Equivariant formality of the isotropy action on Z 2 ⊕ Z 2 -symmetric spaces
aa r X i v : . [ m a t h . A T ] F e b EQUIVARIANT FORMALITY OF THE ISOTROPYACTION ON Z ⊕ Z -SYMMETRIC SPACES MANUEL AMANN AND ANDREAS KOLLROSS
Abstract.
Compact symmetric spaces are probably one of the mostprominent class of formal spaces, i.e. of spaces where the rational ho-motopy type is a formal consequence of the rational cohomology algebra.As a generalisation, it is even known that their isotropy action is equiv-ariantly formal.In this article we show that ( Z ⊕ Z )-symmetric spaces are equivari-antly formal and formal in the sense of Sullivan, in particular. Moreover,we give a short alternative proof of equivariant formality in the case ofsymmetric spaces with our new approach. Introduction
An important notion in Rational Homotopy Theory is the concept of formality , which, roughly speaking, expresses the property of a space that itsrational homotopy type can be formally derived from its rational cohomologyalgebra; that is, all rational information is contained already in the rationalcohomology algebra. In particular, obstructions to formality like Masseyproducts vanish. There are several prominent classes of formal manifolds:compact K¨ahler manifolds, symmetric spaces, homogeneous spaces of equalrank just to mention a few. (Nonetheless, even amongst homogeneous spacesthe lack of formality should be generic—see [1] for several classes of non-formal examples.)In this article we are interested in certain homogeneous spaces
G/K ofcompact connected Lie groups. An a priori unrelated concept is equivariantformality of an action of a compact Lie group K on a manifold M , whichstates that the Borel fibration
M ֒ → M × K E K → B K is totally non-homologous to zero , i.e. the map induced by the fiber inclusion H ∗ ( M × K E K ; Q ) → H ∗ ( M ; Q ) is a surjection.In the special case of the isotropy action of K on G/K , i.e. the actiongiven by left multiplication of the isotropy group K on G/K , there is a well-known characterisation (see [4, Theorem A]) which yields that equivariantformality of K y G/K implies the formality of
G/K . Accordingly, severalclasses of isotropy actions are known to be equivariantly formal, namely the
Date : February 18th, 2020.2010
Mathematics Subject Classification.
Key words and phrases. generalized symmetric spaces, ( Z ⊕ Z )-symmetric spaces,formality, equivariant formality. MANUEL AMANN AND ANDREAS KOLLROSS ones on symmetric spaces or, more generally, on k -symmetric spaces (see[8], [9]).In this note we aim to find further classes of homogeneous spaces withequivariantly formal isotropy actions and to pave a way towards Question 0.1.
Let G be a compact connected Lie group and let σ be anabelian Lie group of automorphisms of G . Does it hold true that the isotropyaction on G/G σ , where G σ denotes the identity component of the fixed pointset of σ , is equivariantly formal? (Clearly, one may ask this question first for the formality of G/G σ .) Weverify the general question for small groups σ . Theorem A.
Question 0.1 can be answered in the affirmative whenever | σ | ≤ . The conjecture is trivial for the trivial group. For | σ | = 2, it exactly statesthe well-known equivariant formality of the isotropy action on symmetricspaces. For | σ | = 4 we have to distinguish the two cases σ = Z or σ = Z ⊕ Z . The first case, as well as the cases | σ | = 3 , , , Z ⊕ Z )-symmetric space isequivariantly formal. The proof proceeds by stepping through the classifi-cation of these spaces and first proving their formality. We then provide ageneral criterion which, building on the formality of a homogeneous space G/K , reveals its isotropy action as equivariantly formal. Finally, it onlyremains to realise that the arguments we provided for formality are actuallystrong enough to already yield equivariant formality in the light of this newcriterion.Note that due to [15, p. 56, Proposition] a ( Z ⊕ Z )-symmetric spacedefined in analogy to a symmetric space (see [15, Definition 1.1]) is alreadyhomogeneous. Hence, in the light of given homogeneity, Question 0.1 allowsfor an obvious reformulation.Observe further that in [15, Proposition 4.3] it is claimed that such a( Z ⊕ Z )-structure is geometrically formal (the product of harmonic formsis again harmonic) once there exists a compatible metric. This would implyformality as well, in generality, however, does not occur (see for example[14]).We remark that our main result has recently been proven by Sam HaghShenas Noshari for the case of ( Z ⊕ Z )-symmetric spaces in his dissertation[11]. His approach, however, is different from ours, directly proving equi-variant formality of the isotropy action without drawing on the classificationof the latter spaces. We hope, however, that our approach can be seen as ageneral approach which is useful in many similar situations. We illustratethis by a quick reproof of the equivariant formality of the isotropy action onsymmetric spaces. Structure of the article.
In Section 1 we review the classification of( Z ⊕ Z )-symmetric spaces. In Section 2 we provide the necessary conceptsfrom Rational Homotopy Theory. The proof of the main result then startswith a case-by-case check of the formality of ( Z ⊕ Z )-symmetric spaces SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 3 in Section 3. In order to extend this to equivariant formality, we provide acriterion in Section 4. We finish the proof of the equivariant formality of theisotropy action on ( Z ⊕ Z )-symmetric spaces in Section 5. As a service tothe reader we provide a new and short proof for the equivariant formalityof the isotropy action on symmetric spaces (building on their well-knownformality) in Section 6. Acknowledgements.
The authors thank Sam Hagh Shenas Noshari forseveral helpful discussions and Peter Quast for pointing out the article [15].The first named author was supported both by a Heisenberg grant andhis research grant AM 342/4-1 of the German Research Foundation; he ismoreover associated to the DFG Priority Programme 2026.1. Z ⊕ Z -symmetric spaces A compact homogeneous space M = G/H is called ( Z ⊕ Z )-symmetricif there are two commuting automorphisms α , β with α = β = 1 of G suchthat ( G α ∩ G β ) ⊆ H ⊆ G α ∩ G β .The classification of ( Z ⊕ Z )-symmetric spaces was carried out in [3,Theorem 14; Tables 1, 2, 3, 4] for the classical cases and in [13, Theorems1.2, 1.3; Table 1] for exceptional Lie groups G .We say that a ( Z ⊕ Z )-symmetric space G/H is decomposable if the Liealgebras of G and H decompose as g = g × g and h = h × h , where h i ⊆ g i and where g , g are nontrivial—otherwise it is called indecomposable . Forour purposes, it suffices to consider indecomposable ( Z ⊕ Z )-symmetricspaces. Proposition 1.1.
Let M = G/H be an indecomposable ( Z ⊕ Z ) -symmetricspace, where G is a connected compact Lie group and H is a connectedclosed subgroup. Then there is an automorphism φ of G such that thepair ( G, φ ( H )) is one of the following:(1) G is a torus and H a closed connected subgroup;(2) G is simple and G/H is a ( Z ⊕ Z ) -symmetric space;(3) G is simple and G/H is a symmetric space in the usual sense;(4) G is simple and H = G ;(5) up to finite coverings, G = L × L , where L is a simple compact Liegroup and H = { ( g, g ) | g ∈ K } , where K ⊂ L is such that L/K isa symmetric space in the usual sense;(6) up to finite coverings, G = L × L , where L is a simple compact Liegroup and H = { ( g, g ) | g ∈ L } ;(7) up to finite coverings, G = L × L × L × L and H = { ( g, g, g, g ) | g ∈ L } , where L is a simple compact Lie group.Proof. For the Lie algebra of G , we have the decomposition g = g ⊕ g ⊕ · · · ⊕ g k , where g is an abelian ideal and g , . . . , g k are simple ideals.The group σ acts on g by automorphisms. Since the above decompositionis uniquely determined and a Lie algebra automorphism maps the center ofa Lie algebra onto itself, it follows that we have Case (1) if g is nontrivial. MANUEL AMANN AND ANDREAS KOLLROSS
Otherwise, we may assume g = g ⊕ . . . ⊕ g k is semisimple. Since anyLie algebra automorphism maps a simple ideal g i of g onto an isomorphicsimple ideal g j of g , it follows that one can define an action of g on the set I := { , . . . , k } by permutations by setting σ ( i ) = j whenever σ ( X ) ∈ g j for X ∈ g i . The orbits of this action on I can have 1, 2, or 4 elements. Thus,since we are dealing with indecomposable spaces, we have k = 1, 2, or 4.Assume k = 1. Then g = g is simple and we have Cases (2), (3), or (4),depending on whether the effectivity kernel of the action of σ on G has 1,2, or 4 elements.Now assume k = 2. Then g = g ⊕ g , where g , g are two isomorphicsimple ideals. Let α, β denote two generators of σ ∼ = Z [ α ] ⊕ Z [ β ]. Then atleast one of α, β maps g to g , say α ( g ) = g . Now the other generator β either leaves g invariant or maps g to g . In the first case define γ := β , inthe latter case define γ := α ◦ β . In either case, α and γ are two generatorsof σ and γ is an automorphism of g . By choosing the isomorphism betweenthe two simple summands appropriately, we may assume that γ | g = α ◦ γ | g ◦ α − . If now γ acts on g as the identity, then we are in Case (6);otherwise, γ acts as a nontrivial involutive automorphism on g and we arein Case (5).If we have k = 4, we are in the remaining Case (7). (cid:3) Tools from rational homotopy theory
Rational Homotopy Theory deals with the so-called rational homotopytype of nilpotent spaces. In this short note we shall not try to review thistheory, but we point the reader to [6] and [7] as a reference. The article buildson Sullivan models (Λ V, d) of nilpotent spaces. Recall the main theorem ofRational Homotopy Theory, which, up to duality, identifies the underlyinggraded vector space V with the rational homotopy groups π ≥ ( X ) ⊗ Q of X (with a special interpretation using the Malcev completion in degree 1).In particular, the construction of a Sullivan model of homogeneous spacesupon which we draw heavily can be found in [6, Chapter 15 (f)] or [7, Chapter3.4.2]. Recall that finite H -spaces have minimal Sullivan models which aregraded free exterior algebras generated by finitely many elements. Due to thelong exact sequence in homotopy applied to the universal fibration B G → E G → G of a compact Lie group G (together with the fact that V is dualto rational homotopy groups and a lacunary argument for the differential)we derive that H ∗ ( B G ; Q ) is actually a free polynomial algebra generated ineven degrees. A usually non-minimal Sullivan model for the homogeneousspace G/K of compact Lie groups with K connected (these spaces are knownto be simple whence accessible to rational homotopy theory, and they arethe only homogeneous spaces which we consider throughout this article) isconstructed as ( H ∗ ( B K ) ⊗ Λ V G , d)where (Λ V G , ∼ = H ∗ ( G ; Q ) and the differential d vanishes on H ∗ ( B K ).Thus the differential is induced (and extended as a derivation) by its be-haviour on a homogeneous basis ( v i ) of V G . One obtains thatd v = H ∗ ( B φ )( v (+1) ) SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 5 where φ : K ֒ → G is the inclusion and v (+1) denotes the suspension, a (+1)-degree shift of v .Throughout this article we shall construct the models of the homogeneousspaces as cited above and always use rational coefficients in cohomology.Let us finally recall the concept of formality here. A nilpotent spaceis formal if there is a chain/a zig-zag of quasi-isomorphisms, i.e. of mor-phisms of cochain algebras inducing isomorphisms on cohomology, betweenthe cochain algebra of polynomial differential forms A PL ( X ) and the co-homology H ∗ ( X ; Q ). As formality does not depend on the field extensionof Q we may replace A PL ( X ) by de Rham forms A DR ( M ) on a smoothmanifold. Another simplification may be formulated using minimal Sullivanmodels (Λ V, d) ∼ −→ (A DR ( M ). Then M is formal if and only if there is aquasi-isomorphism (Λ V, d) ∼ −→ H ∗ ( M ; R ).Note that a finite product is formal if and only if so are its factors—weshall actually only use the completely obvious implication that the formalityof the factors implies the formality of the product.It is well-known that a homogeneous space G/H (with G , H compact and H connected) is formal if and only if G/T H is formal. This can be easilyderived from [2, Theorem A] applied to the fibration H/T H ֒ → G/T H → G/H using the fact that
H/T H satisfies the Halperin conjecture, i.e. its cohomol-ogy (basically for degree reasons) has no non-trivial derivations of negativedegree. Corollary 2.1.
Let
H, H ′ ⊆ G . Suppose that the maximal torus T H ⊆ H isconjugate in G to the maximal torus T H ′ ⊆ H ′ . Then G/H is formal if andonly if
G/H ′ is formal. Proof . Conjugation induces a diffeomorphism
G/T H ∼ = G/T H ′ . (cid:3) Recall further the concept of a pure space ; that is a space admittingthe subsequent decomposition of its minimal Sullivan model: let (Λ V, d) besuch a model, then d | V even = 0 and d V odd ∈ Λ V even . (We may extend thisdefinition to non-minimal Sullivan models.) The model of a homogeneousspace constructed above clearly is of this form (and hence so is its minimalmodel). For such models formality implies a well-known strong structuralresult (for example see [1]): Proposition 2.2.
Let (Λ V, d) be a pure minimal Sullivan algebra of formaldimension d . Then up to isomorphism it is of the form (Λ V, d) ∼ = (Λ V ′ , d) ⊗ (Λ h x i i ≤ i ≤ k , with (Λ V ′ , d) elliptic and with k ≥ . In particular, it is formal if and onlyif (Λ V ′ , d) has positive Euler characteristic.In the case of a compact homogeneous space G/K the number k equalsthe corank rk G − rk K of K ⊆ G . (cid:3) We call a rationally elliptic space of positive Euler characteristic positivelyelliptic . MANUEL AMANN AND ANDREAS KOLLROSS Formality: Stepping through the cases
Reducing to the case of simple G . We deal with the different casesin Proposition 1.1. ad 1: If G is a torus and H a closed connected subgroup, then it is againa torus, and so will be G/H . ad 5: If G = L × L , where L is a simple compact Lie group and H = { ( g, g ) | g ∈ K } with K ⊂ L such that L/K is a symmetric space, thenwe can argue as follows: The resulting space is constructed via inclusions H = K ⊆ L ⊆ L × L = G , where the latter is a diagonal inclusion. It followsthat the Sullivan model of G/H is of the form(Λ h x , . . . , x l i , ⊗ ( A, d)where the first factor is a model of the antidiagonal { ( l, l − ) | l ∈ L } in L × L ,and ( A, d) is a model of L/K . Hence it is a product of formal algebras. ad 6: If G = L × L , where L is a simple compact Lie group and H = { ( g, g ) | g ∈ L } , then the quotient is just diffeomorphic to the antidiagonalwhich again is diffeomorphic to L . ad 7: If G = L × L × L × L and H = { ( g, g, g, g ) | g ∈ L } , where L is asimple compact Lie group, then G/H is diffeomorphic to L × L × L realisedas a complement of the diagonal in L × L × L × L .3.2. Coverings.
We remark that if
G/K is a homogeneous space of com-pact Lie groups G , K , and both G and K are connected, then G/K is asimple space (see [7, Proposition 1.62, p. 31]) and the techniques of rationalhomotopy theory apply. In particular, a Sullivan model of
G/K can be con-structed as described in Section 2. However, this implies that once G ′ /K ′ isa homogeneous space of connected Lie groups and a finite covering of G/K (both G , K connected), then the model we construct for it is identical tothe one of G/K . Indeed, for this we note that the inclusions of the maxi-mal tori T H ֒ → T G respectively T H ′ ֒ → T G ′ which completely determine therespective models (together with the given Weyl groups, which remain thesame under finite coverings) are uniquely determined by local data, namely,by the inclusions of the corresponding Lie algebras t H = t H ′ ֒ → t G = t G ′ dueto the commutativity and surjectivity properties of the exponential map oncompact Lie groups.Hence, the spaces G/H and G ′ /H ′ which may differ by a finite coveringare rationally not distinct as long as all groups are connected; hence, in thefollowing, we may pick a representative out of the corresponding equivalenceclass.3.3. Homogeneous spaces of equal rank groups.
These spaces are so-called positively elliptic spaces with cohomology concentrated in even de-grees given as the quotient of a polynomial algebra by a regular sequence(see [6, Proposition 32.2, p. 436]. In particular, they are formal.The following ( Z ⊕ Z )-spaces fall in this category: • SU ( a + b + c ) / S ( U ( a ) U ( b ) U ( c )); • SU ( a + b + c + d ) / S ( U ( a ) U ( b ) U ( c ) U ( d )); • SO ( a + b + c ) / SO ( a ) SO ( b ) SO ( c ), where at most one of a, b, c is odd; SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 7 • SO ( a + b + c + d ) / SO ( a ) SO ( b ) SO ( c ) SO ( d ), where at most oneof a, b, c, d is odd; • Sp ( a + b + c ) / Sp ( a ) Sp ( b ) Sp ( c ); • SO (2 a + 2 b ) / U ( a ) U ( b ); • Sp ( a + b + c + d ) / Sp ( a ) Sp ( b ) Sp ( c ) Sp ( d ); • E / U (3) U (3); • E / Spin (6)
Spin (4) U (1); • E / U (5) U (1); • E / Spin (8) SO (2) U (1); • all ( Z ⊕ Z )-spaces of E , E , F and G except for E / F , E / Sp (4), E / Spin (8).3.4.
Spaces sharing the maximal torus with a symmetric space.
Asa consequence of Corollary 2.1 we note that any space
G/H such that H shares its maximal torus with H ′ such that G/H ′ is a symmetric space isformal.As for ( Z ⊕ Z )-symmetric spaces we apply this to the following: • SU (2 k ) / U ( k ), where U ( k ) ⊂ Sp ( k ) ⊂ SU (2 k ); • SU (2 a + 2 b ) / Sp ( a ) Sp ( b ), where Sp ( a ) Sp ( b ) ⊂ S ( U (2 a ) U (2 b )); • SO ( a + b + c ) / SO ( a ) SO ( b ) SO ( c ), where a, b are odd and c is even,we have SO ( a ) SO ( b ) SO ( c ) ⊂ SO ( a ) SO ( b + c ) ⊂ SO ( a + b + c ); • SO ( a + b + c + d ) / SO ( a ) SO ( b ) SO ( c ) SO ( d ), where a, b are odd and c, d are even, we have SO ( a ) SO ( b ) SO ( c ) SO ( d ) ⊂ SO ( a ) SO ( b + c + d ) ⊂ SO ( a + b + c + d ); • Sp ( a + b ) / U ( a ) U ( b ), here we have U ( a ) U ( b ) ⊂ U ( a + b ) ⊂ Sp ( a + b ); • E / U (4), where U (4) ⊂ Sp (4) ⊂ E ; • E / Sp (2) Sp (2), where Sp (2) Sp (2) ⊂ Sp (4) ⊂ E ; • E / Sp (3) Sp (1), where Sp (3) Sp (1) ⊂ Sp (4) ⊂ E .3.5. The remaining spaces.
The remaining ( Z ⊕ Z )-spaces not coveredby the above two cases are the following: • SU (2 n ) / U ( n ) • SU ( n + m ) / SO ( n ) SO ( m ) • SO ( a + b + c ) / SO ( a ) SO ( b ) SO ( c ), where all three of a, b, c are odd • SO ( a + b + c + d ) / SO ( a ) SO ( b ) SO ( c ) SO ( d ), where three or fourof a, b, c, d are odd • SO (2 n ) / SO ( n ) • SO (4 n ) / Sp ( n ) • Sp (2 n ) / Sp ( n ) • Sp ( n ) / SO ( n ) • E / Spin (9) • E / Sp (4) • E / SO (8) • E / F In the following we shall deal with these cases.
MANUEL AMANN AND ANDREAS KOLLROSS
Diagonal double block inclusions. Case 1.
Let us consider thespace Sp (2 n ) / Sp ( n ). We use the convention sp ( n ) = (cid:26)(cid:18) A B − ¯ B ¯ A (cid:19) | A, B ∈ C n × n , A = − A ∗ , B = B t (cid:27) . Now using the inclusions sp ( n ) ⊂ sp ( n ) ⊕ sp ( n ) ⊂ sp (2 n ), we obtain thesubalgebra A B A B − ¯ B A − ¯ B A | A, B ∈ C n × n , A = − A ∗ , B = B t ⊂ sp (2 n ) . Hence the space Sp (2 n ) / Sp ( n ) is given by the two-block diagonal inclu-sion of Sp ( n ) into Sp (2 n ).Each standard inclusion Sp ( n ) ֒ → Sp (2 n ) is rationally 2 n -connected. Asin the following cases, this can be verified by direct computation or by citing[16, Chapter 6]. For the convenience of the reader we sketch the computa-tion. We prove the analog statement for the induced map between classifyingspaces. For this we identify cohomology generators with polynomials in thecohomology generators t i of the classifying space of the maximal torus in-variant under the action of the Weyl group. It is then easy to see thatcorresponding polynomials are mapped to each other.This implies that the induced map on rational homotopy groups of theconsidered inclusion Sp ( n ) ֒ → Sp (2 n ) is multiplication by two in degrees1 ≤ i ≤ n − n + 1 ≤ i ≤ n −
1, as the rational homotopy groups of Sp ( n ) are concentrated belowdegree 4 n + 1. This implies that π ∗ ( Sp (2 n ) / Sp ( n )) ⊗ Q is concentrated inodd degrees. More precisely, considering the model(Λ( V BSp ( n ) ⊕ V Sp (2 n ) ) , d)(cf. [6, Proposition 15.16, p. 219]) of this homogeneous space where(Λ V BSp ( n ) ,
0) and (Λ V Sp (2 n ) ,
0) are minimal models for
BSp ( n ) and Sp (2 n )respectively. We deduce that(Λ( V BSp ( n ) ⊕ V Sp (2 n ) ) , d) ≃ (Λ h v n +1 , . . . , v n − i , v i of degree i corresponding to the rational homotopy groupsof Sp (2 n ) in degree i .This can be seen as follows: The linear part d of the differential d cor-responds to the transgression in the dual long exact homotopy sequence ofthe fibration Sp (2 n ) ֒ → Sp (2 n ) / Sp ( n ) → BSp ( n ). The generators of theunderlying vector spaces of the minimal models identify with the rationalhomotopy groups up to duality. It follows that the vector spaces V BSp ( n ) and V Sp (2 n ) are graded isomorphic via d (raising degree by +1) in degreesbelow 4 n + 1. In other words, up to a contractible algebra a Sullivan modelof the homogeneous space is concentrated in degrees at least 4 n + 1, i.e. sois its minimal Sullivan model. Moreover, the differential on the remaininggenerators becomes 0. Consequently, Sp (2 n ) / Sp ( n ) is formal. SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 9 Case 2.
For the case SU (2 n ) / U ( n ) we recall its construction as a ( Z ⊕ Z )-symmetric space: We have the group diagram SU (2 n ) Sp ( n ) ' (cid:7) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ S ( U ( n ) × U ( n )) k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ U ( n ) = Sp ( n ) ∩ S ( U ( n ) × U ( n )) j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ & (cid:6) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (We remark that this corrects a putative error in [3].) Due to this diagram wesee that the map H ∗ ( BSU (2 n )) → H ∗ ( BU ( n )) factors over H ∗ ( BSp ( n )).In particular, we directly see that, since H ∗ ( BSp ( n )) is concentrated indegrees divisible by 4, generators of H ∗ ( BSU ( n )) of degree congruent to2 modulo 4 map to 0. Hence at most n generators of H ∗ ( BSU (2 n )) canhave a non-trivial image. Since this number equals the rank of U ( n ), thesegenerators indeed must map to a regular sequence in H ∗ ( BU ( n )), since thecohomology of the homogeneous space is finite dimensional. It follows that H ∗ ( SU (2 n ) / U ( n )) splits as the product of a positively elliptic algebra anda free algebra in n generators of odd degree whence the space is formal. Case 3.
Basically the same arguments as in the first case apply to SO (2 n ) / SO ( n ) with the double block standard inclusion. Here, however,we have to differ two cases.First, we assume that n is odd. The cohomology of BSO (2 n ) is generatedby elementary symmetric polynomials in the t i (except for the top degreeone) where the t i generate H ∗ ( B T ) with T ⊆ SO (2 n ) the maximal torus,together with the polynomial t · t · . . . · t n in degree 2 n − t · . . . · t n ). Using the analog description for BSO ( n ) we derive thatthe standard inclusion SO ( n ) ֒ → SO (2 n ) injects all the rational homotopygroups of SO ( n ). They lie in degrees 3 , , . . . , n − SO (2 n ). Interpretingthese results for the double blockwise inclusion, we derive that in these lowdegrees the induced map on rational homotopy groups is just multiplicationwith 2 again. Thus, by taking the quotient with a contractible algebra, weobtain that(Λ( V BSO ( n ) ⊕ V SO (2 n ) ) , d) ≃ (Λ h v ′ n − , v n +1 , v n +5 , . . . , v n − i , n is even. With the analogous arguments wederive that all the rational homotopy groups of SO ( n ) which are repre-sented by symmetric polynomials in the t i are mapped bijectively to thecorresponding ones in SO (2 n ). There remains the class x n represented by t · . . . · t n/ and the minimal model looks like(Λ( V BSO ( n ) ⊕ V SO (2 n ) ) , d) ≃ (Λ h x n , v ′ n − , v n +1 , v n +5 , . . . , v n − i , d) with d( v ′ n − ) = x n and d vanishing on all other generators. (The firstproperty is due to the fact that t · . . . · t n maps to t · . . . · t n/ under themorphism induced on the cohomologies of classifying spaces by diagonalinclusion.) Thus, the space rationally is the product of odd-dimensionalspheres with exactly one even-dimensional sphere and again formal. Case 4.
We consider SO (4 n ) / Sp ( n ). The morphism induced in the co-homology of classifying spaces H ∗ ( BSO (4 n )) → H ∗ ( BSp ( n )) is induced onformal roots by t i ˜ t i/ , t i +1
7→ − ˜ t i/ for i ≡ H ∗ ( B T SO (4 n ) ) = Q [ t , . . . , t n ], H ∗ ( B T Sp ( n ) ) = Q [˜ t , . . . , ˜ t n ]. In particular, this means thatthe k -th elementary symmetric polynomial in the t i is mapped to a non-trivial multiple of the k -th elementary polynomial in the ˜ t i up to a perturba-tion by another symmetric polynomial generated by the first k − n non-trivial rational homotopy groups. In other words, it followsthat the map induced in the cohomology of classifying spaces is surjective,and that H ∗ ( SO (4 n ) / Sp ( n )) is an exterior algebra, which is intrinsicallyformal. Case 5.
Let us now deal with Sp ( n ) / SO ( n ) in a similar manner. Theinclusion of SO ( n ) ⊆ Sp ( n ) is induced by the componentwise inclusion R → H . That is, on the torus we have the induced morphism t t , t t , t t , . . . , t n t ⌊ n/ ⌋ and t i
0. This implies that elementarysymmetric polynomials in the t i map to elementary symmetric polynomialsin the t i although we lose several summands:Let us first assume that n is odd. Then these considerations imply thatthe inclusion SO ( n ) ⊆ Sp ( n ) is rationally 4(( n − / n is even, then the same arguments imply that the quotient is just aproduct of an even-dimensional sphere (with volume form corresponding tothe product t · . . . · t n/ —the t i from H ∗ ( BSO ( n ))—) and a free algebra.In any case the space is formal.3.7. The space SU ( n + m ) / SO ( n ) × SO ( m ) . For this case we may assumethat both n and m are odd. Indeed, otherwise the group SO ( n ) × SO ( m )shares its maximal torus with SO ( n + m ). The space SU ( n + m ) / SO ( n + m )is symmetric. Since formality only depends on the inclusion of the maximaltorus, SU ( n + m ) / SO ( n ) × SO ( m ) is also formal.Thus suppose that n, m are odd, i.e. we consider SU (2 n +2 m +2) / SO (2 n +1) × SO (2 m + 1). We compute the Sullivan model of the space. (We mayassume that the maximal torus of the first denominator factor is embed-ded into complex coordinates 1 to 2 n , the one of the second factor intocoordinates 2 n + 3 to 2 n + 2 m + 2.) We obtain the Sullivan model(Λ( V BSO (2 n +1) ⊕ V BSO (2 m +1) ⊕ V SU (2 n +2 m +2) ) , d) SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 11 with the differential induced by t t , t
7→ − t , t t , t
7→ − t , . . . ,t n − t n − , t n t n − ,t n +1 , t n +2 ,t n +3 t n +3 , t n +4
7→ − t n +3 , . . . ,t n +2 m +1 t n +2 m +1 , t n +2 m +2
7→ − t n +2 m +1 where t , t , t , . . . , t n − generates the cohomology of the torus of SO (2 n +1) and t n +3 , t n +5 , . . . , t n +2 m +1 generates the cohomology of the torus of SO (2 m + 1).Since the rational homotopy groups of BSO (2 n + 1) respectively the onesof BSO (2 m + 1) are concentrated in degrees divisible by four, so is theirrational cohomology, and we derive that d v = 0 for v ∈ V SU (2 n +2 m +2) ofdegree congruent to 1 modulo 4. It follows thatd v = d v = d v = . . . = d v n +2 m +2) − = 0The differential of the generator in top degree 2(2 n + 2 m + 2) − t t · . . . · t n +2 m +2
7→ ± t t t · . . . · t n − · · · t n +3 t n +5 · . . . · t n +2 m +1 Consequently, also its differential vanishes. Thus the model splits as(Λ( V BSO (2 n +1) ⊕ V BSO (2 m +1) ⊕ V SU (2 n +2 m +2) ) , d) ≃ (Λ( V BSO (2 n +1) ⊕ V BSO (2 m +1) ⊕ h v , v , v , . . . , v n +2 m +2) − i ) , d) ⊗ (Λ h v n +2 m +2) − , v n +2 m +2) − i , SU (2 n +2 m ) / ( Sp ( n ) × Sp ( m )). The latter space shares its maximal torus with thesymmetric space SU (2 n + 2 m ) / Sp ( n + m ); thus it is formal.Alternatively, due to the observation above this model is of the form(Λ( V BSO (2 n +1) ⊕ V BSO (2 m +1) ⊕ V SU (2 n +2 m +2) ) , d) ≃ (Λ( V BSO (2 n +1) ⊕ V BSO (2 m +1) ⊕ h v , v , v , . . . , v n +2 m +2) − i ) , d) ⊗ (Λ v , v , . . . , h v n +2 m +2) − , v n +2 m +2) − i , SU (2 n + 2 m + 2) / SO (2 n + 1) × SO (2 m + 1) is the product of two formal Sullivan algebras and formal,consequently.3.8. The spaces SO ( a + b + c ) / SO ( a ) SO ( b ) SO ( c ) and SO ( a + b + c + d ) / SO ( a ) SO ( b ) SO ( c ) SO ( d ) . As remarked above, we may assume for thefirst space that all three of the a, b, c are odd; for the second space three orfour of the a, b, c, d are odd. (Otherwise we would deal with an equal rankpair or a space sharing its torus with a symmetric pair.)Let us first deal with the first space. The inclusion of the stabiliser groupis blockwise. The stabiliser group has corank 1. Combining this informa-tion a similar approach as in the last section actually yields, that the topelementary symmetric polynomial in the t i from SO ( a + b + c ), namely t · . . . · t a + b + c − / , maps to zero under the map induced on classifyingspaces by the inclusion of the stabiliser. Consequently, the remaining coho-mology generators of the numerator need to map to a regular sequence in H ∗ ( B ( SO ( a ) SO ( b ) SO ( c ), since cohomology is finite-dimensional. It followsthat the minimal model for the homogeneous space splits as this positively el-liptic space times the free algebra generated by the generator of SO ( a + b + c )corresponding to t · . . . · t a + b + c − / up to degree shift.Let us now consider the second space and first assume that all of a, b, c, d are odd. Then the stabiliser has corank two and, similar to the last case,now the top two rational homotopy groups of SO ( a + b + c + d ) map to zero.Thus the homogeneous space rationally is the product of a positively ellipticspace and the free algebra generated by two elements. In particular, again,it is formal.Suppose now that a, b, c are odd, d is even. Then the corank is 1, the toprational homotopy group restricts to zero; again we have a similar splittingand formality.3.9. The spaces E / F , E / Sp (4) , E / SO (8) . In order to prove the for-mality of E / F we shall merely use two pieces of information.(1) the rational homotopy groups of E and F ,(2) the fact that the inclusion of simple simply-connected compact Liegroups induces an isomorphism in third rational cohomology, i.e. isrationally 4-connected.Let us elaborate on this briefly.The rational homotopy groups of E are concentrated in degrees3 , , , , , , F we have one-dimensional rationalhomotopy groups in degrees 3 , , , E / F . It isgiven by (Λ( V BF ⊕ V E ) , d)and the differential vanishes on the first summand. Write V BF = h v , v , v , v i The cohomology is finite-dimensional. This implies that there is an elementin V E mapping under the differential to v k + . . . , one (not necessarily adifferent one) to v k + . . . , one to v k + . . . , one to v k + . . . for k , k , k , k ∈ N .Comparing degrees with the generators of V E , i.e. the rational homotopygroups of E , we obtain that k = k = 1, k ∈ { , } , k ∈ { , , , , } . SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 13 Since the inclusion of the denominator group induces an isomorphism onthird cohomology, we can assume that k = 1.It follows that E / F has the rational homotopy type of either a prod-uct of three odd-dimensional spheres (dimensions 19 , ,
35) or of threeodd-dimensional spheres (dimensions 11 , ,
27) with a factor of the form(Λ h v, v ′ i , d) with d( v ) = 0, d v ′ = v , deg v = 12 , deg v ′ = 35. In any casethe manifold is formal.As for the formality of E / Sp (4) and E / SO (8), it suffices to observe thatboth subgroups Sp (4) and SO (8) share one of their respective maximal toriwith the subgroup F ⊂ E considered above. To see this for Sp (4), weobserve that both subgroups Sp (4) and F are contained in E ⊂ E andarise as the (connected component) of the fixed point set of an automorphismof E . It follows from [5, Chapitre II] that two maximal tori of Sp (4) and F ,respectively, are conjugate by an element of E (see Corollary 2.1). Thesame argument may be applied to the subgroups Sp (4) and SO (8), whichare both contained in SU (8) ⊂ E .3.10. The spaces E / Spin (9) , E / Sp (2) Sp (2) . It again remains to ob-serve that
Spin (9) and Sp (2) Sp (2) share a common maximal torus (up tocovering). For this we may write E / Sp (2) Sp (2) = E / Spin (5)
Spin (5)due to due to the exceptional Lie group isomorphism Sp (2) ∼ = Spin (5). Weobserve that there are a symmetric space E / Sp (4) and inclusions( U (4) ⊆ ) Sp (4) ⊇ Sp (2) × Sp (2)All of these groups have rank 4 whence they share a common maximal torusin E .As for sharing the maximal torus with Spin (9) we draw on the sym-metric space E / Spin (10) U (1). The group Sp (2) Sp (2) in the form of Spin (5)
Spin (5) is contained in this
Spin (10)-subgroup and shares a maxi-mal torus with
Spin (9) in E . (This follows easily from the block inclusionsof SO (5) SO (5) ⊆ SO (10) ⊆ SO (9).)4. A criterion for equivariant formality
We present a characterisation of the equivariant cohomology of the isotropyaction which builds on formality of
G/K and provides an additional condi-tion to check. Note that π ∗ ( B G ) = Hom( π ∗ ( B G ) , Q ) denote dual rationalhomotopy groups. The first inclusion expresses the fact that rational coho-mology of B G is a polynomial algebra generated by spherical cohomologyclasses, i.e. by dual homotopy groups. Theorem 4.1.
Let K y G/K be the isotropy action such that
G/K isformal. Suppose further that dim ker (cid:0) π ∗ ( B G ) ֒ → H ∗ ( B G ) → H ∗ ( B K ) (cid:1) = rk G − rk K Then the isotropy action is equivariantly formal.
Proof . Under the given assumptions we show that the induced map onrational cohomology of the fibre inclusion of the fibration
G/K ֒ → E K × K G/K → B K is surjective. We form a model of the total space as a relative model of thisfibration, i.e. actually as a biquotient model (cf. [12], [7, Chapter 3.4.2]).This yields the Sullivan algebra( H ∗ ( B K ) ⊗ H ∗ ( B K ) ⊗ H ∗ ( G ) , d)encoding the rational homotopy type of the Borel construction, with itscohomology being equivariant cohomology. The differential is given byd | H ∗ ( B K ) ⊗ H ∗ ( B K ) = 0, d( x ) = ( H ∗ ( B φ )( x (+1) ) , − H ∗ ( B φ )( x (+1) )) and ex-tended as a derivation where x is a spherical cohomology class, i.e. an al-gebra generator of H ∗ ( B G ), and φ : K → G denotes the inclusion. Here, x (+1) is the suspension, i.e. a degree shift by +1.Clearly, the algebra generators of H ∗ ( B G ) correspond to its rational ho-motopy groups. That is, the condition from the assertion guarantees thatalso in the model of the Borel construction the same (rk G − rk K )-manygenerators of H ∗ ( B G ) map to zero.Now by construction of the model of the total space as a (minimal) relativeSullivan model the fibre inclusion is modelled by the projection(Λ( H ∗ ( B K ) ⊗ H ∗ ( B K ) ⊗ H ∗ ( G )) , d) → (Λ( H ∗ ( B K ) ⊗ H ∗ ( B K ) ⊗ H ∗ ( G )) , d) ⊗ H ∗ ( B K ) Q (where we divide by the base H ∗ ( B K ) encoding the action).Recall the classical observation that, by construction, both the Sullivanmodel of the Borel construction and the model of G/K are pure. Moreover,if such a space is formal, then it is necessarily isomorphic to the product ofa rationally elliptic space of positive Euler characteristic and a free algebragenerated in odd degrees (see Proposition 2.2).Hence, since
G/K is formal, the cohomology of this positively ellipticfactor is generated by its H ∗ ( B K ). Similarly, we observe that we havea subalgebra of equivariant cohomology generated by H ∗ ( B K ) ⊗ H ∗ ( B K )with the second factor mapping onto its analog H ∗ ( B K ) on the fibre modelof G/K . Hence, the fibre projection surjects onto the positively elliptic co-homology. It remains to see that it also surjects onto the free part generatedin odd degrees.By construction this factor for the model of the fibre is generated by(rk G − rk K )-many odd-degree elements. The crucial observation whichwill finish the proof is that the isomorphism which yielded the productsplitting is actually the identity , since (rk G − rk K )-many rational homotopygroups/generators of the model map to zero under the differential which ona generator is just H ∗ ( B φ ) up to degree shift by +1. Consequently, also onthe model of the Borel construction the exactly same (rk G − rk K )-manyodd degree generators map to zero under the differential as well. Since themodels are pure, it follows that the projection, which again is the identityon H ∗ ( G ) is surjective on the cohomology generated by these odd-degreeelements.Summarizing, we have seen the following: Since one H ∗ ( B K ) from theBorel construction surjects onto the H ∗ ( B K ) of the fibre (via the identity),it follows that the fibre projection is surjective on the cohomology generatedby even degree elements. It is also surjective on the cohomology generated SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 15 by odd-degree elements (and hence surjective in total), since( H ∗ ( B K ) ⊗ H ∗ ( B K ) ⊗ H ∗ ( G ) , d)splits as a product of an algebra with cohomology generated by H ∗ ( B K ) ⊗ H ∗ ( B K ) and a free algebra generated by exactly the same (rk G − rk K )-many generators as we find in the analog splitting on( H ∗ ( B K ) ⊗ H ∗ ( B K ) ⊗ H ∗ ( G ) , d) ⊗ H ∗ ( B K ) Q .Consequently, in total the fibre projection induces a surjective morphismon cohomology. Hence the isotropy action is equivariantly formal. (cid:3) Let us refine this criterion a little more in the form of a corrollary tothe actual proof of it. The content of it is indebted to the fact that thecondition of the theorem will often be hard to check and maybe only holdup to, for example, taking the quotient with a contractible algebra, sincethe models we construct are usually highly non-minimal. The differentialof the model of
G/K is induced by φ : H ∗ ( B G ) → H ∗ ( B K ). The map inTheorem 4.1 hence controls this differential on generators. It hence requiresthat the dimension of the cohomology spanned by elements from V equalsthe rank difference. We refine this condition.We denote by ( H ∗ G ⊗ H ∗ B K, d) the standard model of G/K (which isusually not minimal) with ( H ∗ G,
0) = (Λ V,
0) a minimal model of G . Weidentify V with dual rational homotopy groups of G . Hence the morphismfrom Theorem 4.1 transcribes to ψ = { ψ i } i : (cid:0) V (+1) → H ∗ ( B K ) (cid:1) where ψ i is the morphism in degree i , i.e. from V i +1 → H i ( B K ).Theorem 4.1 transcribes to and is slightly extended by Corollary 4.2.
Let K y G/K be the isotropy action such that
G/K isformal. Suppose further that there exists a homogeneous basis ( v j ) of V (+1) such that v , . . . , v rk G − rk K satisfy that ψ ( v j ) ∈ H ∗ ( B K ) · Λ h ψ ( v s ) i s In the following corollary we apply Corollary 4.3 in the case when in themodel of G/K the subalgebra H ∗ ( B K ) is contractible. Hence rk G − rk H many generators of V (due to the pureness of the model) necessarily mapinto this contractible algebra. Extending these generators by a basis of H ∗ ( B K ) brings us in the situation to apply the previous corollary. Corollary 4.3. If H ∗ ( G/K ) is a free algebra, then the isotropy action isequivariantly formal. Proof . Such a space is intrinsically formal. Its homotopy Euler character-istic on the one hand equals the number of generators of the free algebra,on the other hand χ π ( G/H ) = rk G − rk H . The rest follows from Corollary4.2. (cid:3) Our subsequent discussion of equivariant formality will draw on the crite-rion as presented in the corollary together with the next simple observation.The following proposition is also well-known (see [10, Proposition C.26])and follows basically from the general form of the following commutativediagram (which we already specialise to our concrete case) with horizontaland vertical fibrations. ∗ (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) K/T / / (cid:127) _ (cid:15) (cid:15) K/T (cid:127) _ (cid:15) (cid:15) G/K (cid:31) (cid:127) / / (cid:15) (cid:15) E T × T G/K / / (cid:15) (cid:15) B T (cid:15) (cid:15) G/K (cid:31) (cid:127) / / E K × K G/K / / B K Proposition 4.4. If a compact connected Lie group G acts on a compactmanifold M , then its action is equivariantly formal if and only if so is theinduced action of its maximal torus T . (cid:3) Again, we obtain an analog of Corollary 2.1—see also [4, Theorem 2.3.2,p. 4]. Corollary 4.5. Suppose that the maximal tori T K ⊆ K and T K ′ ⊆ K ′ areconjugated in G . Then the isotropy action K y G/K is equivariantly formalif and only if the isotropy action K ′ y G/K ′ is equivariantly formal. Proof . We first observe that K y G/K is equivariantly formal if and onlyif so is T K y G/K . Now we use the symmetry between left hand and righthand actions to deduce that the right action of K on the homogeneous space T K \ G is equivariantly formal if and only if so is the right action T K y T K \ G .Since T K and T K ′ are conjugate, this action is equivalent to T K ′ y T K ′ \ G which is equivalent to the ordinary isotropy action T K ′ y G/T K ′ . This oneis equivariantly formal if and only if so is K ′ y G/K ′ . (cid:3) Equivariant formality of Z ⊕ Z -symmetric spaces Considering the Cases 1, 5, 6, 7 from Proposition 1.1 which we alreadytook into account for formality we reduce to the case of a simple numeratorgroup G . For this it only remains to observe that in all these 4 cases theSullivan model was of the form (Λ h x , . . . , x l i , ⊗ ( A, d) (with A possiblytrivial).If A is trivial, then the cohomology of this algebra differs from its equi-variant cohomology only by another free factor corresponding to H ∗ ( B K ) SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 17 (from the left K -action), and the fibre projection in the Borel fibration issurjective.In the case where A = Q , the algebra ( A, d) is a model of a symmetricspace, and we draw on the equivariant formality of the isotropy action onthere.Due to Proposition 4.4, the fact that equal rank homogeneous spaceshave equivariantly formal isotropy actions (as the E -page of the Leray–Serre spectral sequence associated to the Borel fibration is concentrated ineven bidegrees only; respectively, even more generally, since homogeneousspaces satisfy the Halperin conjecture), and the result that so do symmetricspaces (see [8]) it suffices in view of Corollary 4.5 to focus on the verysame remaining cases from Subsection 3.5 as we did for formality. Havingproved the formality of ( Z ⊕ Z )-symmetric spaces in view of Theorem 4.1it remains to check the induced map on cohomology algebra generators π ∗ ( B G ) ⊗ Q ֒ → H ∗ ( B G ) → H ∗ ( B K )and the dimension of its kernel. Moreover, we may again restrict to onlydealing with one representative out of those spaces with identical maximaltorus of the stabiliser groups.However, we may recur to Subsections 3.6, 3.7, 3.8, 3.9, 3.10 where wecarefully already did this analysis. Let us quickly recall the crucial observa-tions from there: • The spaces Sp (2 n ) / Sp ( n ), SO (2 n ) / SO ( n ), SO (4 n ) / Sp ( n ), Sp (2 m ) / SO ( m ) all came from block inclusions. In particular, weobserved that in nearly all cases these spaces had a free cohomologyalgebra (generated in odd degrees). Hence, due to Corollary 4.3 theisotropy action is equivariantly formal.It only remains to consider the only two exceptions from this,namely, the spaces SO (2 n ) / SO ( n ) and Sp ( n ) / SO ( n ), both with n even. In these cases the cohomology of G/K splits as the product ofthe cohomology of an even-dimensional sphere and odd-dimensionalspheres. Moreover, in the respective models there is exactly one gen-erator (namely the n -th elementary symmetric polynomial in the t i , t · . . . · t n , in the case of SO (2 n ) / SO ( n ) respectively the n/ t i in the case of Sp ( n ) / SO ( n ))mapping into the ideal (and then actually the subalgebra) generatedby the “volume form”, t · . . . · t n/ of the even-dimensional sphere in H ∗ ( G/K ). Hence all the rk G − rk K − H ∗ ( G/K )which generate the free factor actually have vanishing differentialsup to a contractible algebra. Hence, applying Corollary 4.2, we seethat the isotropy action is equivariantly formal in this case as well. • As for the space SU (2 n ) / U ( n ) we observed that it split as the prod-uct of a positively elliptic algebra an a free algebra in n generatorsof odd degree. We further observed that these n elements come fromthe generators of H ∗ ( BSU (2 n )) of degree congruent to 2 modulo 4.Their image in H ∗ ( BU ( n )) is trivial. Hence, due to Theorem 4.1 we may lift them to closed forms on the Borel construction, and theisotropy action is equivariantly formal. • In the case of the space SU ( n + m ) / SO ( n ) × SO ( m ) we observedthat its minimal model splits as a positively elliptic factor times afree algebra generated by the elements v , v , v , . . . , v n +2 m +2) − , v n +2 m +2) − , v n +2 m +2) − Thus it remains to observe that the we proved that the differentialon all these elements vanished. Then we apply Corollary 4.2. • The spaces SO ( a + b + c ) / SO ( a ) SO ( b ) SO ( c ) and SO ( a + b + c + d ) / SO ( a ) SO ( b ) SO ( c ) SO ( d ) result from standard block inclusion.The free factors of their cohomology algebras are generated by onerespectively by two respectively by one elements (depending on theparities of the a, b, c ). We recall that their differentials vanish andapply Corollary 4.2. • The space E / F . The cohomology of E is generated in degrees3 , , , , , , 35, the one of F in degrees 3 , , , 23. Extend-ing the discussion of this case from above, we observe that either thegenerators in degree 19, 27 and 35 map into the ideal generated bya contractible algebra (generated by those in degree 3 and 4, 11 and12, 15 and 16, 23 and 24), or so do the ones in 11, 19 and 27 (withthe contractible algebra given by those in degrees 3 and 4, 15 and16, 23 and 24—we can guarantee that the generator in degree 23maps to the one in degree 24, without additional summands, afterapplying an isomorphism of the model). Hence Corollary 4.3 yieldsthe result in the first case, or we argue as in the second case, namelyadapting the argument from the first item: The free factor of thecohomology of E / F is generated by 3 elements which also havetrivial differentials considered as elements in the model of the Borelconstruction. Hence the fibre projection is surjective, and the actionis equivariantly formal. • The space E / Spin (9). The cohomology of SO (9) is generated indegrees 3 , , , 15, the ones of E in degrees 3 , , , , , 23. Weobserved that the minimal model is generated in degrees 8 , , , h w, y, z, x i , d) with deg w = 8, deg y =9, deg z = 17, deg x = 23, d w = d y = d z = 0 and d x = w . As inthe last item we may hence argue that the free factor is generatedin degrees 9 and 17, and the corresponding generators x and y againalso, up to isomorphism, have vanishing differential considered aselements of the model of the Borel construction. Hence the fibreprojection is surjective in cohomology and the action equivariantlyformal.6. A short alternative proof of the equivariant formality ofthe isotropy action on symmetric spaces We provide a short argument for the equivariant formality of symmetricspaces building upon their classification together with the classical fact thatthey are (geometrically) formal. SOTROPY FORMALITY OF Z ⊕ Z -SYMMETRIC SPACES 19 It is a short and classical argument to prove that compact Lie groups,considered as symmetric spaces ( G × G ) / ∆ G (with the diagonal inclusion∆ G ) have equivariantly formal isotropy actions. In this case, the quotient G ∼ = ( G × G ) / ∆ G is obviously formal, and, rationally, the fibre projectionin the Borel fibration the Borel construction is just( H ∗ ( B G ) ⊗ H ∗ ( B G ) ⊗ H ∗ ( G ) ⊗ H ∗ ( G ) , d) → H ∗ ( G )The differential is such that the cohomology of the anti-diagonal { ( g, − g ) | g ∈ G } in G × G projects surjectively onto H ∗ ( G ).Since equivariant formality behaves well with product actions, and sincehomogeneous spaces G/K with rk G = rk K have equivariantly formal isotropyactions, it remains to prove this for those symmetric spaces with rk K < rk G . These then come out of a short list: • The spaces SU ( n ) / SO ( n ). Let us make a distinction by the parityof n . Suppose first that n is odd. The algebra H ∗ ( SU ( n ) is con-centrated in even degrees 4 , , . . . , n whereas H ∗ ( BSO ( n )) ( n odd)is concentrated in degree divisible by four, namely 4 , , . . . , n − 2. Hence, merely for degree reasons exactly half, i.e. ( n − / H ∗ ( BSU ( n )), namely the ones in de-grees 6 , , . . . , n − , n map to zero under the morphism inducedby the group inclusion. Since the corank of SO ( n ) in SU ( n ) is ex-actly ( n − − ( n − / n − / n − / n is even. We now repeat the argument from n odd observing that now H ∗ ( BSO ( n )) is generated in degrees4 , , . . . , n − n . If n isdivisible by four, exactly the same arguments as above apply: Fordegree reasons the generators in degrees 6 , , . . . , n − n/ − n − − n/ SU ( n ) − rk SO ( n )many. If n ≡ n of H ∗ ( BSO ( n )), namely the one correspondingto t · . . . · t n/ , is hit by the generator of H ∗ ( BSU ( n )), namely theelement corresponding to the n/ t · . . . · t n/ + . . . + t n/ · . . . · t n , in the formal roots, i.e. in thealgebra generators of the cohomology of the classifying space of itsmaximal torus. (Recall that t t , t 7→ − t , t t , t 7→ − t ,etc.) Consequently, up to the contractible algebra formed by thesetwo, all other generators of H ∗ ( BSU ( n )) of degree congruent to 2modulo 4 map to zero. These are n/ − H ∗ ( BSU ( n )) corresponding to t · . . . · t n maps to − t t · . . . · t n/ , which, however, is minus the square of t · . . . · t n/ (which already lies in the image of the map). Hence, in view of Corol-lary 4.2, we have found n/ − rk SU ( n ) − rk SO ( n ) many odd degree generators which lift to closed elements on the Borelconstruction. Hence the isotropy action is equivariantly formal. • The spaces SU (2 n ) / Sp ( n ). The argument is basically the same asfor n odd in the previous case: The algebra H ∗ ( SU (2 n ) is con-centrated in even degrees 4 , , . . . , n whereas H ∗ ( BSp ( n )) is con-centrated in degree divisible by four, namely 4 , , . . . , n . Hence,again merely for degree reasons n − H ∗ ( BSU ( n )), namely the ones in degrees 6 , , . . . , n − Sp ( n ) in SU (2 n ) is exactly n − n − • The spaces SO ( p + q ) / SO ( p ) × SO ( q ) with both p, q odd. Thestabiliser inclusion is blockwise. The subgroup has corank 1. Henceit suffices to observe that the ( p + q ) / t i , the formal roots of H ∗ ( BSO ( p + q )) maps tozero under the map induced by the inclusion on the cohomology ofclassifying spaces—indeed, the cohomology of the classifying space ofthe denominator group is generated by polynomials in the t i . Hencethis element comes from the corresponding cohomology class of theBorel construction, and the isotropy action is equivariantly formal. • The space E / ( Sp (4) / ( ± I )). Since the cohomology of E is gen-erated in degrees 3 , , , , , 23 and the one of the denominatorgroup in degrees 3 , , , 15 we derive as above that H ∗ ( E / ( Sp (4) / ( ± I )) ∼ = (Λ h a, b i , b a ) ⊗ Λ h x, y i with deg a = 8, deg b = 23, deg x = 9, deg y = 17. In the spirit ofCorollary 4.2 we observe that x and y also have trivial differentialsconsidered as elements of the model of the Borel construction modulothe ideal of a contractible algebra. Hence the isotropy action isequivariantly formal. • The space E / F . Since the cohomology of E is generated in de-grees 3 , , , , , 23 an the one of F in degrees 3 , , , 23 wesee as above that the cohomology algebra of E / F is a free algebragenerated in degrees 9 and 17. Due to Corollary 4.3 the isotropyaction is equivariantly formal. References [1] M. Amann. Non-formal homogeneous spaces. Math. Z. , 274(3-4):1299–1325, 2013.[2] M. Amann and V. Kapovitch. On fibrations with formal elliptic fibers. Adv. 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American Mathematical Society, Providence, RI, 1991.Translated from the 1978 Japanese edition by the authors.[17] Z. St¸epie´n. On formality of a class of compact homogeneous spaces. Geom. Dedicata ,93:37–45, 2002.[18] S. Terzich. Real cohomology of generalized symmetric spaces. Fundam. Prikl. Mat. ,7(1):131–157, 2001. Manuel AmannInstitut f¨ur MathematikDifferentialgeometrieUniversit¨at AugsburgUniversit¨atsstraße 1486159 AugsburgGermany