Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 47–50

ON COMPACT HOMOGENEOUS SPACES

WITH NON–VANISHING MASSEY PRODUCTS

Aleksei Tralle^{1}

Abstract. We construct explicitly examples of homogeneous spaces of compact simple Lie groups with non-vanishing Massey products. Keywords. Homogeneous space, Massey product. MS classification. 53C, 55P.

It is well-known that formality or non-formality (in the sense of Sullivan) of a manifold implies strong geometric consequences. For example, if a manifold carries a Kählerian structure, it must be formal [3]. In fact it is not easy to determine whether the given manifold is formal or not (see, e.g. [4]). It is known that if a topological space has nonzero Massey products, it is not formal. Therefore, it is an interesting question to describe various classes of manifolds with non-vanishing Massey products. The first example of that kind in the case of compact homogeneous nilmanifold was obtained by L. Cordero, M. Fernandez and A. Gray in [2]. Here we consider the “opposite” case of homogeneous spaces of compact simple Lie group.

Theorem. There exist compact homogeneous spaces G/H of simple compact Lie groups G with non-vanishing Massey products. In particular, homogeneous space

SU(n + 1)/SU(3) × SU(3), n ≥ 5

possesses non-vanishing Massey products.

Introduce the appropriate notations. As far as the author would like this note to be short, the reader is supposed to be familiar with the standard facts of cohomology theory of homogeneous spaces and rational homotopy theory. We shall use all the notations and facts in accordance with [1, 6] without additional explanations. Everywhere the symbol W (G, T) denotes the Weyl group of a compact Lie group G with respect to its maximal torus T. For a subgroup K ⊂ G we choose its maximal torus T′ in such a way that T′ ⊂ T. For any vector space V the symbol k[V] denotes the polynomial algebra over V and k[V]^{Γ} denotes the ring of Γ-invariants of the discrete group Γ ≤ GL(V). A free graded commutative algebra over a graded vector space V is denoted by Λ(V) and the same notation is used for the exterior algebra over V (but the sense of the notation will be always clear from the context). Everywhere Lie algebras of Lie groups G, H, …

Typeset by AMS-TEX

1This paper is in final form and no version of it will be submitted for publication elsewhere

will be denoted by the corresponding Gothic letters 𝔊, ℌ . . .. We use also the notation k[X_{1}, …, X_{n}] for the polynomial ring of variables X_{1}, …, X_{n} over the field k.

Proof of the theorem. Consider, at first, the minimal case

M = SU(6)/SU(3) × SU(3)

and calculate its Cartan algebra (C, δ). In our case (see [1] for details)

(C, δ) = (ℝ[𝔱′]^{W (A_{2}×A_{2})} × Λ(y_{1}, …, y_{5}), δ), δy_{j} = f̃_{j} = f_{j} |_{𝔱′}, j = 1, …, 5

where δ = 0 on the first term of the tensor product, f_{j} and f̃_{j} are defined by Chevalley isomorphisms

\begin{matrix}ℝ[𝔱]^{W (A_{5})} ≅ & ℝ[f_{1}, …, f_{5}], \\ ℝ[𝔱′]^{W (A_{2}×A_{2})} ≅ & ℝ[u_{1}, u_{2}, v_{1}, v_{2}]. \end{matrix}

Using the explicit expressions for f_{j} and u_{i}, v_{i} [5] one obtains

\begin{matrix}f_{j}(x_{1}, …, x_{6}) = & x_{1}^{j+1} + ⋯ + x_{6}^{j+1}, & x_{1} + ⋯ + x_{6} = 0, & j = 1, …, 6, \\ u_{i} = & x_{1}^{i+1} + x_{2}^{i+1} + x_{3}^{i+1}, & x_{1} + x_{2} + x_{3} = 0, & i = 1, 2, \\ v_{i} = & x_{4}^{i+1} + x_{5}^{i+1} + x_{6}^{i+1}, & x_{4} + x_{5} + x_{6} = 0, & i = 1, 2. \end{matrix}

Then, up to scalar multiplies, which are not important in our considerations

f̃_{1} = u_{1} + v_{1}, f̃_{2} = u_{2} + v_{2}, f̃_{3} = u_{1}^{2} + v_{1}^{2},
f̃_{4} = u_{1}u_{2} + v_{1}v_{2}, f̃_{5} = u_{1}^{3} + v_{1}^{3} + u_{2}^{2} + v_{2}^{2}

and finally

(C, δ) = (ℝ[u_{1}, u_{2}, v_{1}, v_{2}] ⊗ Λ(y_{1}, …, y_{5}), δ), δu_{i} = δv_{i} = 0, i = 1, 2,
δy_{1} = u_{1} + v_{1}, δy_{2} = u_{2} + v_{2}, δy_{3} = u_{1}^{2} + v_{1}^{2},
δy_{4} = u_{1}u_{2} + v_{1}v_{2}, δy_{5} = u_{1}^{3} + v_{1}^{3} + u_{2}^{2} + v_{2}^{2},

where the degrees of variables have values: deg(u_{1}) = 4, deg(v_{1}) = 4, deg(u_{2}) = deg(v_{2}) = 6, deg(y_{1}) = 3, …, deg(y_{5}) = 11.

Observe that (C, δ) is not a minimal algebra (see [6] for the appropriate notions). Nevertheless, (C, δ) is a free graded commutative algebra and we can apply Sullivan’s method for constructing the minimal model of (C, δ). In what follows we apply the calculations of [6] (Proposition V.7). Denote by V the linear span of {u_{1}, v_{1}, u_{2}, v_{2}, y_{1}, …, y_{5}}:

V = L(u_{1}, u_{2}, v_{1}, v_{2}, y_{1}, …, y_{5})

and introduce a new derivation d′, defined on V by the formula

d′ : V^{n} → V^{n+1}, d′ = π ∘ δ, π : V^{n+1} ⊕ L^{++}(V)^{n+1} → V^{n+1},

(L^{++}(V) being the ideal of decomposable elements in Λ(V), and π being the projection on the first summand). Then

d′u_{i} = d′v_{i} = 0 (i = 1, 2), d′y_{1} = u_{1} + v_{1}, d′y_{2} = u_{2} + v_{2}, d′y_{j} = 0 (j > 2).

Consider the direct sums

V = Im d′ ⊕ V′ ⊕ W, Im d′ ⊕ V′ = Ker d′.

Then, obviously,

Ker d′ = L(u_{1}, u_{2}, v_{1}, v_{2}, y_{3}, y_{4}, y_{5}),
Im d′ = L(u_{1} + v_{1}, u_{2} + v_{2}).

Thus

L(u_{1}, u_{2}, v_{1}, v_{2}, y_{3}, y_{4}, y_{5}) = L(u_{1} + v_{1}, u_{2} + v_{2}) + L(v_{1}, v_{2}) + L(y_{3}, y_{4}, y_{5}),

which implies

V′ = L(v_{1}, v_{2}) ⊕ L(y_{3}, y_{4}, y_{5}), W = L(y_{1}, y_{2}),
W′ = d(W) = L(δy_{1}, δy_{2}) = L(u_{1} + v_{1}, u_{2} + v_{2}).

Consider the algebra

C̄ = ℝ[u_{1} + v_{1}, u_{2} + v_{2}] ⊗ Λ(y_{1}, y_{2}) = Λ(W′ ⊕ W)

and generate the ideal ⟨C̄^{+}⟩ in Λ(V) by C̄^{+} (it means by the elements of positive degrees). It is easy to calculate that

(Λ(V′), δ) ≅ (Λ(V)/⟨C̄^{+}⟩, δ) ≅ (ℝ[u_{1}, u_{2}] ⊗ Λ(y_{3}, y_{4}, y_{6}), D) = (𝔐, D),
Du_{i} = 0, i = 1, 2, Dy_{3} = u_{1}^{2}, Dy_{4} = u_{1} · u_{2}, Dy_{5} = u_{2}^{2}

(up to scalars, of course). By Sullivan’s theorem [6], (𝔐, D) is a minimal model for M. In what follows we need to calculate certain Massey products. Let ā, b̄, c̄ be the cohomology classes in H^{∗}(𝔐, D) corresponding to the elements

a = u_{1}, b = u_{1}, c = u_{2}. As far as ab = u_{1}^{2} = δy_{3}, bc = u_{1}u_{2} = δy_{4} we obtain āb̄ = 0, b̄c̄ = 0. Then

(1)  ay_{4} − y_{3}c = u_{1}y_{4} − u_{2}y_{3}

is a cocycle, representing the Massey triple product ⟨ā, b̄, c̄⟩ ∈ H^{11}(𝔐, D)/āH^{7}(𝔐, D)+ c̄H^{5}(𝔐, D). Obviously, (1) can not be a coboundary, because any non-zero coboundary in (𝔐, D) is of u_{1}- or u_{2}-degree greater than 1. Moreover, cohomology class u_{1}y_{4}−u_{2}y_{3} ∉ āH^{7}(𝔐, D) + c̄H^{5}(𝔐, D) (it is also evident, as far as otherwise there would hold the equality u_{1}y_{4} − u_{2}y_{3} = u_{1}h_{1} + u_{2}h_{2} + δw, and taking into consideration the degrees deg(u_{i}) and deg(y_{j}) one would come to the previous argument). Thus

⟨ā, b̄, c̄⟩ ≠ 0

and the theorem is proved in the case SU(6)/SU(3) × SU(3). To finish the proof it is enough to notice, that for n ≥ 5 the proof in fact does not differ from the particular case above, because the Cartan algebra in the general case differs from (C, δ) only by the exterior variables y_{6}, … for which δy_{6}, …, … ∈ (f_{3}, f_{4}, f_{5}), but the latter do not change ⟨ā, b̄, c̄⟩ (here we denote by ( , , ) the ideal generated by f_{3}, f_{4}, f_{5}. □

Acknowledgements

The author is grateful to Professor S. Salamon for encouragement and valuable discussions during the Conference in Opava.

References

1. Borel A., Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. Math. 57 (1953), 115–207.

2. Cordero L., Fernandez M., Gray A., Symplectic manifolds with no Kähler structure, Topology 25 (1986), 375–380.

3. Deligne P., Griffiths P., Morgan J., Sullivan D., The real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–254.

4. Greub V., Halperin S., Vanstone R., Connections, Curvature and Cohomology, v. 3, Academic Press, New York, 1976.

5. Flatto L., Invariants of finite reflection groups, L’Enseign. Math. 24 (1978), 237–293.

6. Lehmann D., Theorie homotopique des formes differentielles (d’apres D. Sullivan), Asterisque, Soc. Math. France 45 (1977).

Aleksei Tralle Institute of Mathematics University of Szczecin Wielkopolska 15, 70–451 Szczecin Poland

Received 28 September 1992