Minimal models of oriented Grassmannians and applications

Goutam Mukherjee Stat-Math Unit Indian Statistical Institute 203 Barrackpore Trunk Road, Calcutta 700 035, INDIA and Parameswaran Sankaran Chennai Mathematical Institute 92 G.N. Chetty Road, Chennai 600 017, INDIA

               E-mail: goutam@isical.ac.in sankaran@smi.ernet.in

Abstract: We construct the minimal models for the oriented Grassmann manifold G̃_{n,k} of all oriented k dimensional vector subspaces of ℝ^{n} and show that they are formal. As an application we obtain a classification of real flag manifolds according to nilpotence, which was first established by H.Glover and W.Homer, We also establish a result of K. Varadarajan that the classifying space BO(k) is nilpotent if and only if k is odd.

1991 A.M.S. Subject Classification:- 55 P 62, 55 P 99.

Key words and phrases: Grassmann manifolds, flag manifolds, rational homotopy theory, minimal models, formality, nilpotence.

1 Introduction

The purpose of this paper is to give an explicit description of minimal models of oriented Grassmann manifolds. The construction of minimal models of compact simply connected homogeneous manifolds is well understood from the work of Sullivan [15] and others. (Cf. [4], [7]). However, we have not been able to find explicit reference for the description, depending only on the parameters n and k, 1 ≤ k < n, of minimal model of a oriented Grassmann manifold G̃_{n,k} of oriented k-vector subspaces of ℝ^{n}. It is our hope that such a description will be useful in answering many questions about Grassmannians (oriented as well as unoriented). Using our description of the minimal model, we prove that G̃_{n,k} is formal. Of course, this is a well-known result since the oriented Grassmann manifolds are Riemannian symmetric spaces (see p.326,[15], p.158,[8] and [9]). (Cf. Remark 2 below.) We apply our results to show that the action of the fundamental group of G_{n,k}, the Grassmann manifold of k planes in ℝ^{n} on π_{k}(G_{n,k}) is not nilpotent in case k is even. We deduce a result of H.Glover and W.Homer that the real flag manifold G(n_{1}, ⋯, n_{s}) = O(n)/(O(n_{1}) × ⋯ × O(n_{s})), n = ∑_{1≤i≤s} n_{i} is not nilpotent when one of the n_{i} is even. We also obtain a new proof of a result of K.Varadarajan that the classifying space BO(k) is nilpotent if and only if k is odd.

We hope that our method of explicit construction and proof, more than the results themselves, will be of some interest. Our approach to nilpotence via the theory of minimal models is probably new.

We now state the result of this paper. Write k = 2s or 2s + 1, n − k = 2t or 2t + 1, 1 ≤ s, t ∈ ℤ, n ≥ 2k.

Let P denote the polynomial algebra ℝ[p_{1}, ⋯, p_{s}] where each p_{j} is homogeneous of degree |p_{j}| = 4j. Define homogeneous elements h_{j} ∈ P of degree 4j by the equation (1 + p_{1} + ⋯ + p_{s})(1 + h_{1} + ⋯ + h_{j} + ⋯) = 1. Thus h_{j}, j ≥ 1, is a certain polynomial in p_{1}, ⋯, p_{s}.

Introduce elements σ_{k}, τ_{n−k} of degree k and n−k such that σ_{k}^{2} = p_{s}, if k is even, σ_{k} = 0 if k is odd; τ_{n−k} = 0 if n−k is odd, otherwise it is an indeterminate. Let A be the algebra got by adjoining σ_{k}, τ_{n−k} to P. Note that A is a polynomial algebra in even degree generators.

Let M := M_{n,k} denote the commutative differential graded algebra over the reals defined as follows. Recall that commutativity is the graded sense: for homogeneous elements u and v, uv = (−1)^{|u||v|}vu.

Case 1: Let n = 2m, k = 2s, n − k = 2t, s ≥ t. Let M = A[u_{0}, v_{0}, ⋯, v_{s−1}], |v_{j}| = 4(t + j) − 1, 0 ≤ j < s, |u_{0}| = 2m − 1. The differential d on M is defined as d(A) = 0, d(v_{j}) = h_{t+j}, 1 ≤ j < s, d(v_{0}) = h_{t} − τ_{n−k}^{2}, and d(u_{0}) = σ_{k}τ_{n−k}.

Case 2a: Let n = 2m + 1, k = 2s, n − k = 2t + 1, k ≤ m. Let M = A[v_{1}, ⋯, v_{s}], where |v_{j}| = 4(t + j) − 1, d(v_{j}) = h_{t+j}, 1 ≤ j ≤ s, d(A) = 0.

Case 2b: Let n = 2m + 1, k = 2s + 1, n − k = 2t, k ≤ m. Define M = A[v_{0}, v_{1}, ⋯, v_{s}], where |v_{j}| = 4(t + j) − 1, 0 ≤ j ≤ s, d(A) = 0, d(v_{0}) = h_{t} − τ_{n−k}^{2}, d(v_{j}) = h_{t+j}, 1 ≤ j ≤ s.

Case 3: Let n = 2m + 2, k = 2s + 1, n − k = 2t + 1. Let M = A[v_{0}, v_{1}, ⋯, v_{s}], where |v_{j}| = 4(t + j) − 1, 1 ≤ j ≤ s, |v_{0}| = 2m + 1, d(A) = 0, d(v_{0}) = 0, d(v_{j}) = h_{t+j}, 1 ≤ j ≤ s.

Main Theorem: Let 2 ≤ k ≤ [n/2]. With notation as above, the commutative differential graded algebra M_{n,k} is the minimal model for the oriented Grassmann manifold G̃_{n,k}.

The minimal model of G̃_{n,1} ≡ S^{n−1}, the (n − 1)-sphere, is well-known (see [3]). Since G̃_{n,k} ≡ G̃_{n,n−k}, the hypothesis that k ≤ [n/2] is not a restriction.

The paper is organised as follows: In §2 we recall a basic theorem needed in the construction of minimal models of homogeneous spaces. In §3 we prove the Main Theorem stated above and deduce the formality of the oriented Grassmann manifolds. In §4 we obtain results on nilpotence of flag manifolds (theorem 6) and the classifying space for the orthogonal group (theorem 7).

Acknowledgements: We thank K. Varadarajan for his interest in this problem and for making available to us his paper [17]. We thank J.Korbaš for his comments on an earlier version of this paper and, in particular, for pointing out to us the work of H.Glover and W.Homer [6].

2 Minimal Models of homogeneous spaces

Let G be a connected compact simple Lie group. Let H be a closed connected subgroup of G. One has the following description of the minimal model of the smooth homogeneous manifold G/H. Let T ⊂ G be a maximal torus in G such that S := T ∩ H is a maximal torus of H. Denote by W the Weyl group of G with respect to T and by W′ the Weyl group of H with respect to S. Let m = dim T, and let r = dim S. The group W acts on T and hence on the real cohomology algebra H^{∗}(BT; ℝ) of the classifying space of T which is a polynomial algebra over ℝ in m generators each having degree 2. The W -invariant subalgebra can be identified with the real cohomology algebra of BG. The cohomology algebra H^{∗}(BG; ℝ) is a polynomial algebra ℝ[F_{1}, ⋯, F_{m}] in homogeneous elements F_{j} having even degrees. (See Borel [2].) Similarly, H^{∗}(BH; ℝ) = ℝ[x_{1}, ⋯, x_{r}]. Let ρ : H^{∗}(BG; ℝ)⟶H^{∗}(BH; ℝ) denote the map induced by the inclusion H ⊂ G. Let f_{i} = ρ(F_{i}), 1 ≤ i ≤ m. Now let C = C(G, H) denote the differential graded algebra (d.g.a) H^{∗}(BH; ℝ)[u_{1}, ⋯, u_{m}] where |u_{i}| = |f_{i}| − 1, and the differential d is defined as d(H^{∗}(BH; ℝ)) = 0, and d(u_{i}) = f_{i}, 1 ≤ i ≤ m. Note that since |u_{i}| is odd, graded commutativity implies that u_{i}u_{j} = −u_{j}u_{i}, and, in particular, that u_{j}^{2} = 0.

Theorem 1. (i) (H.Cartan) With notation as above, the minimal model M_{G/H} of G/H is isomorphic to the minimal model of the d.g.a. (C(G, H), d).

(ii) (Cf. Example (ii),(v), p.317, [15].) The space G/H is formal if for some integer s, 1 ≤ s ≤ m, the sequence f_{1}, ⋯, f_{s} is a regular sequence in H^{∗}(BH; ℝ), and the elements f_{s+1}, ⋯, f_{m} belong to the ideal generated by f_{1}, ⋯, f_{s}. □

A proof is sketched in §4, Chapter 5 of [16].

Remark 2. It is known that the sequence f_{1} ⋯, f_{m} is a regular sequence in H^{∗}(BH; ℝ) when H is of maximal rank in G. Hence when H is of maximal rank, G/H is formal. See Theorem 4.16, Ch. 5, [16].

3 Minimal Model of G̃_{n,k}

In this section we prove the Main Theorem stated in the introduction. We apply theorem 1 to the case G = SO(n), H = SO(k) × SO(n − k). Write n = 2m or 2m + 1, and k = 2s or 2s+1, n − k = 2t or 2t + 1, m, s, t being integers. We shall assume that k ≥ 2, since minimal models of spheres are well-known. We take T to be the standard maximal torus which consists of elements t = [t_{1}, ⋯, t_{m}] ∈ SO(2m) ⊂ SO(n), t_{j} ∈ ℝ, where t(e_{i}) = cos(2πt_{j})e_{i}+sin(2πt_{j})e_{i+1} and t(e_{i+1}) = −sin(2πt_{j})e_{i}+cos(2πt_{j})e_{i+1} where i = 2j − 1, 1 ≤ i ≤ 2m. (Here the e_{i} denote the standard basis of ℝ^{n}. ) When n is odd or k is even, H is of maximal rank, m. When n is even and k is odd, S := H ∩ T is of dimension r = s + t = m − 1. For w ∈ W, and t = [t_{1}, ⋯, t_{m}] ∈ T, w · t ∈ T is obtained by a permutation of the t_{j} and changing certain of the t_{j} to −t_{j}. When n = 2m the number of sign changes is to be even. From this it is easy to compute the W -invariant subalgebra of H^{∗}(BT; ℝ) = ℝ[t_{1}, ⋯, t_{m}], |t_{j}| = 2. Let P_{j} denote the j-th elementary symmetric polynomial in t_{1}^{2}, ⋯, t_{m}^{2}, and let σ_{m} = t_{1} ⋯ t_{m}. Then, H^{∗}(BSO(2m); ℝ) = ℝ[P_{1}, ⋯, P_{m−1}, σ_{m}], and H^{∗}(BSO(2m + 1); ℝ) = ℝ[P_{1}, ⋯, P_{m}]. Note that P_{m} = σ_{m}^{2} ∈ H^{∗}(BSO(2m); ℝ). The element (−1)^{r}P_{r} is the Pontrjagin class of the canonical n plane bundle over BSO(n); when n = 2m + 1 the (integral) Euler class of the canonical bundle is of order 2 and hence it vanishes in real cohomology (cf. [12]). The calculation of H^{∗}(BH; ℝ) is similar. One has H^{∗}(BH; ℝ) = ℝ[p_{1}, ⋯, p_{s}, q_{1}, ⋯, q_{t}, σ_{k}, τ_{n−k}] where σ_{k} = 0 (resp. σ_{k}^{2} = p_{s}) if k is odd (resp. even), and τ_{n−k} = 0 (resp. τ_{n−k}^{2} = q_{t}) when n − k is odd (resp. even). The restriction map ρ : H^{∗}(BG; ℝ)⟶H^{∗}(BH; ℝ) is given by

(i) ρ(P_{r}) = ∑_{i+j=r} p_{i}q_{j} =: f_{r}, 1 ≤ r ≤ s + t,

(ii) ρ(σ_{m}) = \{ \begin{matrix}σ_{k} · τ_{n−k} =: θ & if (n, k) = (2m, 2s), \\ 0 & otherwise. \end{matrix}

(It is understood that p_{0} = q_{0} = 1.)

It can be shown that when (n, k) = (2m, 2s) the elements θ, f_{1}, ⋯, f_{m−1} form a regular sequence in the ring R := H^{∗}(BH; ℝ). To see this, we note that, R/⟨θ⟩ is isomorphic to the polynomial ring over ℝ in p_{1}, ⋯, p_{s}, q_{1}, ⋯, q_{t} modulo the ideal generated by p_{s}q_{t} = f_{m}. (This is because θ^{2} = p_{s}q_{t}.) Since, by Prop. 23.7, [3], f_{1}, ⋯, f_{m} forms a regular sequence in the polynomial algebra ℝ[p_{1}, ⋯, p_{s}, q_{1}, ⋯, q_{t}] it follows that θ, f_{1}, ⋯, f_{m−1} forms a regular sequence in H^{∗}(BH; ℝ). Theorem 1(ii) shows in particular that the space G̃_{n,k} is formal when n and k are both even. Similarly it is seen that f_{1}, ⋯, f_{m} is a regular sequence in H^{∗}(BH; ℝ) when n = 2m+1. Thus we conclude that G̃_{n,k} is formal when n is odd or k is even. Note that H is of maximal rank in G except when n is even and k odd. Therefore the formality of G̃_{n,k} when n is odd or k even follows from Remark 2. In any case, as remarked in the introduction, G̃_{n,k} is formal since it is a Riemannian symmetric space. We shall verify formality of G̃_{n,k} directly for all values of n and k.

Let P = ℝ[p_{1}, ⋯, p_{s}] be a polynomial algebra where |p_{j}| = 4j, 1 ≤ j ≤ s and let h_{r} ∈ P be defined by (1 + h_{1} + h_{2} + ⋯ + h_{r} + ⋯) = (1 + p_{1} + ⋯ + p_{s})^{−1} where |h_{j}| = 4j. We have the following

Lemma 3. For any non-negative integer t, the elements h_{t+1}, ⋯, h_{t+s} forms a regular sequence in the polynomial algebra P = ℝ[p_{1}, ⋯, p_{s}].

Proof: Let P = ℝ[p_{1}, ⋯, p_{s}]. When s = 1, the lemma is obviously true for any t. Inductively assume the statement holds for any t when s is replaced by s − 1.

By induction hypothesis, for any t, h̄_{t+1}, ⋯, h̄_{t+s−1} is a regular sequence in P̄ := P/⟨p_{s}⟩ ≅ ℝ[p_{1}, ⋯, p_{s−1}]. Equivalently, h_{t+1}, ⋯, h_{t+s−1}, p_{s} is a regular sequence in P. In particular, p_{s} mod ⟨h_{t+1}, ⋯, h_{t+s−1}⟩ is not a zero divisor in P/⟨h_{t+1}, ⋯, h_{t+s−1}⟩. Note that h_{t+s} + h_{t+s−1}p_{1} +⋯+h_{t}p_{s} = 0 in P. Hence h_{t+s} ≡ h_{t}p_{s} modulo the ideal ⟨h_{t+1}, ⋯, h_{t+s−1}⟩ ⊂ P.

When t = 0, it is clear that the ideal ⟨h_{t+1}, ⋯, h_{t+s−1}⟩ = ⟨p_{1}, ⋯, p_{s−1}⟩ and h_{t+s} ≡ h_{0}p_{s} = p_{s} is clearly not a zero divisor in P/⟨h_{t+1}, ⋯, h_{t+s−1}⟩ in this case. Assume that t ≥ 1 and that the lemma holds when t is replaced by t − 1. Hence h_{t}, ⋯, h_{t+s−1} is a regular sequence. It follows that h_{t} is not a zero divisor in P/⟨h_{t+1}, ⋯, h_{t+s−1}⟩. It follows that h_{t+s} ≡ h_{t}p_{s} modulo ⟨h_{t+1}, ⋯, h_{t+s−1}⟩ is not a zero divisor in P/⟨h_{t+1}, ⋯, h_{t+s−1}⟩. The lemma follows.

We shall now establsih the Main Theorem stated in the introduction. Let 2 ≤ k ≤ [n/2].

Case 1: Let n = 2m, k = 2s, n − k = 2t, 1 ≤ s ≤ t. In this case the commutative d.g.a. C_{n,k} := C(SO(n), SO(k)×SO(n−k)) is H^{∗}(BH; ℝ)[u_{0}, ⋯, u_{m−1}], where d(H^{∗}(BH; ℝ)) = 0 and du_{r} = f_{r} for 1 ≤ r < m, d(u_{0}) = θ. Thus, writing u_{m} = θ · u_{0}, one has du_{m} = θ.du_{0} = θ^{2} = p_{s}q_{t} =: f_{m}, and hence (1 + du_{1} + ⋯ + du_{m}) = (1 + f_{1} + ⋯ + f_{m}) = (1 + p_{1} + ⋯ + p_{s})(1 + q_{1} + ⋯ + q_{t}). Writing (1 + h_{1} + ⋯) = (1 + p_{1} ⋯ p_{s})^{−1}, one has h_{r} = ∑_{||α||=r}(_{α}^{|α|})(−1)^{|α|}p^{α}, where α = (α_{1}, ⋯, α_{s}) is a sequence of non-negative integers, ||α|| = ∑_{1≤i≤s} iα_{i}, |α| = ∑_{i} α_{i}, (_{α}^{|α|}) denotes the multinomial coefficient |α|!/(α_{1}! ⋯ α_{s}!), and p_{α} denotes the monomial ∏_{1≤i≤s} p_{i}^{α_{i}}. One has the following inhomogeneous equation in C_{n,k}: 1 + q_{1} + ⋯ + q_{t} = (1 + du_{1} + ⋯ + du_{m})(1 + h_{1} + ⋯). In particular we obtain, for 1 ≤ j ≤ s,

h_{t+j} = −(h_{t+j−1}du_{1} + ⋯ + h_{1}du_{t+j−1} + du_{t+j}).

When n − k is even, q_{t} = τ_{n−k}^{2} and hence

h_{t} − τ_{n−k}^{2} = −(h_{t−1}du_{1} + ⋯ + du_{t}).

Let A = ℝ[p_{1}, ⋯, p_{s−1}, σ_{k}, τ_{n−k}] ⊂ H^{∗}(BH; ℝ). Let M_{n,k} = A[u_{0}, v_{0}, ⋯, v_{s−1}] denote the commutative d.g.a. over ℝ where, |v_{j}| = |h_{t+j}| − 1 = 4(t + j) − 1, 0 ≤ j < s, and |u_{0}| = |θ| − 1 = 2m − 1. The differential d on M_{n,k} is defined as follows: d(v_{j}) = h_{t+j}, 1 ≤ j < s, d(v_{0}) = h_{t} − τ_{n−k}^{2}, d(u_{0}) = σ_{k}τ_{n−k}, and d(A) = 0. Clearly M_{n,k} is a free d.g.a. over ℝ.

Note that since t ≥ s, h_{t+j} ∈ A is decomposable for j ≥ 1. Also, since p_{s} = σ_{k}^{2}, h_{s} ∈ A is decomposable. It follows that M_{n,k} is minimal as a d.g.a. over ℝ. We shall prove that M_{n,k} is a model for the d.g.a. C_{n,k}. From theorem 1, it will follow that M_{n,k} is a minimal model for G̃_{n,k}.

Let ϕ : M_{n,k}⟶C_{n,k} the A-algebra homomorphism defined by ϕ(u_{0}) = u_{0}, ϕ(v_{j}) = −(h_{t+j−1}u_{1} + ⋯ + u_{t+j}), 1 ≤ j < s, ϕ(v_{o}) = −(h_{t−1}u_{1} + ⋯ + u_{t}). Then ϕ is a morphism of d.g.a.’s. Indeed, d(ϕ(u_{0})) = d(u_{0}) = θ = ϕ(θ) = ϕ(d(v_{0})), and, for 1 ≤ j < s, one has d(ϕ(v_{j})) = −d(h_{t+j−1}u_{1} + ⋯ + u_{t+j}) = −(h_{t+j−1}du_{1} + ⋯ + du_{t+j}) = h_{t+j} = ϕ(h_{t+j}) = ϕ(d(v_{j})), since d(h_{r}) = 0 as h_{r} ∈ ℝ[p_{1}, ⋯, p_{s}] ⊂ A. Similarly, d(ϕ(v_{0})) = h_{t} − τ_{n−k}^{2} = ϕ(d(v_{0})). To show that the chain map ϕ induces an isomorphism in cohomology, first observe that H^{∗}(C_{n,k}, d) ≅ H^{∗}(G̃_{n,k}; ℝ). This is because (C_{n,k}, d) is a model for the space G̃_{n,k} (see Theorem1). Alternatively, one applies a Koszul complex argument (cf.

Ch.XXI,§4,[11]) and uses the fact that θ, f_{1}, ⋯, f_{m−1} is a regular sequence to see that the cohomology of C_{n,k} is H^{∗}(BH; ℝ)/⟨θ, f_{1}, ⋯, f_{m−1}⟩ = H^{∗}(G̃_{n,k}; ℝ). Using the relation (1 + f_{1} + ⋯ + f_{m})(1 + h_{1} + ⋯) = (1 + q_{1} + ⋯ + q_{t}), we see that 0 = f_{r} = ∑_{i+j=r} p_{i}q_{j}, 1 ≤ r ≤ t, in H^{∗}(BH; ℝ)/⟨σ_{k}τ_{n−k}, f_{1}, ⋯, f_{m−1}⟩ ≅ H^{∗}(G̃_{n,k}; ℝ). In particular, q_{j} = h_{j} for any j, 1 ≤ j ≤ t and τ_{n−k}^{2} = q_{t} = h_{t} in H^{∗}(BH; ℝ)/⟨σ_{k}τ_{n−k}, f_{1}, ⋯, f_{m}⟩. Hence

H^{∗}(G̃_{n,k}; ℝ) ≅ ℝ[p_{1}, ⋯, p_{s−1}, σ_{k}, τ_{n−k}]/⟨h_{t+1}, ⋯, h_{t+s−1}, σ_{k}τ_{n−k}, h_{t} − τ^{2}⟩.

Using lemma 3, one sees that τ_{n−k}, h_{t} − τ_{n−k}^{2}, h_{t+1}, ⋯, h_{t+s−1} and σ_{k}, h_{t} − τ_{n−k}^{2}, h_{t+1}, ⋯, h_{t+s−1} are regular sequences in A. From Lemma 23.6 of [3] it follows that θ, τ_{n−k}^{2} − h_{t}, h_{t+1}, ⋯, h_{m−1} is a regular sequence in A. Again by applying a Koszul complex argument we obtain that

H^{∗}(M_{n,k}, d) ≅ A/⟨τ_{n−k}^{2} − h_{t}, h_{t+1}, ⋯, h_{m−1}, σ_{k}τ_{n−k}⟩ ≅ H^{∗}(G̃_{n,k}; ℝ).

Under our identifications, the map ϕ actually induces the identity map of H^{∗}(G̃_{n,k}; ℝ). This proves that M_{n,k} is quasi isomorphic to C_{n,k} and hence it is the minimal model of G̃_{n,k} in this case.

Case 2: Let n = 2m + 1 = 2s + 2t + 1, k = 2s, or 2s + 1. We assume that k ≤ m (equivalently s ≤ t with equality only if k = 2s). In this case C_{n,k} = H^{∗}(BH; ℝ)[u_{1}, ⋯, u_{m}], where d(H^{∗}(BH; ℝ)) = 0, du_{j} = f_{j}, 1 ≤ j ≤ m.

Let A ⊂ H^{∗}(BH; ℝ) be the polynomial algebra over ℝ in generators p_{1}, ⋯ p_{s−1}, σ_{k} (resp. p_{1}, ⋯, p_{s}, τ_{n−k}) for k even (resp.k odd). Thus p_{s} = σ_{k}^{2} when k = 2s.

Subcase (a): Let k = 2s. Let M_{n,k} = A[v_{1}, ⋯, v_{s}] be the d.g.a. over ℝ where |v_{j}| = |h_{j+t}| − 1 = 4(j + t) − 1 and d(v_{j}) = h_{t+j}, 1 ≤ j ≤ s, d(A) = 0. The free d.g.a. M_{n,k} is minimal since the h_{t+j} are decomposable for j ≥ 1. The A-algebra map ϕ : M_{n,k}⟶C_{n,k} defined be ϕ(v_{j}) = −(h_{j+t−1}u_{1} + ⋯ + u_{j+t}), 1 ≤ j ≤ s is a morphism of d.g.a.’s. Also, the elements h_{t+j}, 1 ≤ j ≤ s form a regular sequence in A. Therefore arguing as in case 1 above, we conclude that ϕ is a quasi isomorphism. Hence M_{n,k} is a minimal model of G̃_{n,k}.

Subcase (b): Let k = 2s+1. Let M_{n,k} = A[v_{0}, v_{1}, ⋯, v_{s}] be the commutative d.g.a. over ℝ where |v_{j}| = |h_{t+j}|−1 = 4(t+j)−1, 0 ≤ j ≤ s and d(v_{0}) = h_{t}−τ_{n−k}^{2}, d(v_{j}) = h_{t+j}, 1 ≤ j ≤ s, and d(A) = 0. Since s < t, h_{t+j} is decomposable for j ≥ 0. Therefore M_{n,k} is minimal. The A-algebra map ϕ : M_{n,k}⟶C_{n,k} defined by ϕ(v_{j}) = −(h_{t+j−1}u_{1} + ⋯ + u_{j}), 0 ≤ j ≤ s is a morphism of d.g.a. over ℝ. Using the fact that h_{t}−τ_{n−k}^{2}, h_{t+1}, ⋯, h_{m} is a regular sequence in A = ℝ[p_{1}, ⋯, p_{s}, τ], we conclude, as before, that M_{n,k} is a minimal model for G̃_{n,k}.

Case 3: Let n = 2m + 2, k = 2s + 1, n − k = 2t + 1, 1 ≤ s ≤ t. In this case the subgroup SO(k) × SO(n − k) is not of maximal rank in SO(n). The c.d.g.a. C_{n,k} has the description H^{∗}(BH; ℝ)[u_{1}, ⋯, u_{m}, u_{0}] with |u_{j}| = 4(t + j) − 1, |u_{0}| = 2m + 1 = n − 1, and du_{j} = f_{j}, 1 ≤ j ≤ m, and du_{0} = 0.

Let A = P = ℝ[p_{1}, ⋯, p_{s}] ⊂ H^{∗}(BH; ℝ). Let M_{n,k} denote the d.g.a. A[v_{1}, ⋯, v_{s}, v_{0}] where |v_{j}| = 4(t + j) − 1, 1 ≤ j ≤ s, |v_{0}| = 2m + 1, d(v_{j}) = h_{j+t}, 1 ≤ j ≤ s, dv_{0} = 0, and d(A) = 0. As before, M_{n,k} is free and minimal.

The A-algebra map ϕ : M_{n,k}⟶C_{n,k} defined by ϕ(v_{j}) = u_{j}, 0 ≤ j ≤ s, is a chain map as can be verified as in case 1. Note that the cohomology of M_{n,k} can again be computed using the Koszul complex of A with respect to the sequence h_{t+1}, ⋯, h_{t+s}, 0 ∈ A. Again using the fact that h_{t+1}, ⋯, h_{t+s} is a regular sequence, a simple calculation leads to:

H^{∗}(M_{n,k}; ℝ) = A[u_{0}]/⟨h_{t+1}, ⋯, h_{t+s}⟩.

This is also the cohomology of C_{n,k} and as in case 1, we see that ϕ induces isomorphism in cohomology. Hence M_{n,k} is the minimal model of G̃_{n,k}.

This completes the proof of the Main Theorem. □

Corollary 4. The oriented Grassmannian G̃_{n,k} is formal for all 1 ≤ k < n. In particular all Massey products in H^{∗}(G̃_{n,k}; ℝ) vanish. □.

Proof: First let n = 2s + 2t + 2, k = 2s + 1. With notation as above, note that the A-algebra map M_{n,k}⟶A[u_{0}]/⟨h_{t+1}, ⋯, h_{t+s}⟩ = H^{∗}(M_{n,k}; ℝ) defined by v_{0} ↦ u_{0}, v_{j} ↦ 0, 1 ≤ j ≤ s is a map of d.g.a.’s where the differential on H^{∗}(M_{n,k}; ℝ) is defined to be zero. Hence G̃_{n,k} is formal.

Same argument as above shows that G̃_{n,k} is formal for all parities of n and k. □

Corollary 5. Let dim_{ℝ}(π_{r}(G̃_{n,k}) ⊗_{ℤ} ℝ) = π_{r}.

(i) Let n = 2s + 2t, k = 2s, 1 ≤ s ≤ t. Then ∑_{r≥0} π_{r}z^{r} = 1 + z^{n−1} + z^{2s} + z^{2t} + z^{4t−1} + ∑_{1≤j<s}(z^{4j} + z^{4(t+j)−1}).

(ii)(a) Let n = 2s + 2t + 1, k = 2s, s ≤ t. Then ∑_{r≥0} π_{r}z^{r} = 1 + z^{2s} + z^{4(s+t)−1} + ∑_{1≤j<s}(z^{4j} + z^{4(t+j)−1}).

(ii)(b) Let n = 2s + 2t + 1, k = 2s + 1, s < t. Then ∑_{r≥0} π_{r}z^{r} = 1 + z^{2t} + z^{4t−1} + ∑_{1≤j≤s}(z^{4j} + z^{4(t+j)−1}).

(iii) Let n = 2s + 2t + 2, k = 2s + 1, 1 ≤ s ≤ t. Then ∑_{r≥0} π_{r}z^{r} = 1 + z^{n−1} + ∑_{1≤j≤s}(z^{4j} + z^{4(t+j)−1}).

Proof: This follows from the above description of the minimal model of G̃_{n,k} and the fact that Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ) is isomorphic to the r-th degree component of the graded vector space M_{n,k}/D where D denotes the ideal M_{n,k}^{+}.M_{n,k}^{+} of “decomposable elements”. □

4 Nilpotence of Grassmannians and related spaces

Let X be a path connected topological space with base point x. Recall that X is called nilpotent if the fundamental group π = π_{1}(X, x) of X is nilpotent as a group and all the higher homotopy groups of X are nilpotent as modules over the integral group ring ℤπ. That is, denoting the augmentation ideal of ℤπ by I, X is nilpotent if and only if π is nilpotent and, for each n ≥ 2, there exists an integer N = N(n) such that I^{N}.π_{n}(X, x) = 0.

A path connected topological space X is said to be of finite ℚ-type if H^{n}(X; ℚ) is finite dimensional for all n ≥ 1. If X is nilpotent, then X is of finite ℚ type if and only if H_{1}(X; ℚ) and π_{n}(X) ⊗ ℚ are finite dimensional for all n ≥ 2. When X is a nilpotent spaces of finite ℚ-type, one associates to X the minimal model M_{X} of the Sullivan-de Rham complex of X which is a c.d.g.a. over ℚ. The minimal model M_{X} contains all the rational homotopy information of X in this case. That is, M_{X} and M_{Y} are quasi isomorphic if and only if X and Y are of same rational homotopy type, where both X and Y are of finite ℚ-type. We refer the reader to Chapter 2, [1] for details. (See also §9 of [4].) In case X is a smooth manifold, it is more convenient to work with the real homotopy theory via the minimal model of the de Rham complex of X.

Let n_{1}, ⋯, n_{s} be a sequence of positive integers, and let n = ∑_{1≤i≤s} n_{i}. Denote by X = G(n_{1}, ⋯, n_{s}) the flag manifold of consisting of flags (V_{1}, ⋯, V_{s}) where V_{i} is an n_{i} dimensional vector subspace of ℝ^{n} such that V_{i} ⊥ V_{j} if i ≠ j. The flag manifold X can be identified with the coset space O(n)/(O(n_{1}) × ⋯ × O(n_{s})) so that it is naturally a smooth compact manifold of dimension ∑_{1≤i<j≤s} n_{i}n_{j}. When s = 2, it is identified with the Grassmannian G_{n,n_{1}}. The universal covering of the flag manifold is the oriented flag manifold X̃ = G̃(n_{1}, ⋯, n_{s}) = SO(n)/(SO(n_{1}) × ⋯ × SO(n_{s})) which consists of “oriented flags”, that is, flags (V_{1}, ⋯, V_{s}) together with orientations on each vector space V_{i} so that the direct sum orientation on ∑ V_{i} = ℝ^{n} coincides with the standard orientation on ℝ^{n}. The natural map p : X̃⟶X that forgets the orientations on oriented flags of X̃ is the covering projection. The deck transformation group is generated by the involutions α_{i}, 1 ≤ i < s, which reverses the orientation on i-th and s-th vector space in each oriented flag, and is isomorphic to (ℤ/2)^{s−1}. We note that, when n_{i} and n_{s} are odd, α_{i} can be realized as multiplication on the left by the element A_{i} ∈ SO(n) where A_{i} has, as a block diagonal matrix j-th block down the diagonal the identity matrix of size n_{j} if j ≠ i, s and the i-th and s-th block being negative identity matrices of sizes n_{i} and n_{s} respectively.

In this section we prove

Theorem 6. (Glover-Homer [6]) Let X = G(n_{1}, ⋯, n_{s}), s ≥ 2, denote the real flag manifold. Then the following are equivalent:

(i) all the n_{i} are odd, (ii) X is simple, (iii) X is nilpotent.

Theorem 7. (K.Varadarajan) The classifying space BO(k) is nilpotent if and only if k is odd.

We need the following lemma (cf. lemma 7, §3, Ch. 7 of [14]). We shall denote the free homotopy classes from the n-sphere to Y by π_{n}(Y). Note that when Y is a simply connected space, π_{n}(Y) may be identified with π_{n}(Y, y).

Lemma 8. Let p : X̃⟶X be the universal covering projection of a path connected finite CW complex, and let r ≥ 2. Then p induces an isomorphism between π_{r}(X̃) and π_{r}(X, x) which is compatible with the action of the deck transformation group on π_{r}(X̃) and that of π_{1}(X, x) on π_{r}(X, x). □

Proof of Theorem 6: (i) ⟹ (ii): Assume that all the n_{i} are odd. Since SO(n) is connected, maps induced on X̃ by multiplication by elements of SO(n) are all homotopic to the identity map. In particular the α_{i}, 1 ≤ i < s, are homotopic to the identity map of X̃. Since the α_{i} generate the deck transformation group of the universal covering projection X̃⟶X, it follows from Lemma 8 that the action of π_{1}(X) ≅ (ℤ/2)^{s−1} on any π_{r}(X) is trivial and so the space X is simple.

It is evident that (ii) implies (iii).

(iii) ⟹ (i): We will first assume that s = 2 so that X is the Grassmann manifold X = G_{n,n_{1}}. For simplicity of notation let k = n_{1}. Assume that at least one of the integers k, n − k is even. We will prove that X is not nilpotent. Indeed, let r be an even integer in the set {k, n − k}. We will prove that the action of the generator of π_{1}(X) ≅ ℤ/2 on π_{r}(X) ⊗_{ℤ} ℝ has −1 as an eigenvalue.

When r = n − k > k it will be convenient to denote by σ_{r} the element that was denoted τ_{n−k} in §3.

We first note that the deck transformation α := α_{1} of the covering projection G̃_{n,k}⟶G_{n,k} is not homotopic to the identity. In fact α reverses the orientation on the canonical kplane bundle over X̃ = G̃_{n,k}. When r is even, the Euler class σ_{r} of the canonical oriented r-plane bundle is not torsion and in real cohomology one has α^{∗}(σ_{r}) = −σ_{r}. The map α induces an endomorphism α̂ of the d.g.a. M_{n,k} which is unique upto chain homotopy. The map α̂ induces an involution [α], which depends only on the chain homotopy class of α̂, of the graded ℝ-vector space M_{n,k}/D such that the following diagram commutes, where the vertical arrows are natural isomorphisms (cf. [5]). Here α^{∗} denotes the ℝ-linear involution on Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ) induced by α.

(M_{n,k}/D)^{r}  ⟶^{[α]}  (M_{n,k}/D)^{r}
↓  ↓
Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ) ⟶^{α^{∗}} Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ)

Note that by Corollary 5 the vector space Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ) is non-zero. In fact the class σ_{r} is not in the ideal D of decomposable elements of M_{n,k}. Also [α](σ_{r}) = −σ_{r} since in de Rham cohomology α^{∗} maps the Euler class of the canonical r-plane bundle to its negative. Hence it follows that the −1 eigenspace of α^{∗} : Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ)⟶Hom_{ℤ}(π_{r}(G̃_{n,k}), ℝ) is non-zero. It follows that −1 is an eigenvalue for the action of the generator of the fundamental group of G_{n,k} on π_{r}(G_{n,k}) ⊗_{ℤ} ℝ. Hence G_{n,k} is not nilpotent.

Now let s ≥ 3 and assume without loss of generality that n_{1} is even. One has a natural inclusion j : Y := G(n_{1}, n_{2}) ⟶ X and a projection map q : X ⟶ G(n_{1}, n_{2}+⋯+n_{s}) =: Z. Explicitly, j(A) = (A, E_{2}, ⋯, E_{s}) and q(V_{1}, ⋯, V_{s}) = V_{1} where E_{r} denotes the span of the standard basis vectors e_{t}, n_{1} + ⋯ n_{r−1} + 1 ≤ t ≤ n_{1} + ⋯ + n_{r}. Note that q ∘ j is the natural inclusion of Y into Z induced by the inclusion of ℝ^{n_{1}+n_{2}} into ℝ^{n} and hence induces an isomorphism of fundamental groups. Denote by f : Ỹ ⟶Z̃ a lift of q ∘ j ∘ p to Z̃ := G̃_{n,n_{1}+n_{2}} where p : Ỹ := G̃_{n_{1}+n_{2},n_{1}}⟶Y is the universal covering projection. The map f pulls back the canonical n_{1}-plane bundle on Z̃ to that on Ỹ. Hence it maps the Euler class σ_{n_{1}}(Z̃) to the class ±σ_{n_{1}}(Ỹ). Replacing f by f ∘ α if necessary one may as well assume that f^{∗}(σ_{n_{1}}(Z̃)) = σ_{n_{1}}(Ỹ). As in the case when s = 2, using naturality properties of minimal models, one concludes that the morphism of d.g.a. induced by f maps the class σ_{n_{1}}(Z̃) ∈ M_{Z} to σ_{n_{1}}(wtY_{)} ∈ M_{Y}. This implies that the −1 eigenspace for the action of π_{1}(Y) (via the isomorphism of fundamental groups induced by q ∘ j) on Hom(π_{n_{1}}(Z), ℝ) is non-zero. Hence the action of π_{1}(Y) via the monomorphism of fundamental groups induced by j on Hom(π_{n_{1}}(X), ℝ) must have non-zero −1 eigenspace. This clearly implies that the action of π_{1}(X) on π_{n_{1}}(X) is not nilpotent. □

Remark 9. Theorem 7.2 of [10] can be readily applied to show that (iii) ⟹ (i) once this is known for Grassmannians. Our proof however exhibits a higher homotopy group of X on which the action of the fundamental group is not nilpotent. Alternatively, one can deduce the same result from our result for Grassmannians using the second Claim in the proof of theorem 7.2 of [10]. We record, as a corollary of the the above proof, the following proposition for possible future reference.

Proposition 10. Let X = G(n_{1}, ⋯, n_{s}), s ≥ 2. If n_{i} is an even integer, then π_{n_{i}}(X) is not nilpotent as a π_{1}(X) module. □

Proof of Theorem 7: Note that BO(k) = ⋃_{n≥2k} G_{n,k} where we regard G_{n,k} as the subspace of G_{n+1,k} in the usual way, considering the vector space ℝ^{n} as the subspace of ℝ^{n+1} consisting of those vectors with last coordinate being zero. The inclusion map i_{n} : G_{n,k}⟶BO(k) is an (n−k)-equivalence and hence, given any r ≥ 1, for n > k +r +1, the map i_{n} induces isomorphism of the r-th homotopy groups. Also the action of the fundamental group of G_{n,k} on π_{r}(G_{n,k}, x) is compatible with the action of the fundamental group of BO(k) on π_{r}(BO(k), x) via the map induced by i_{n}. In particular if k is even, choosing r = k and n > 2k + 1, one sees from our proof of Theorem 6 that BO(k) is not nilpotent.

When k is odd, one can always choose n to be even (in addition to n > k + r + 1) to conclude that the fundamental group of BO(k) acts trivially on π_{r}(G̃_{n,k}) for any r ≥ 1. Hence we conclude that the space BO(k) is nilpotent. In fact we have shown that BO(k) is simple. □

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