Some phenomena in tautological rings of manifolds

We prove several basic ring-theoretic results about tautological rings of manifolds W, that is, the rings of generalised Miller--Morita--Mumford classes for fibre bundles with fibre W. Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitely-generated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W. Lastly, we give extensive computations in the tautological rings of CP^2 and S^2 x S^2.

1. Introduction 1.1. Recollections on tautological rings. A smooth fibre bundle π : E → B with closed d-dimensional fibre W equipped with an orientation of the vertical tangent bundle T π E has characteristic classes defined as follows. For each characteristic class c ∈ H k (BSO(d)) of oriented d-dimensional vector bundles, we may form κ c (π) := π c(T π E) ∈ H k−d (B), the generalised Mumford-Morita-Miller class (or κ-class) associated to c, by evaluating c on the vector bundle T π E and integrating the result along the fibres of the map π. This construction may in particular be applied to the universal such fibre bundle, whose base space is the classifying space BDiff + (W ) of the topological group of orientation-preserving diffeomorphisms of W , to give universal characteristic classes κ c ∈ H * (BDiff + (W )). If c has degree d then κ c is a degree zero cohomology class, and may be identified with the characteristic number W c(T W ) of W .
If we work in cohomology with rational coefficients then H * (BSO(d); Q) is generated by the Pontrjagin and Euler classes, and in this case we define the tautological ring R * (W ) ⊂ H * (BDiff + (W ); Q) to be the subring generated by all classes κ c . Our goal is to describe some quantitative and qualitative properties of these rings, for certain manifolds W . Before doing so, we introduce some variants. The topological group Diff + (W, * ) of diffeomorphisms of W which fix a marked point * ∈ W has a homomorphism to GL + d (R) by sending a diffeomorphism ϕ to its differential Dϕ * at the marked point. On classifying spaces this gives a map s : BDiff + (W, * ) −→ BGL + d (R) BSO(d) and for each c ∈ H * (BSO(d); Q) we may also form s * c ∈ H * (BDiff + (W, * ); Q). We let the tautological ring fixing a point R * (W, * ) ⊂ H * (BDiff + (W, * ); Q) be the subring generated by all the classes κ c and s * c.
Finally, if BDiff + (W, D d ) is the classifying space of the group of diffeomorphisms of W which are the identity near a marked disc D d ⊂ W , then we let the tautological ring fixing a disc R * (W, D d ) ⊂ H * (BDiff + (W, D d ); Q) be the subring generated by all the classes κ c . The inclusions of diffeomorphism groups therefore yield Q-algebra homomorphisms R * (W ) −→ R * (W, * ) −→ R * (W, D d ) whose composition is surjective.
These rings have been studied by Grigoriev [Gri17], and by Galatius, Grigoriev, and the author [GGRW17], mainly for the manifolds W = # g S n × S n with n odd. This is the natural generalisation of the case of oriented surfaces, i.e. n = 1, which has been studied in great detail: see e.g. [Mum83,Loo95,Fab99,Mor03]. Our purpose here is to explain to what extent those results apply to more general manifolds. We will only consider even-dimensional manifolds. For odd-dimensional manifolds the classes κ c have odd degree and so anticommute and are nilpotent, and tautological rings in this situation seem to have a different flavour.
1.2. Finiteness. Our first result concerns conditions under which the rings R * (W ) and R * (W, * ) are suitably finite.
Theorem A. Let W be a closed smooth oriented 2n-manifold, and assume that either (H1) H * (W ; Q) is non-zero only in even degrees, or (H2) H * (W ; Q) is non-zero only in degrees 0, 2n and odd degrees, and χ(W ) = 0. Then (i) R * (W ) is a finitely-generated Q-algebra, and (ii) R * (W, * ) is a finitely-generated R * (W )-module.
The result under hypothesis (H2) generalises a theorem of Grigoriev [Gri17], and proceeds by establishing the same basic source of relations among κ-classes found by Grigoriev. In the case 2n = 2 this source of relations had been established by the author [RW12], using ideas of Morita [Mor89a,Mor89b]. As the later results of [Gri17] and the results of [GGRW17] are deduced almost entirely from this basic source of relations, the same results largely follow assuming only hypothesis (H2). For example, for g > 1, k odd, and n ≥ k, it follows that Q[κ ep1 , κ ep2 , . . . , κ epn−1 ] −→ R * (# g S k × S 2n−k )/ √ 0 is surjective, which was obtained in [GGRW17] only in the case k = n. We give details of this in Section 4.1.
The result under hypothesis (H1) is entirely new and its method of proof is novel. We consider a fibre bundle W → E π → B as determining a parametrised spectrum over B, and hence its rational "cochains" as giving a parametrised HQ-module spectrum over B. We then use the notion of Schur-finiteness from the theory of motives to obtain a Cayley-Hamilton-type trace identity for endomorphisms of this cochain object, which establishes concrete relations among κ-classes. Later we shall describe some explicit calculations done using these relations.
(i) χ(W ) = 0 and the fixed set W T is connected, or (ii) the fixed set W T is discrete and non-empty. Then Kdim(R * (W )) ≥ k.
For example, if W 2n is a quasitoric manifold then case (ii) gives the estimate Kdim(R * (W )) ≥ n. As another example, if W = # g S n × S n with n odd then it is a consequence of the localisation theorem in equivariant cohomology (which we shall discuss in Section 3.1) that any torus action on W has connected fixed set, so by case (i) restricting the SO(n) × SO(n)-action on W constructed in [GGRW17,§4] to a maximal torus (which has rank n − 1) we obtain Kdim(R * (W )) ≥ n − 1 for g > 1, which recovers the calculation of that paper. This example admits many variants: the construction of [GGRW17,§4] can be easily modified to give a SO(k) × SO(2n − k)-action on # g S k × S 2n−k , so for any odd k and any n we have Kdim(R * (# g S k × S 2n−k )) ≥ n − 1.
We shall say more about this example in Section 4.1.

Examples.
In the last section of the paper we exhibit several phenomena in tautological rings by calculations for specific manifolds. The following result is complementary to Theorem A, and shows that the hypotheses of that theorem cannot be completely removed.
Theorem C. There are closed smooth manifolds W for which R * (W )/ √ 0 is not finitely-generated as a Q-algebra. There are examples of any dimension 4k + 2 ≥ 6, and in dimensions 4k + 2 ≥ 14 such manifolds can also be assumed to be simplyconnected.
To show the effectiveness of the relations between κ-classes arising in the proof of Theorem A, we apply them to the simplest manifold whose tautological ring is not yet known, namely CP 2 . These relations, along with relations associated to the Hirzebruch L-classes coming from index theory, give the following.
Theorem D. The ring R * (CP 2 ) has Krull dimension 2. The ring R * (CP 2 , D 4 ) is a vector space of dimension at most 7 over Q.
In fact, we show that the ring R * (CP 2 )/ √ 0 is equal to either , whose variety is the union of a line and a plane, or Q[κ p 2 1 , κ ep1 , κ p 4 1 ]/(4κ p 2 1 − 7κ ep1 ), whose variety is a plane. It would be interesting to determine which case occurs, and very interesting if it is the first case.
Finally, we give a calculation which shows that the lower bound of Corollary B is not always sharp. The 3-torus cannot act effectively on S 2 × S 2 , and yet Theorem E. The ring R * (S 2 × S 2 ) has Krull dimension 3 or 4.
The lower bound on the Krull dimension comes from a 1-parameter family of 2-torus actions, to which the method of proof of Corollary B is applied. The upper bound comes from the relations between κ-classes which we found in the proof of Theorem A.
Acknowledgements. I am grateful to Søren Galatius for an enlightening discussion of the ideas in Section 2 of this paper, and to Jens Reinhold and Dexter Chua for spotting several errors. I would also like to thank the anonymous referee for their useful suggestions. I was partially supported by EPSRC grant EP/M027783/1.

Tautological relations and finite generation
Unless specified, all cohomology in this paper will be taken with Q coefficients.
In this section we describe some techniques for obtaining relations between tautological classes, which for some manifolds W suffice to establish that R * (W ) is finitely-generated. The techniques we will introduce are perhaps more important than any particular application that can be made, but Theorem A will be a consequence.
2.1. Integrality. One consequence of conclusion (ii) of Theorem A is that R * (W, * ) is integral over R * (W ). In fact, this integrality statement implies the two finiteness statements, as follows.
For the sake of clarity, we will first formulate and prove a purely algebraic statement of which this proposition is a consequence.
Lemma 2.2. Let π : B → E be a homomorphism of Q-algebras, g : E → B be a homomorphism of B-modules (where E is made into an B-module via π), and C ⊂ E be a finitely-generated subalgebra, generated by {c i } i∈I . Let R ⊂ B be the subalgebra generated by g(C). If each c i is integral over π(R) ⊂ E, with for some a i,j ∈ R, then R is generated by the finitely-many elements Proof. By definition R is generated by the elements g( c ki i ), so we must show that these lie in the subring generated by the indicated elements. By assumption we may write c ki i as a Q[π(a i,j )]-linear combination of terms c j i with j < n i , and so we may write c ki i as a Q[π(a i,j )]-linear combination of terms c mi i with m i < n i . As g is B-and hence R-linear, we may therefore write g( c ki i ) as a Q[a i,j ]-linear combination of terms g( c mi i ) with m i < n i . Thus g( c ki i ) lies in the subring generated by the indicated elements.
Proof of Proposition 2.1. The universal fibre bundle with fibre W may be identified with the natural projection This gives a Q-algebra homomorphism by pullback and an H * (BDiff + (W ); Q)-module homomorphism by fibre integration. As we have described in the introduction, taking the differential at the marked point gives a map s : BDiff + (W, * ) → BGL + d (R) BSO(d), which classifies the vertical tangent bundle of the universal fibre bundle p.
For the second part, as H * (BSO(d)) is a finitely-generated Q-algebra, we know that R * (W, * ) is a finitely-generated R * (W )-algebra, so under the integrality assumption it follows that R * (W, * ) is in fact finitely-generated as a R * (W )-module.
Thus in order to prove Theorem A we shall actually show that R * (W, * ) is integral over R * (W ).

2.2.
Outline. To motivate the proof of Theorem A let us first explain its proof under hypothesis (H1) and an additional assumption: that the universal smooth oriented fibre bundle W → E π → B = BDiff + (W ) satisfies the Leray-Hirsch property in rational cohomology, i.e. that π 1 (B) acts trivially on H * (W ) and the Serre spectral sequence for π : E → B collapses. (The proof of Theorem A under hypothesis (H1) is a technical device which allows the following argument to be made without this additional assumption.) Under this assumption, H * (E) is a free finitely-generated H * (B)-module, say with basisx 1 , . . . ,x k ∈ H * (E) lifting a basis x 1 , . . . , x k for H * (W ). Furthermore, as W has all its cohomology in even degrees, H ev (E) is a free finitely-generated module over the commutative ring H ev (B), with basis thex i . For , and by the Cayley-Hamilton theorem (for finite modules over a commutative ring, alias the determinantal trick) we have χ x (x) = 0 ∈ H ev (E). Furthermore, the coefficients of the characteristic polynomial χ x (z) may be expressed as polynomials in the elements which make sense as H ev (E) is a finite free H ev (B)-module. The following lemma relates such traces to fibre-integration and the Euler class of the vertical tangent bundle.
Lemma 2.3. For any x ∈ H ev (E) we have We apply the above discussion to x = c(T π E) for c ∈ H * (BSO(2n)) a characteristic class of oriented 2n-dimensional vector bundles. Then the polynomial χ x (z) is monic, has coefficients in the subring generated by the κ ec i = π e(T π E) · c(T π E) i , and satisfies χ x (c(T π E)) = 0. Thus we deduce that c(T π E) = s * c is integral over R * (W ) and hence that R * (W, * ) is integral over R * (W ). Theorem A in the case we are considering follows by applying Proposition 2.1. It remains to prove this lemma.
Proof of Lemma 2.3. Rational cohomology classes are determined by their evaluations against rational homology classes, and any rational homology class is carried on a map from a smooth oriented manifold. So we may assume that π : E → B is a fibre bundle over a smooth oriented manifold, still satisfying the Leray-Hirsch property.
The pairing is non-singular, as in the basisx i its matrix X agrees modulo the ideal H * >0 (B) of H * (B) with that of the intersection form of W in the basis x i , so det(X) ∈ H * (B) is a unit modulo H * >0 (B), and hence is a unit as the ideal H * >0 (B) is nilpotent. Let us writex ∨ i for the dual H * (B)-module basis of H * (E) with respect to this pairing, characterised by x i ,x ∨ j = δ ij . Then for any x ∈ H ev (E) we have so to establish the claimed formula we must show that ix i ·x ∨ i = e(T π E) ∈ H ev (E). The diagonal map ∆ : E → E × B E is a map of smooth oriented manifolds, whose normal bundle is identified with T π E. Thus the Euler class e(T π E) ∈ H d (E) may be described as ∆ * ∆ ! (1). It is therefore enough to show that This is the parametrised analogue of the classical formula [MS74, Theorem 11.11] for the Poincaré dual of the diagonal, and we shall prove it in the same way. For As the classes (b ·x ∨ j ) ⊗x k generate H * (E × B E) as a Q-module, it follows from Poincaré duality for E × B E that ∆ ! (1) = ix i ⊗x ∨ i , as required. 2.3. Parametrised spectra and Schur functors. The technical device we shall use to attempt the argument of the previous section without the Leray-Hirsch assumption is to consider a fibre bundle as a parametrised manifold over its base, and make the argument in the parametrised setting. In order to do so, we shall suppose that B is a connected CW-complex, and work in a symmetric monoidal category (Sp (apart from this map, the notation (−) ! will always denote Gysin maps). The functor r * is strong monoidal.
Our argument applies more generally than to oriented fibre bundles: for now, we let π : E → B be a Hurewicz fibration (later we will add a finiteness hypothesis to 1 However, our arguments are not model-dependent and can be applied in the ∞-categorical formalism of Ando, Blumberg, Gepner, Hopkins, and Rezk [ABG + 14, ABG11], and presumably even in more naïve models of parametrised spectra. the fibres of π). This defines a parametrised spectrum Σ ∞ B E ∈ Sp /B ; we shall abuse notation and continue to call it E. Note that r ! (E) ∈ Sp is the suspension spectrum Σ ∞ E + .
The ring spectrum HQ has a 2-periodic version and we write π * (HP Q) = Q[t ±1 ] with t ∈ π 2 (HP Q). The constant parametrised spectra H B Q := r * (HQ) and HP B Q := r * (HP Q) define ring objects in Sp /B , and the main objects we will consider are the function objects These are again ring objects, using the fibrewise diagonal map on E and the multiplication on H B Q and HP B Q; we write µ for the multiplication on either object. The map E → * gives ring maps H B Q → C and HP B Q → CP , making them H B Q-and HP B Q-modules respectively. Let us write (H B Q-mod, ⊗, H B Q) for the homotopy category of H B Q-module spectra, with derived smash product of H B Q-modules as the symmetric monoidal structure and unit H B Q; similarly write (HP B Q-mod, ⊗, HP B Q) for the homotopy category of HP B Q-module spectra. We have C ∈ H B Q-mod and CP ∈ HP B Q-mod, and we can calculate Both H B Q-mod and HP B Q-mod are Q-linear tensor categories (i.e. categories enriched in Q-modules, equipped with a symmetric monoidal structure which is an enriched functor) which are idempotent complete (the retract associated to an endomorphism e : X → X which is idempotent up to homotopy may be taken to be the homotopy colimit of a diagram X e → X e → X e → · · · of modules over the appropriate ring object).
We must now recall a little representation theory of symmetric groups; we need nothing beyond Lecture 4 of [FH91]. To each partition λ of n there is associated an irreducible representation S λ of Σ n , with character χ λ . This character takes rational (in fact, integer) values, so we may form the element which is central (as χ λ is a class function) and idempotent (the coefficient dim S λ n! is chosen to make this so). For any object X in a Q-linear tensor category (D, ⊗, 1), the action of the nth symmetric group Σ n on X ⊗n yields a map of Q-algebras so e(d λ ) is an idempotent endomorphism of X ⊗n in D; if D is idempotent complete then we write S λ (X) for the corresponding retract of X ⊗n in D: this defines the Schur functor S λ (−) on D. In this paper the trivial and sign representations will play the most prominent role, and we write or, if we wish to emphasise the ambient category, ∧ n D and Sym n D . The categories H B Q-mod and HP B Q-mod are idempotent complete Q-linear tensor categories, so there are defined Schur functors on both categories. Furthermore, let us write (V Q , ⊗ Q , Q) for the symmetric monoidal category of graded Q-modules, ]-modules (where t has degree 2). These are also idempotent complete Q-linear tensor categories. Taking homotopy groups defines functors π * (−) : which are strong monoidal (by the Künneth theorem, as every graded Q-or Q[t ±1 ]module is free). Taking derived homotopy fibres at b ∈ B defines functors is also strong monoidal. In particular, all of the above functors preserve Schur functors.
Lemma 2.4. Let X ∈ H B Q-mod or HP B Q-mod be such that for each fibre X b we have S λ (π * (X b )) = 0. Then S λ (X) * .
Proof. As taking homotopy groups preserves Schur functors, we have π * (S λ (X b )) = S λ (π * (X b )) which vanishes by assumption. Thus S λ (X b ) * , so as taking derived fibres preserves Schur functors it follows that S λ (X) b * . Thus the map from S λ (X) to the terminal object is an equivalence on derived fibres, and hence an equivalence, as taking derived fibres reflects isomorphisms.
2.4. Duals, trace, and transfer. We recall the framework of categorical traces, from [DP80]. If (C, ⊗, 1) is a symmetric monoidal category and X ∈ C is an object, a strong dual of X is an object X ∨ ∈ C and morphisms ε : are the identity maps of X and X ∨ respectively. If f : X → Y is a map of objects having strong duals, then the dual of f is If f : X → X is an endomorphism of X, the trace of f is the composition This agrees with the perhaps more obvious choice but the first definition is that of [DP80]. Generalising this more obvious choice, if f : A ⊗ X → B ⊗ X is a morphism then the trace of f over X is the composition If X is in addition equipped with a comultiplication d : X → X ⊗ X then the transfer of f is First, consider the symmetric monoidal category given by the homotopy category of (Sp /B , ∧ B , S 0 B ).
Lemma 2.5. If π : E → B is a Hurewicz fibration and its fibre has the homotopy type of a finite CW-complex, then its associated parametrised spectrum Σ ∞ B E is a strongly dualisable object in the homotopy category of parametrised spectra.
Suppose then that π : E → B is a Hurewicz fibration and its fibre has the homotopy type of a finite CW-complex. Recall that we abuse notation by writing E for Σ ∞ B E. The fibrewise suspension of the fibrewise diagonal map ∆ : E → E × B E gives a comultiplication on the object E, we may thus form On applying r ! : In particular for c ∈ H * (BSO(2n)) we have that trf * π (c(T π E)) = κ ec is a tautological class.
Let us now consider the symmetric monoidal categories (H B Q-mod, ⊗, H B Q) and (HP B Q-mod, ⊗, HP B Q).
Corollary 2.6. If π : E → B is a Hurewicz fibration and its fibre has the homotopy type of a finite CW-complex, then Proof. The functor has a monoidality given by the adjoint of the morphism given by evaluation and product. This is a strong monoidality: the induced morphism on derived fibres is a weak equivalence (by the Künneth theorem, as every π * (HQ) = Q-module is free). As E ∈ Ho(Sp /B ) is strongly dualisable by Lemma 2.5, so is C = F B (E, H B Q), because strong monoidal functors preserve (strong) duals. The argument for CP is identical.

Schur-finiteness and trace identities. Deligne has introduced [Del02, §1]
the notion of Schur-finiteness of an object X in an idempotent complete Q-linear tensor category to be the property that S λ (X) is trivial for some partition λ n. In this section we consider this notion applied to the category (HP B Q-mod, ⊗, HP B Q) and so consider an HP B Q-module X such that S λ (X) * , and let us in addition suppose that X is dualisable in HP B Q-mod. Let us write X ∨ for the dual of X, with duality structure given by η : Given an endomorphism f : X → X, we may form the endomorphism and take the trace over the last (n − 1) copies of X, i.e. apply the construction Tr X ⊗n−1 (−) described in the previous section, to obtain an endomorphism of X. This endomorphism is null because the idempotent e(d λ ) : X ⊗n → X ⊗n factors through S λ (X) which is contractible by assumption.
We now translate this into formulas. We have d λ = dim S λ n! σ∈Σn χ λ (σ) · σ so the essential calculation is to describe the endomorphism of X obtained from σ • (X ⊗ f ⊗n−1 ) : X ⊗n → X ⊗n by taking the trace over the last (n − 1) copies of X. This is a universal construction in idempotent complete Q-linear tensor categories, and has been worked out by Abramsky [Abr05, Proposition 3]. In the notation of that proposition, one takes A 1 = B 1 = X and U 2 = · · · = U n = X, then f 1 = Id X and f 2 = · · · = f n = f , and π = σ. The trace of σ • (X ⊗ f ⊗n−1 ) over the last (n − 1) copies of X is then given by Here p −1 σ = Id X , and g 1 is given by composing the f i along the cycle in σ starting at 1: as f 1 = Id X , if this cycle is (1, p 2 , . . . , p k ) then this gives g 1 = f •k−1 ; L(σ) is the set of cycles in the permutation σ which do not contain 1, and for such a cycle l = (p 1 , p 2 , . . . , p k ) we have s l := Tr(f p k • · · · • f p1 ) = Tr(f •k ).
We originally learnt this idea from the thesis of del Padrone [Pad06] (see [Pad06, Proposition 2.2.4] for a closely related result).
2.6. Proof of Theorem A under the first hypothesis. In this case we will work with the periodic chains CP . For each b ∈ B we have and so by 2-periodicity we have an isomorphism Here the right-hand side is to be interpreted as the tensor product of graded Q-modules, which in degree k is −i+2n=k Under hypothesis (H1) the cohomology H * (W ) is concentrated in even degrees, so if it has total degree k then we have ] = 0 and so it follows from Lemma 2.4 that ∧ k+1 CP * . As π : E → B is a fibre bundle with compact fibres Corollary 2.6 applies to it, so CP is dualisable in HP B Q-mod and hence the discussion of the previous section applies. Thus, as CP is a ring object in HP B Q-mod, for any multiplication by x yields an endomorphismx : CP → CP , and in this case composing the map (2.2) with 1 ∈ [HP B Q, CP ] HP B Q-mod gives the identity Corollary 2.7. The polynomial is monic of degree k and satisfies ρ x (x) = 0 ∈ H 2 * (E).
The Becker-Gottlieb transfer trf * π is the map on cohomology induced by the composition shows that trf * π (t) is the trace oft ∨ : CP ∨ → CP ∨ , which is the same as the trace oft : CP → CP : thus In particular we have Tr(x •i ) = trf * π (x i ). Substituting this into (2.3) therefore shows that The coefficient of x k is the sum over the k!-many (k + 1)-cycles σ ∈ Σ k+1 of sign(σ) = (−1) k , which is k!(−1) k . Thus after dividing by this coefficient we see that ρ x (x) = 0 as required.
Applying this to the universal fibre bundle p : BDiff + (W, * ) → BDiff + (W ) and the cohomology class s * c, and using the identity trf * and hence that R * (W, * ) is integral over R * (W ). Theorem A under hypothesis (H1) follows by applying Proposition 2.1.

2.7.
Proof of Theorem A under the second hypothesis. In this case we will work with the non-periodic chains C. We shall first prove the following generalisation of a theorem of Grigoriev [Gri17].
To begin with, we prove the following extension of Corollary 2.6, which is the appropriate form of Poincaré duality in our setting.
Lemma 2.9. If π : E → B is a Hurewicz fibration over a CW-complex, its fibre F has the homotopy type of a finite Poincaré complex of dimension n, and π 1 (B) acts trivially on H n (F ), then an orientation of F determines an identification of the dual of C with Σ n C in H B Q-mod.
Proof. This is proved in Section 3.1 of [HLLRW17] in dual form, where, passing to rational coefficients, the equivalence is expressed as The domain of this morphism is Σ n C and as We now consider a smooth oriented fibre bundle π : E → B as in the statement of Theorem 2.8, with B a CW-complex; this satisfies the hypotheses of the previous lemma. Fibre integration π ! : H * (E) → H * −2n (B) is realised in the category H B Q-mod by the morphism π ! : C → Σ −2n H B Q dual to the unit ι : H B Q → C, using the self-duality of C described in Lemma 2.9. We may thus define an H B Qmodule D by the homotopy fibre sequence realises capping with the fundamental class so is surjective. By the associated long exact sequence we have and the remaining even homotopy groups are zero. Now the map realises the unit so is an isomorphism, and it follows that π odd (D b ) = H −odd (W ) and the even homotopy groups of D b vanish. As the (d + 1)-st symmetric power of the graded Q-module H −odd (W ) vanishes, the rest follows from Lemma 2.4.

The map
so taking homotopy cofibres of these two maps gives a morphism then it lifts to a map to D and hence determines a mapā : The class π ! (a · b) may therefore be represented by Hence the class π ! (a · b) N may be written as as the degree k ofā is even so no sign is incurred in rearranging the factors. As k is even, the mapā N : (Σ k H B Q) ⊗N → D ⊗N factors through Sym N (D) which is contractible as long as N ≥ d + 1. Hence π ! (a · b) d+1 = 0. Similarly, if a = b then we can chooseb =ā : Σ k H B Q → D in which case the map a N ⊗ā N : which is contractible as long as 2N ≥ d + 1. Hence π ! (a 2 ) d+1 2 = 0. Now that we have Theorem 2.8, the entirety of Section 5 of [Gri17] goes through with only notational changes, as this only uses the statement of Grigoriev's theorem. In particular, for p ∈ H * (BSO(2n)) of even degree and χ = χ(W ) = 0, the analogue of [Gri17, Example 5.19] gives the relation sign(σ) · κ ec l(γ 2 ) · · · κ ec l(γ q(σ) ) · c l(γ1)−1 ∈ R * (W, * ) for each c ∈ H * (BSO(2n)), where k = dim Q H * (W ). These may of course be pushed forward to obtain relations in R * (W ). More generally, the trace identity technique of Section 2.5 may be used to find relations among tautological classes for any manifold. Recall that given a fibre bundle W → E π → B we have formed an associated object CP ∈ HP B Q-mod. Let us write d ev = dim Q H ev (W ) and d odd = dim Q H odd (W ). The first ingredient is the following consequence of a calculation of Deligne.

Proof. By Lemma 2.4 it is enough to verify that
where the latter Schur functor is taken in V Q . By [Del02, Corollary 1.9] a (Z/2-)graded vector space is annihilated by S λ (−) under the given assumption on its (super)dimension.
It is a simple exercise with the Murnaghan-Nakayama rule to show that the character χ λ vanishes on all n-cycles if both d ev > 0 and d odd > 0. As d ev cannot be zero, because H 0 (W ) = 0, it follows that this relation is a monic polynomial in c (after perhaps scaling by a rational number) if and only if d odd = 0. (This accounts for why we restricted to manifolds with only even rational cohomology in the first case of Theorem A.)

Torus actions
In this section we suppose that we have a smooth action of the torus T = (S 1 ) k on a d-dimensional orientable manifold W . We write W T for the fixed set of this action. The Borel construction gives a smooth fibre bundle and the action of T on the tangent bundle T W → W gives a vector bundle T T W := T W/ /T → W/ /T , which is the vertical tangent bundle of the smooth fibre bundle π. Following the usual notation of equivariant cohomology we write . , x k ] and H * T (W ) = H * (W/ /T ; Q). As (3.1) is a smooth fibre bundle, there is a ring homomorphism ρ : R * (W ) → H * T , and we denote by R * T ≤ H * T its image. Pulling back π along itself gives a smooth fibre bundle over W/ /T with canonical section, and so a ring homomorphism ρ * : R * (W, * ) → H * T (W ), and we denote by R * T ( * ) ≤ H * T (W ) its image. Our goal in this section is to describe conditions on the manifold W and the action of T on W which allow us to estimate the Krull dimension of R * (W ) as Kdim(R * (W )) ≥ k. We will regularly use the following standard piece of commutative algebra: when one ring is integral over another they have the same Krull dimension, by the "going up" and "going down" theorems [AM69, Ch. 5]. Our most general result is as follows.
Theorem 3.1. Let T act smoothly and effectively on a connected closed orientable manifold W . Let V 1 , V 2 , . . . , V p be an enumeration of the T -representations arising as normal spaces to points on W T , and let B i denote the Euler characteristic of the subspace of W T consisting of those path components having normal representation V i .
If some y i ∈ Q[y 1 , y 2 , . . . , y p ] is integral over the subring generated by p i=1 B i y n i , n = 1, 2, 3, . . . , then H * T is integral over R * T . In particular Kdim(R * (W )) ≥ k. It is perhaps not clear when the hypothesis of this theorem is likely to hold. The following lemma, which we learnt from [BCES16], gives a simple criterion.
Lemma 3.2. Suppose that we have discarded the B i which are zero, and that this is not all of them. If the remaining numbers B 1 , B 2 , . . . , B p have all partial sums non-zero, then Q[y 1 , y 2 , . . . , y p ] is finite over the subring generated by Claim. If (B + ) = (y 1 , y 2 , . . . , y p ) then Q[y 1 , . . . , y p ] is finite over B.
Our proof of this claim follows the discussion at [Jef]. Under the assumption the quotient ring Q[y 1 , y 2 , . . . , y p ]/(B + ) has every y i nilpotent, so is a finite Q-module; let z 1 , z 2 , . . . , z m ∈ Q[y 1 , y 2 , . . . , y p ] be lifts of these finitely-many generators, which can be taken to be homogeneous as the ideal (B + ) is homogeneous. We claim that these generate Q[y 1 , y 2 , . . . , y p ] as a B-module; let M ⊂ Q[y 1 , y 2 , . . . , y p ] be the B-submodule that they generate.
As the z i are homogeneous, and B is generated by homogeneous elements, M is a graded submodule of Q[y 1 , y 2 , . . . , y p ] with the monomial-length grading. Suppose p ∈ Q[y 1 , y 2 , . . . , y p ] is an element of minimal grading which does not lie in M . Then we may write , y 2 , . . . , y p ], and b j ∈ (B + ). But the b j have strictly positive degree, so the V j have strictly smaller degree than p so must lie in M , and hence p does too, which proves the claim.
In order to prove the lemma we must therefore show that (y 1 , y 2 , . . . , y p ) = (0, 0, . . . , 0) is the only simultaneous solution to the equations p i=1 B i y n i = 0 for n ∈ N. If (y 1 , y 2 , . . . , y p ) ∈ Q p is a solution, then grouping terms with y i = y j together we obtain distinct rational numbersȳ i solving the equations where eachB i is a partial sum of the B i , and hence non-zero by assumption. But this means that the vector (B 1ȳ1 , . . . ,B qȳq ) is in the kernel of the (transposed) Vandermonde matrix associated to (ȳ 1 , . . . ,ȳ q ), so as theȳ i are all distinct it follows that (B 1ȳ1 , . . . ,B qȳq ) = 0, and as theB i are all non-zero it follows thatȳ i = 0 as required.
The following corollary, whilst not so powerful as Theorem 3.1, is often easier to apply as one does not need to classify the normal representations at the fixed set.
. . , x ] generated by the i=1 A i x n i onto the subring B ⊂ Q[y 1 , y 2 , . . . , y p ] generated by the p i=1 B i y n i . If x i is integral over A then there is a polynomial q(x) = a i x i with coefficients in A such that q(x i ) = 0. Then q (y) = φ(a i )y i is a polynomial over B such that q (φ(x i )) = 0, so y j = φ(x i ) is integral over B, and hence Theorem 3.1 applies.
Example 3.4. There are several standard conditions which oblige a torus action on a manifold W to have connected fixed-set. For example (i) Let W have dimension 2n, and suppose that all its cohomology apart from H 0 (W ; Q) and H 2n (W ; Q) lies in odd degrees, and that there is some cohomology in odd degrees. Then W T is connected (by the localisation theorem in equivariant cohomology, which we will describe in the following section). (ii) If W has trivial even-dimensional rational homotopy groups, then W T is empty or connected [Hsi75, Theorem IV.5].
In such cases χ(W T ) = χ(W ), so if this is non-zero then the hypotheses of Corollary 3.3 are satisfied.
Example 3.5. Suppose that the action of T k on W has isolated fixed points, or more generally that all A i are equal and non-zero. Then the subring generated by the i=1 A i x n i is the subring of symmetric polynomials in Q[x 1 , x 2 , . . . , x ], and every x i is integral over this so the hypotheses of Corollary 3.3 are satisfied.
This immediately implies that if W 2n is a quasitoric manifold (that is, the "toric manifolds" of [DJ91]) then Kdim(R * (W )) ≥ n, as such manifolds by definition have an action of an n-torus with isolated fixed points. Slightly more subtly, if G/K is a homogeneous space of rank zero (i.e. rk(G) = rk(K)) then a common maximal torus T of G and K acts on G/K with fixed points given by the finite set (W G (T ) · K)/K ⊂ G/K, where W G (T ) := N G (T )/T denotes the (finite) Weyl group of G, so Kdim(R * (G/K)) ≥ rk(G).
3.1. The localisation theorem. We now prepare for the proof of Theorem 3.1. Let X 1 , X 2 , . . . , X be the components of the fixed set W T , with d i := dim(X i ), let ν Xi be the normal bundle of X i in W , and let ν i be the T -representation which arises as each fibre of ν Xi . Let us write A i := χ(X i ), and write V 1 , V 2 , . . . , V p for an enumeration of the T -representations ν i which arise. Then we have B i = j s.t. νj =Vi A j . Let us write ρ i : H * T (W ) → H * T (X i ) for the restriction map in equivariant cohomology, and π ! : for the fibre integration maps. As the T -action on X i is trivial we have X i / /T = BT × X i , and so the fibre integration map (π i ) ! is simply given by slant product with the fundamental class of X i . As T acts on the normal bundle ν Xi → X i , there is an induced vector bundle ν T Xi is an isomorphism. Even more is true: Atiyah See [AP93, p. 366] for a textbook exposition of the localisation theorem.
3.2. Proof of Theorem 3.1. Using the diagram (3.3) to compute κ ep I = π ! (e(T T W )p I (T T W )) ∈ S −1 H * T , which we know lies in the subring H * T , gives When we multiply by e(T X i ) and integrate over X i the latter terms do not contribute, so as Xi e(T X i ) = χ(X i ) = A i we get Grouping these terms by the representation types V j instead gives Applying this to p I = p n j we find that T for all j and n.
Applying the hypothesis of the theorem for each j, we find that there exists an i such that all p j (V i ) lie in a common integral extension R * T ⊆ R ⊆ H * T . On the one hand R is integral over R * T . On the other hand by a theorem of Venkov [Ven59] the ring H * T is finite over the subring generated by the p j (V i ) (because V i is a faithful representation of T , by the standard lemma given below), and hence is finite (and so integral) over R . It follows that H * T is integral over R * T , so in particular they have the same Krull dimension, namely k.
Lemma 3.6. If T acts effectively and smoothly on a connected closed manifold W , then any T -representation arising as the normal space to a point on W T is faithful.
Proof. We may choose a T -invariant Riemannian metric on W , so the exponential map exp : T W → W is equivariant; the restriction of the exponential map to a fibre T x W → W is a diffeomorphism when restricted to a neighbourhood of 0 ∈ T x W .
If the action of T on the normal space V to W T at x had a non-trivial kernel {e} < T ≤ T then the T -action on T x W = T (W T )⊕V is trivial. By exponentiating, it follows that T fixes an open neighbourhood of x ∈ W . Thus the fixed set W T is a submanifold of W which contains an open subset; as W is connected it follows that it is the whole of W . This contradicts the action being effective.

3.
3. An extension. The discussion so far gives a technique more general Theorem 3.1, but difficult to formalise in a single result. It is best described through an example.
Proposition 3.7. Let T act effectively on W with two fixed components X 1 and X 2 . Suppose that χ(X 1 ) = −χ(X 2 ) = 0 but that the normal T -representations ν 1 and ν 2 at X 1 and X 2 have all Pontrjagin classes distinct (when they are non-zero). Then Kdim(R * (W )) ≥ k.
Proof. We have that for all j and n, and p j (ν 1 ) − p j (ν 2 ) = 0 ∈ R * T . Hence Therefore after inverting the finite set . , x k ], we find that the p j (ν 1 ) lie in S −1 R * T , and hence by Venkov's theorem [Ven59] that S −1 H * T is a finite S −1 R * Tmodule. As S −1 H * T still has Krull dimension k (there is a maximal ideal m of H * T not containing the product of the finitely-many elements in S-as the intersection of all maximal ideals is zero-whence (S −1 H * T ) S −1 m ∼ = (H * T ) m so S −1 m is a maximal ideal of S −1 H * T of height k), it follows that S −1 R * T has Krull dimension k and so Kdim(R * T ) ≥ k. Hence Kdim(R * (W )) ≥ k.

4.1.
Manifolds with mostly odd cohomology. Let W be a 2n-dimensional manifold whose cohomology is only non-trivial in degrees 0, 2n, and odd degrees, let d = dim Q H odd (W ), and suppose χ(W ) = 2 − d = 0. Then by Theorem A the Q-algebra R * (W ) is finitely-generated and R * (W, * ) is a finite R * (W )-module. Furthermore, by our method of proof, Grigoriev's theorem holds for these manifolds (our Theorem 2.8). Therefore the results of Sections 2 and 3 of [GGRW17] hold for W as well, as Grigoriev's theorem was the only external input. So if d > 2 then By Example 3.4 (i), if T = (S 1 ) k acts on such a manifold W then the fixed set W T is connected, so Kdim(R * (W )) ≥ k. The construction of [GGRW17, Section 4.1] can be mimicked to obtain an action of SO(k) × SO(2n − k) on # g S k × S 2n−k for any k, and the calculation of the characteristic classes κ epi for the associated bundle is entirely analogous.
We obtain the following generalisation of the results of [GGRW17].
Corollary 4.1. For k odd and g > 1 we have and hence R * (# g S k × S 2n−k , D 2n ) is a finite-dimensional Q-vector space.
As in [GGRW17] results can be obtained for g = 0 or 1 too, but we shall not write them out here.

Quasitoric manifolds.
A quasitoric manifold W 2n has by definition a smooth action of T = (S 1 ) n with isolated fixed points, so has Kdim(R * (W )) ≥ n by Corollary 3.3 . Furthermore, the integral cohomology of W is supported in even degrees, so its rational cohomology is too, and therefore by Theorem A the Q-algebra R * (W ) is finitely-generated and R * (W, * ) is a finite R * (W )-module. 4.3. Non-finite generation. We shall give some examples of manifolds W for which R * (W ), and in fact even R * (W )/ √ 0, is not finitely-generated. We shall do so by constructing actions of a torus T on W and showing that the tautological subring R * T ≤ H * T is not finitely-generated. As H * T is an integral domain the natural surjection R * (W ) → R * T factors through R * (W )/ √ 0, which therefore shows that R * (W )/ √ 0 is not finitely-generated. Before attempting this method there is an important observation to be made.
Observation 4.2. Let T = (S 1 ) k act on W satisfying the hypotheses of Theorem 3.1; then that theorem shows that the inclusion R * T → H * T is integral. As H * T is Noetherian, and H * (BT ; H * (W )) is a finitely-generated H * T -module, it follows from the Serre spectral sequence for the Borel construction that H * T (W ) is a finitely-generated H * T -module and hence is integral over H * T . Therefore the morphism R * T → H * T → H * T (W ) is integral, so R * T → R * T ( * ) is integral too. It then follows from applying Lemma 2.2 as in the proof of Proposition 2.1 that R * T → R * T ( * ) is finite and R * T is a finitely-generated Q-algebra.
So to pursue the programme we have suggested one should only try to use torus actions which do not satisfy the hypotheses of Theorem 3.1. The following allows us to construct manifolds with torus actions having prescribed normal representations and Euler characteristics of its fixed sets.
Construction 4.3. Fix a positive odd integer n and an even integer k. Let Σ(k) 2n be the 2n-manifold of Euler characteristic k obtained as # g S n × S n (if k is nonpositive) or g S 2n (if k is positive). Let H(k) 2n+1 be the manifold with boundary Σ(k) 2n given by g S n × D n+1 or g D 2n+1 respectively. Let T be a torus, and suppose we are given even integers B 1 , B 2 , . . . , B p and distinct faithful complex T -representations V 1 , V 2 , . . . , V p , which are all of the same dimension and which have no trivial subrepresentations. Then we can form the manifold which has a T -action on the right-hand factors. We may then let M be the disjoint union M = M (1) M (2) · · · M (p).
As V i is representation having no trivial subrepresentations, T acts freely on S(V i ) and its only fixed point on D(V i ) is 0. Thus M (i) T = Σ(B i ) 2n × {0}, and the normal representation at these fixed points is given by V i .
Each V i may be written as a sum L 1 ⊕ · · · ⊕ L m of 1-dimensional complex Trepresentations; if a unit vector v ∈ S(V i ) is written in components as (l 1 , . . . , l m ) with all l j non-zero, then a t ∈ T which stabilises it must act trivially on each L j , so must act trivially on V i , so t must be the identity as V i is a faithful Trepresentation. Thus such a v ∈ S(V i ) must lie in a free orbit, so in particular each path component of M (i) has a free orbit. If one prefers a connected manifold, such free orbits in two different path components have tubular neighbourhoods T -equivariantly diffeomorphic to T × D 2n+2m−rk(T ) , which can therefore be cut out and the remaining pieces glued together T -equivariantly along the common boundaries T × S 2n+2m−rk(T )−1 . Doing this enough times yields a connected Tmanifold with the same fixed-point data, and hence by localisation with the same characteristic classes.
Lemma 4.4. The T -manifold M so obtained has κ p I = 0 and Proof. The second statement follows from (3.4 ). An analogous calculation shows that .
The bundle ν Xi → X i is trivial, so the equivariant bundle ν T Xi is isomorphic to the pullback of V i to X i / /T = BT × X i . Thus the total Pontrjagin class satisfies which pushes forward to zero (as dim(X i ) = 2n > 0), so κ p I = 0.
We now give our example.
Example 4.5. Let T = (S 1 ) 2 and V 1 be the 2-dimensional complex T -representation with weights {x 1 + x 2 , x 2 }, and V 2 be the 2-dimensional complex T -representation with weights {x 1 , x 2 }. Construction 4.3 with B 1 = 2 and B 2 = −2 yields a Tmanifold W (which may be chosen to have any dimension at least 6 and congruent to 2 modulo 4) having κ p I = 0 and . For the chosen representations the total Pontrjagin classes are 2 ). Here p 2 (V 1 ) = p 2 (V 2 ) = 0 and 1 , so the only non-zero κ ep I in this quotient ring are x 2 all have different degrees and are non-zero as they are not divisible by x 2 2 . On the other hand, multiplication of any two positive-degree elements in S is zero, as each positivedegree element is divisible by x 2 so a product is divisible by x 2 2 . Thus S is infinitelygenerated, so R * T is too, and hence R * (W )/ √ 0 is too.
Let us record some observations about this example.
Remark 4.6. If we suppose that n ≥ 5 is odd and the T -manifolds M (2, V 1 ) and M (−2, V 2 ) are glued along a free orbit as suggested above, then the (2n+4)-manifold M obtained is simply-connected and has the same integral homology as Remark 4.7. Although this tautological ring is not finitely-generated, Proposition 3.7 applies to this torus action and gives Kdim(R * (W )) ≥ 2. (Specifically, we have Remark 4.8. Choosing * ∈ X 2 gives a map R * (W, * ) → H * T , whose image is generated by the κ ep I = 2(p I (V 1 ) − p I (V 2 )) along with the characteristic classes of the representation V 2 , which are e(V 2 ) = x 1 x 2 , p 1 (V 2 ) = −(x 2 1 + x 2 2 ), and p 2 (V 2 ) = x 2 1 x 2 2 . Rearranging a little shows that this is the subring generated by e(V 2 ) and the p j (V i ), so is finitely generated. (Similarly if we choose * ∈ X 1 .) This raises the interesting possibility that R * (W, * ) might be finitely-generated in more generality than R * (W ) is.
4.4. The complex projective plane. Let us consider the manifold CP 2 , whose cohomology is supported in even degrees. Thus by Theorem A the Q-algebra R * (CP 2 ) is finitely-generated and R * (CP 2 , * ) is a finite R * (CP 2 )-module. We will explain estimates on the generators for these algebras, using the relations developed in Section 2.5. The computations were done with assistance from Maple™.
(4.4) (More generally, one could fully polarise this relation, by writing c = u + t · v + s · w, expanding out and taking the coefficient of ts: this gives a trilinear form in the variables (u, v, w) which vanishes for all u, v, w ∈ H * (BSO(4)) = Q[p 1 , e], and there is no reason to take these to be linear terms. However, we will not pursue this here.) The relations (4.1), (4.2), (4.3) and (4.4), multiplied by monomials in Q[p 1 , e] and pushed forward, show that certain κ e a p b 1 ∈ R * (CP 2 ) are decomposable. Specifically κ xp 3 1 is decomposable for any monomial x = 1, e, p 1 κ xep 2 1 is decomposable for any monomial x = 1, e, p 1 κ xe 2 p1 is decomposable for any monomial x = 1, e, p 1 κ xe 3 is decomposable for any monomial x = 1, e, p 1 .
the last of which shows that the generator κ p 4 1 may be eliminated from the ring Q[e, p 1 , κ p 2 1 , κ p 4 1 , κ ep1 ]/ √ J. One may also deduce from these relations that κ ep1 and κ p 2 1 are integral over Q[e, p 1 ], so that R * (CP 2 , * )/ √ 0 is finite over Q[e, p 1 ].
4.4.2. Fixing a disc. As passing from R * (CP 2 , * ) to R * (CP 2 , D 4 ) in particular kills e and p 1 , we deduce from the above that Corollary 4.11. R * (CP 2 , D 4 ) is a finite-dimensional Q-vector space.
In fact, setting K = J + (e, p 1 ) and simplifying, we find that K is generated by
Remark 4.14. In [Fin77,Fin78] there is given an analysis of S 1 -actions on simplyconnected 4-manifolds, from which it is possible to deduce-through a very laborious consideration of cases and analysis of fixed-point data-that for any circle action on CP 2 we have 4κ e 2 = κ ep1 and so by the first Hirzebruch relation we have 4κ p 2 1 − 7κ ep1 = 0. Alternatively, this may be proved using Hsiang's splitting theorem for the S 1 -equivariant cohomology of CP 2 [Hsi75, Theorem VI.1]. 4.4.4. The tautological variety. We find it quite revealing to consider the (reduced) tautological ring R * (CP 2 )/ √ 0 by considering its associated variety V CP 2 . The choice of generators κ p 2 1 , κ p 4 1 , and κ ep1 of R * (CP 2 ) presents V CP 2 as a subvariety of A 3 , and it follows from Corollary 4.10 that V CP 2 is contained in the union of the plane P := {4κ p 2 1 − 7κ ep1 = 0} and the line L := {κ p 2 1 − 2κ ep1 = 0, 316κ 3 ep1 − 343κ p 4 1 = 0}. Furthermore, it follows from the calculation of Section 4.4.3 that V CP 2 contains P, so the variety V CP 2 is either P or P ∪ L. It would be extremely interesting if L ⊂ V CP 2 , but no method for showing this seems to be available. (Each circle action on CP 2 gives a homomorphism R * (CP 2 )/ √ 0 → Q[x 1 ] and hence a morphism A 1 → V CP 2 , but by Remark 4.14 all such morphisms have image in P.) Similarly, by the calculation of Section 4.4.1 the four elements e, p 1 , κ p 2 1 , and κ ep1 generate R * (CP 2 , * )/ √ 0, which presents the associated variety V (CP 2 , * ) as a subvariety of A 4 . Eliminating the variable κ p 4 1 from the radical ideal described in Section 4.4.1 shows that V (CP 2 , * ) is contained in the union of the plane {4κ p 2 1 − 7κ ep1 = 0, κ ep1 − 4p 1 + 4e = 0} and the lines {κ p 2 1 − 2κ ep1 = 0, e = 0, κ ep1 − 7p 1 = 0} {κ p 2 1 − 2κ ep1 = 0, 2κ ep1 − 7e = 0, 5κ ep1 − 7p 1 = 0}. It follows from the calculation of Section 4.4.3 that the plane is contained in V (CP 2 , * ) . 4.5. The manifold S 2 × S 2 . The cohomology of S 2 × S 2 is supported in even degrees. Thus by Theorem A the algebra R * (S 2 × S 2 ) is finitely-generated and R * (S 2 × S 2 , * ) is a finite R * (S 2 × S 2 )-module.