Tautological rings and stabilisation

We construct a ring homomorphism comparing the tautological ring, fixing a point, of a closed smooth manifold with that of its stabilisation by $S^{2a} \times S^{2b}$.


Introduction and statement of result
1.1. Tautological rings. For a connected closed oriented smooth d-manifold N , the universal smooth fibre bundle with fibre N may be described in terms of classifying spaces as Here Diff + (N ) denotes the topological group of orientation-preserving diffeomorphisms of N , and Diff + (N, ) denotes the subgroup of those diffeomorphisms fixing a marked point ∈ N . Assigning to a diffeomorphism fixing ∈ N its derivative at this point gives a map D : BDiff + (N, ) −→ BGL + d (R) BSO(d).
Using this we may pull back any cohomology class c ∈ H * (BSO(d); Q) to give a class on BDiff + (N, ), which we continue to denote by c. We may then define classes κ c := π c ∈ H |c|−d (BDiff + (N ); Q) by integration along the fibres of the map π. These are known as tautological classes, κ-classes, or generalised Miller-Morita-Mumford classes. If |c| = d then the degree zero cohomology class κ c is simply a characteristic number of N ; the higher degree κ c 's may be considered as analogues of characteristic numbers for families of manifolds.
The tautological ring R * (N ) ⊂ H * (BDiff + (N ); Q) is the subring generated by the classes κ c . We may pull the classes κ c back along π and hence also consider them as cohomology classes on BDiff + (N, ), where we continue to denote them by κ c . A variant of R * (N ), the tautological ring fixing a point R * (N, ) ⊂ H * (BDiff + (N, ); Q) is the subring generated by the classes κ c as well as the classes c.
Context. The rings R * (N ) have been extensively studied in the case that N is an oriented surface, as in this case BDiff + (N ) is a model for the moduli space of Riemann surfaces, cf. [Mum83,Loo95,Fab99,Mor03]. For manifolds of higher dimension they have recently been studied by Grigoriev, Galatius, and the author [Gri17, GGRW17,RW18], and in the case of 4-manifolds by Baraglia [Bar20]. In a different direction the vanishing of tautological classes for various aspherical manifolds has been shown by Bustamante, Farrell, and Jiang [BFJ16], and by Hebestreit, Land, Lück, and the author [HLLRW17]. A variant of tautological rings for Poincaré complexes rather than manifolds has been studied by Prigge [Pri19].
1.2. Main result. The main result of this note concerns the case d = 2(a + b), and gives an explicit ring homomorphism R * (N #S 2a × S 2b , ) → R * (N, ). This is rather surprising because-as far as we can tell-there is no corresponding map BDiff + (N, ) → BDiff + (N #S 2a × S 2b , ), even at the level of rational cohomology groups.
In order to state our result we must first explain our conventions for describing the classes c. When d = 2n we have H * (BSO(2n); Q) = Q[e, p 1 , p 2 , . . . , p n−1 ], the polynomial ring of the Euler class and Pontrjagin classes. There is a further Pontrjagin class, p n , which agrees with e 2 . Using this we may write any monomial in this ring as either p I or ep I , with I = (i 1 , i 2 , . . . , i r ) having 1 ≤ i j ≤ n and p I = p i1 · · · p ir . Theorem 1.1. Let N be a 2(a + b)-dimensional manifold. Then the formula gives a well-defined and surjective ring homomorphism.
subring generated by the classes κ c . There are natural ring homomorphisms sending κ c to κ c , and the second map sending c to 0. Considering Diff + (N, D d ) as the group of diffeomorphisms of N \ int(D d ) fixing the boundary, there are natural maps BDiff + (N, D d ) → BDiff + (N #M, D d ), and these induce ring homomorphisms sending κ c to κ c . Theorem 1.1 may be viewed as a refinement of this map which does not require an entire disc to be fixed, but only a point. (ii) Tautological rings can equally well be defined for homeomorphism groups of topological manifolds, but for our method smoothness is used in an essential way (we use that Diff + (R d ) is homotopy equivalent to a compact Lie group).
Our method can more generally be used to compare tautological rings of N and N #M when M is a 2n-manifold with an n-torus action satisfying certain cohomological hypotheses. In Section 2 we develop our construction in this generality, in Section 3 we verify the cohomological hypotheses in the case M = S 2a × S 2b , thereby proving Theorem 1.1, and in Section 4 we explain the analogous result in the case M = CP 2 .
Acknowledgements. The author was partially supported by the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 756444), and by a Philip Leverhulme Prize from the Leverhulme Trust.
2. The general construction 2.1. Parametrised connect-sum. Let (M, m 0 ) and (N, n 0 ) be d-dimensional connected manifolds with marked points, and choose charts ϕ M : R d → M and ϕ N : R d → N around these marked points.
More generally for a subset X ⊂ M \ ϕ M (R d ) let Diff + (M, ϕ M , X) denote the subgroup which fixes X pointwise.
These are just slightly unusual models for the group of diffeomorphisms fixing a point, as follows.
Proof sketch. Combine the facts (i) that the inclusion SO(d) → GL + d (R) into the space of invertible matrices with positive determinant is an equivalence, and (ii) that the space of Riemannian metrics on M is contractible.
Taking the derivative at the marked point m 0 = ϕ M (0) gives a homomorphism and similarly with n 0 = ϕ N (0) gives Let r : SO(d) → SO(d) be given by conjugating by a reflection: the induced map Br : BSO(d) → BSO(d) corresponds to reversing orientation. For compactness we write D m0 := r • D m0 . Using these maps we can form a homotopy pullback square The space G (for "glue") is equipped with the following data: Proof of Proposition 2.3. We form E M #N by gluing together To produce the cobordism data we first make a local construction. Let us write U for the inner product space R d with an orientation det(U ) ∈ Λ d U . Consider the elementary cobordism W loc between S 0 × D(U ) and D 1 × S(U ), realised SO(U )equivariantly by a codimension zero submanifold with corners W loc ⊂ R × U equipped with the Morse function f (z, u) = |u| 2 −z 2 and the orientation ∂ ∂z ∧det(U ). Remove the unit disc around (0, 0) from W loc to obtain W loc as shown in Figure  1, whose new boundary component is S(U ⊕ R). We consider W loc as a SO(U )equivariant oriented cobordism of manifolds with boundary As  The kernel of the differential Df defines a SO(U )-equivariant d-dimensional oriented subbundle τ of T W loc on the complement of the point (0, 0), restricting to the oriented tangent bundles of {−1} × D(U ) and D 1 × S(U ), and the oppositelyoriented tangent bundle over {+1} × D(U ), respectively, as these are level sets of f . The bundle τ restricts to an oriented vector bundle of the same name on W loc , and hence to an oriented vector bundle on S(U ⊕ R), which we now identify.
At the point (z, u) ∈ S(R ⊕ U ) the differential D (z,u) f : R ⊕ U → R is given by inner product with 2(−z, u), so its kernel is identified with the tangent space T (−z,u) S(R ⊕ U ) though with the opposite orientation. As (z, u) → (−z, u) gives a (orientation-reversing) diffeomorphism of S(R ⊕ U ) commuting with the SO(U )action, there is SO(U )-equivariant identification of oriented vector bundles between τ | S(R⊕U ) and T S(R ⊕ U ).
We implant this local construction as follows. Applying the local construction fibrewise the oriented orthogonal vector bundle V → G gives a bundle of cobordisms of manifolds with boundary over G, equipped with a vector bundle τ (V ) which agrees with the tangent bundle over the incoming and outgoing boundaries. We then construct W as This is a bundle of oriented cobordisms E M E N E M #N S(R⊕V ), and contains G × [0, 1] × X ⊂ [0, 1] × E M \ int(D(V )). The bundle τ (V ) extends to a vector bundle on W by taking the vertical tangent bundle of E M \ int(D(V )) E N \ int(D(V )). This establishes (ii).
As mentioned in the introduction, we use that e 2 = p n to write monomials in this ring as either p I or ep I , with I = (i 1 , i 2 , . . . , i r ) having 1 ≤ i j ≤ n. Proof. We have T q S(R ⊕ V ) ⊕ R ∼ = R ⊕ q * V and hence the cohomology classes p i (T q S(R ⊕ V )) = q * p i (V ) are pulled back from the base. Then q p I (T q S(R ⊕ V )) = q q * (p I (V )) = 0 by the projection formula (i.e. the fact that fibre integration is a map of modules over the cohomology of the base), and similarly Corollary 2.5. There are identities in H * (G; Q).
Proof. Consider the bundle of cobordisms W : E M E N E M #N S(R ⊕ V ) constructed in Proposition 2.3 (ii), with its oriented 2n-dimensional vector bundle which restricts to the vertical tangent bundles over the two ends. By Stokes' theorem, for any c ∈ H * (BSO(2n); Q) we therefore have The result follows by using Lemma 2.4.

Torus actions.
If the n-torus T acts on the 2n-manifold M fixing m 0 ∈ M and X = m 1 ∈ M , then by choosing ϕ M to be an equivariant orthogonal chart around m 0 (obtained for example by exponentiating with respect to a T -invariant Riemannian metric) we have homomorphisms We may then form the following commutative cube, in which the front face is (2.1), the map i T : BT → BSO(2n) is B(D m0 • φ), and the remaining faces are developed by taking homotopy pullbacks.
When we pull this back to BDiff + T (N, ϕ N ) the classes κ c (π N ) can be written as (i N ) * κ c , and the classes p I can be written as (i N ) * (D n0 ) * p I . The classes κ c (π M ) pulled back to BDiff + T (N, ϕ N ) may be written as (D T n0 ) * φ * κ c . Finally, Proposition 2.3 (ii) implies that Dm 1 commutes up to homotopy, which with the cube above shows that the composition To proceed we require D m0 • φ : T → SO(2n) to be injective, in which case it is the inclusion of a maximal torus.
Lemma 2.7. The homomorphism D m0 • φ is injective if and only if m 0 ∈ M is an isolated fixed point of this torus action.
Proof. If this homomorphism is not injective then its image is a torus of rank ≤ n−1, which may therefore be conjugated into the maximal torus of SO(2n − 1) ≤ SO(2n): in this case T fixes a tangent vector at m 0 , and since T acts linearly in a chart around this point it follows that m 0 is not an isolated fixed point.
Conversely, if m 0 is not an isolated fixed point then this torus fixes a non-zero vector in T m0 M , so lies in some SO(2n − 1), and hence cannot be injective (by dimension of the maximal torus of SO(2n − 1)).
Thus if m 0 is an isolated fixed point then the map D m0 • φ : T → SO(2n) is the inclusion of a maximal torus. We use this in the following way. The homotopy fibre of the map i N is then SO(2n)/T which has non-zero Euler characteristic (it is the order of the Weyl group of SO(2n)), and therefore just as in Example 1.4 the Becker-Gottlieb transfer shows that the ring homomorphism is injective. Thus: For these to hold we must impose conditions on the torus action on M . We do not try to pursue this in its greatest generality, and instead treat two special cases.

Proof of Theorem 1.1: Stabilisation by S 2a × S 2b
We consider S 2k = (R 2k ) + with the usual SO(2k)-action, and let the standard We write ζ for the corresponding oriented 2(a + b)-dimensional representation of T , and ζ for the same representation with opposite orientation. Proof. The T k -action on S 2k = (R 2k ) + fixes precisely 0 and ∞. The normal representation at 0 is the standard oriented representation T k ≤ SO(2k). The orientation-reversing reflection in the equator interchanges the fixed points 0 and ∞ and commutes with the T k -action, so the normal representation at ∞ is the opposite of the standard representation. Taking products gives the claimed description.
Proof. We will use the localisation theorem in T -equivariant rational cohomology t| xj e j .
As H * T is an integral domain this determines the unlocalised fibre-integration map. We apply this to the T -action on S 2a × S 2b . By Lemma 3.1, tangent representations at all fixed points are isomorphic to ζ, so have Pontrjagin classes p i (ζ), but taking orientation into account the Euler classes of the tangent representations at (0, 0) and (∞, ∞) are e(ζ) and at (0, ∞) and (∞, 0) are e(ζ) = −e(ζ). Thus we have As the oriented tangent representation at m 1 ∈ S 2a × S 2b is isomorphic but with opposite orientation to that at m 0 ∈ S 2a × S 2b , so that D m1 D m0 , we have . By the discussion above the formula is then a well-defined ring homomorphism, which is clearly surjective. This proves Theorem 1.1.
Remark 3.3. In this argument we could have chosen the fixed point m 1 = (∞, ∞), whose tangential T -representation is oriented isomorphic with that at m 0 . This has the disconcerting effect that the formula also gives a well-defined ring homomorphism, where c →c is the automorphism of H * (BSO(2n); Q) induced by conjugation by a reflection.