is i.

Proof. We denote Sp(n)/Sp(n − k) by Y_{n,k} for typographical reasons. There is a sequence of fibrations

\xymatrix{S^{3} \ar[d] & S^{7} \ar[d] & S^{11} \ar[d] &  & S^{4n−5} \ar[d] \\ Y_{n,n} \ar[r] & Y_{n,n−1} \ar[r] & Y_{n,n−2} \ar[r] & ⋯ \ar[r] & Y_{n,2} \ar[r] & Y_{n,1} = S^{4n−1},}

in which each horizontal map is the projection of a fiber bundle and each vertical map is the inclusion of the fiber into the total space. Now one only needs to slightly modify the proof of 3.1 to finish the proof.

Substituting k = n in the above theorem one obtains the cohomology rings of the sympletic groups:

Corollary 4.2. For each n ≥ 1 the integral cohomology ring of the sympletic group Sp(n) is isomorphic as a graded ring to the exterior algebra Λ_{ℤ}[x_{3}, x_{7}, …, x_{4n−1}], in which the degree of each generator x_{i} is i.

Remark 4.3. The method used in this paper does not seem to apply to the orthogonal groups O(n), n ≥ 1 because the corresponding Serre spectral sequences do not collapse, and in fact have lots of nonzero differentials.

References

[H] Allen Hatcher, Vector Bundles and K-Theory, First installment, August 1998, available at http://math.cornell.edu/∼hatcher.

[M] Haynes Miller, Private Communication.

[S] Edwin H. Spanier, Algebraic Topology, McGraw-Hill series in higher mathematics, McGraw-Hill, New York, 1966.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail address: donald@math.mit.edu