An upper bound for the pseudoisotopy stable range

We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n-2)$$\end{document}(2n-2). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, and prove their non-triviality. We also explain how similar methods show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Top}(2n)/\mathrm {O}(2n)$$\end{document}Top(2n)/O(2n) is rationally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4n-5)$$\end{document}(4n-5)-connected.

To explain our approach, let W g,1 := # g S n × S n \int(D 2n ) with 2n ≥ 6, and consider the fibration sequence from the classifying space of the group of diffeomorphisms of W g, 1 to the classifying space of the group of block diffeomorphisms of W g, 1 . The rational cohomology of BDiff ∂ (W g,1 ) has been computed for g 0 by Galatius and the author in [12,13]; the rational cohomology of B Diff ∂ (W g,1 ) has been computed for g 0 by Berglund and Madsen in [1,2] and in a forthcoming revision of [2]. Ebert and the author have shown in [8] that the map i is surjective on rational cohomology in the stable range.
Our approach to Theorem A is motivated by forthcoming work of Berglund and Madsen, in which they show that the map induced by i on rational cohomology is injective in degrees * < 2n and g 0, and more importantly for our current purpose they show that this is sharp, in the following sense. This has implications for the Serre spectral sequence of (0.2), and it is this that we shall exploit to prove Theorem A. As Proposition B is central to our argument, and its proof is not yet available, in Sects. 2 and 3 we will give an independent proof of it, which works for all g ≥ 1 and does not require the computation of both groups. It consists of defining Mumford-Morita-Miller classes for block bundles, which extend those that we have already defined with Ebert in [8], and then showing that a certain such class-namelyκ e 2 −κ p n , which is easily seen to lie in the kernel (0.3)-is not trivial. The construction of these classes and their non-triviality may be of interest independently of Theorem A.
Finally, in Sect. 4 we show how similar methods can be used to show that the space Top(2n)/O(2n) is rationally (4n − 5)-connected as long as 2n > 4.

Proof of Theorem A
By the work of Weiss-Williams [22,Theorem A], there is a certain map is the 1-connected cover of the (smooth) Whitehead spectrum Wh Diff (W g,1 ), which in turn is related to Waldhausen's algebraic K -theory of spaces by a (split) cofibre sequence of spectra This identification requires the stable parameterised h-cobordism theorem [20]. Our strategy is then as follows. We use a theorem of Hsiang-Staffeldt to compute the spectrum cohomology H * (Wh Diff (W g,1 ); Q) in degrees * ≤ 2n. We take care to compute this as a representation of the mapping class group g,1 of W g,1 , in terms of the standard representation is then given by truncating, desuspending, and taking Z/2-invariants, and the cohomology of 1 )) is the free graded-commutative algebra on the result. We now suppose for a contradiction that φ(2n) ≥ 2n − 1, so the map (1.1) is 2nconnected and hence we have a computation of the rational cohomology of 1 ) in degrees * ≤ 2n − 1, as a g,1 -module. We then study the Serre spectral sequence for (0.2), and derive a contradiction.

Rational homology of the Whitehead spectrum
We shall use Corollary 1.2 of Hsiang-Staffeldt [15], which shows that where K is a minimal model for the dga C * ( W g,1 ; Q),K denotes the augmentation ideal, which inherits the structure of a graded Lie algebra with bracket given by [x, y] := x · y − (−1) |x|·|y| y · x, andK ab =K /[K ,K ] is the abelianisation of this graded Lie algebra.
As W g,1 is a suspension, the homology of W g,1 is the tensor algebra on the vector space H g [n − 1]. In particular it is a free (non-commutative) algebra, so is quasiisomorphic to C * ( W g,1 ; Q), and we may take K = H * ( W g,1 ; Q) with trivial differential. It follows thatK ab is the augmentation ideal of the free graded commutative algebra on H g [n − 1], that is Let us write if n is even.
Then we have in degrees * ≤ 2n. Applying the cofibre sequence (1.2), we obtain in degrees * ≤ 2n. The rational homology of Wh Diff s (W g,1 ) is therefore the same, as it is already 1-connected. Thus, dualising, we have in degrees * ≤ 2n −1, for some involution. It follows from Farrell-Hsiang [10] (which considers the case g = 0) that this involution acts as −1 onK * −1 (Z) ⊗ Q, so this summand does not contribute to the invariants. Thus in degrees * ≤ 2n − 1, for some involution on U . Taking the free graded-commutative algebra on this, it follows that in degrees * ≤ 2n − 1.

The Serre spectral sequence argument
The Serre spectral sequence for the fibration (0.2) takes the form Proof The action of g,1 on H n (W g,1 ; Z) preserves the intersection form, determining a homomorphism This is onto if n is even or n = 1, 3, 7, but for the remaining odd n its image is the finite-index subgroup-often denoted g (1, 2) ≤ Sp 2g (Z) in the theory of theta functions-of those symplectic matrices which preserve the standard quadratic form, cf. [2,Example 4.2]. Let us write G for the algebraic group O g,g or Sp 2g , depending on the parity of n, and A g ≤ G(Z) for the image of this homomorphism. As Sp 2g and SO g,g are connected semisimple algebraic groups defined over Q, it follows from a theorem of Borel-Harish-Chandra [5, Theorem 7.8] that A g is a lattice in G(R), and hence by the Borel Density Theorem [3] that A g is Zariski dense in G(R), so also in G(C).
Consider the fibration sequence where B Tor g,1 is defined to be the homotopy fibre. By [2, Proposition 4.1] we have is an isomorphism, and if n ≡ 3 mod 4 then we have an exact sequence In the case n ≡ 3 mod 4, n is odd and Zariski density of Sp 2g (C) and so vanishes by standard invariant theory (for which we refer to [11, §F.2]).
It remains to show that H 1 (A g ; U ) = 0. The representation U is arithmetic, so a theorem of Borel [4,Theorem 1] can be used to identify this with H 1 (A g ; Q) ⊗ U A g as long as g 0; see [9, Proposition 3.9] for a statement of this result adapted to our situation. Hence it is enough to show the vanishing of U A g .
If n is odd then U A g is Sym 2 (H g ) A g , whose complexification is the same as Sym 2 (H g ⊗ C) Sp 2g (C) by Zariski density, and this vanishes by standard invariant theory. If n is even then U A g is ∧ 2 (H g ) A g , whose complexification is ∧ 2 (H g ⊗ C) O g,g (C) , which also vanishes by standard invariant theory (noting that O g,g (C) ∼ = O 2g (C)).

Characteristic classes of block bundles
We should like to give a proof of Proposition B, as it does not require the entire corpus [1,2,12,13] and beyond to see that the kernel (0.3) is non-trivial. We shall show that this kernel is non-trivial by producing an explicit element in it, which will be described in terms of generalised Mumford-Morita-Miller classes. If (π : E → |K |, A) is a smooth oriented block bundle with fibre a closed d-manifold M (we refer to [8, Section 2] for this notation), in [8, Section 3] Ebert and the author have associated to it arises from a smooth fibre bundle then these data reduce to those coming from the bundle structure. In the case d = 2n, we then employed the following ruse: If π came from a smooth fibre bundle with 2n-dimensional fibres, so there was an unstable vertical tangent bundle T π E, then we would have e(T π E) 2 . Therefore, for a monomial p I in Pontrjagin classes, if we definẽ then these classes restrict to the usual κ p I and κ ep I on fibre bundles, and these give all generalised Mumford-Morita-Miller classes on fibre bundles. By way of apology for this ruse, we add to the list above (iv) an Euler class e(T π E) ∈ H d (E; Z).
(Of course e(T π E) is merely notation: there is no d-dimensional bundle T π E of which it is the Euler class.) Using this Euler class, we may then definẽ The symbolκ ep I has the same meaning as before, by Lemma 2.2 (iv) below. The existence of this Euler class is a consequence of the Fibre Inclusion Theorem of [7] (or rather its proof, which constructs a canonical such class), and the fact that the homotopy fibre of π is homotopy equivalent to a Poincaré duality space of dimension d, namely M [8, Proposition 2.8]. As the construction is quite pretty, let us describe it.
Construction 2.1 Embed |K | into R k for some k 0, and let B be a closed regular neighbourhood, so that there is a retraction r : B → |K |. Let B = D(B ) be the double of B , a closed smooth manifold. This has a retraction s : D(B ) → B , and let p : X → B be the Hurewicz fibration obtained by turning π into a fibration π f : E f → |K | and pulling it back along rs. As B and the fibre of p are Poincaré duality spaces, of dimensions k and d respectively, X is too [14], of dimension (d +k). But X × B X = p * (X ) → X is also a fibration over a Poincaré duality space with Poincaré duality fibre, so is again a Poincaré duality space, of dimension (2d + k).
Writing : X → X × B X for the fibrewise diagonal map, which admits an umkehr map ! as source and target are both Poincaré, we define We then define e(T π E) by restriction along It is easy to see that the class so obtained is independent of all choices, and it is shown in [7, §4] that it restricts to the Euler class on the fibre M. The definition given in [7, §4] seems to differ by a sign, but it does not, by Lemma 2.2 (i) below.

Lemma 2.2
The Euler class defined enjoys the following properties: For (ii), note that if (π : E → |K |, A) arises from a smooth fibre bundle then in Construction 2.1 we do not need to replace it by a fibration. The resulting p : X → B is a smooth fibre bundle with vertical tangent bundle T p X , and the map : X → X × B X is a smooth embedding with normal bundle T p X . Hence * ! (1) is the Euler class of T p X , which restricts to the Euler class of T π E. For (iv), we must involve ourselves in the details of the construction of the transfer in [7], with which we assume the reader is familiar. We begin by constructing a commutative diagram In this diagram, B is a Poincaré duality space and p is a Hurewicz fibration with fibre F M d (obtained as in Construction 2.1). W is a smooth oriented manifold of dimension (d + ) with boundary, which is homotopy equivalent to M, and p is a smooth fibre bundle (obtained from the Closed Fibre Smoothing Theorem of [7]). The map p is obtained as the fibrewise double of p , and is a smooth oriented fibre bundle with closed fibres. Finally, the horizontal arrows express each left-hand space as a (fibrewise) retract of the right-hand space. For a fibration p : S → T with fibre homotopy equivalent to a finite CW complex, and a fibrewise map f : S → S, let us write trf * p, f : H * (S) → H * (T ) for the associated transfer map. This is the map denoted τ f in [7]. When f = Id S , we shorten this to trf * p . By the definition of the transfer in [7, §6], we have trf * p = trf * p ,vuts s * t * . By the construction of the transfer for smooth fibre bundles in [7, §5], if we write , which we may write as p ! ((ts) ! (δ * (d ! (1)) · −), so we will be done if (ts) ! (δ * (d ! (1))) is equal to the class e(T p X ) defined by Construction 2.1. Consider the homotopy cartesian squares of Poincaré duality spaces, to which Lemma 2.3 below applies and shows that (The signs can be determined by restricting each square to a single fibre over B.) Thus, writing 1 = (ts) * (1), we have which is e(T p X ), as required. The sign ambiguity is unavoidable under the given hypotheses: changing the orientation of B, say, does not change g ! u * , but changes v * f ! by a sign.

Lemma 2.3 Consider a homotopy cartesian square
Proof Let us write a for the formal dimension of A, and so on. We assume some familiarity with the notion of Poincaré embeddings, for which we refer to [17] for details. It is enough to prove the identity for the larger square By this device, we may suppose [17,Lemma 3.1] that f admits the structure of a Poincaré embedding, with complement K and normal spherical fibration ξ of dimension (d − b − 1). Let u * ξ → A denote the pulled back spherical fibration, and v * K → C denote the homotopy pullback of the map K → D along v. There is then a homotopy commutative cube in which the bottom face is homotopy cocartesian, and the vertical faces are all homotopy cartesian. It follows by Mather's Second Cube Theorem [18,Theorem 25] that the top face is also homotopy cocartesian. We therefore have a map by collapsing v * K , and similarly for K . This gives a homotopy commutative diagram

Proof of Proposition B
We can extend the definition of the classesκ e i p I to block bundles having fibres W g,1 by filling in a disc in each fibre, giving a new block bundle with fibre W g := W g,1 ∪ ∂ D 2n = # g S n × S n . There are therefore defined universal characteristic classesκ e i p I ∈ H * (B Diff ∂ (W g,1 ); Q), by the proof of [8,Theorem 3.4].
In particular, we have a classκ e 2 −κ p n ∈ H 2n (B Diff ∂ (W g,1 ); Q) which vanishes in H 2n (BDiff ∂ (W g,1 ); Q), because e 2 = p n on the total space of a smooth fibre bundle. Proposition B is an immediate consequence of the following. Proposition 3.1 For each g ≥ 1 and each n ≥ 3 there is a block bundle (π : E → |K |, A) with fibre W g,1 , such that .
Proof From Lemma 2.2 (iii) it follows that theκ e i vanish for all i > 0 on all fibre homotopically trivial block bundles. We will therefore construct π to be fibre homotopically trivial, guaranteeing thatκ e 2 (π ) = 0.
We will use the (space-level) surgery fibration of Quinn [19], which following the discussion in [1, Section 3.2], in particular equation (43), may be put in the form hAut ∂ (W g,1 ) (1) .
Thus to construct a fibre homotopically trivial block bundle over B (with some triangulation) it is enough to give a map f : B → map * (W g,1 /∂ W g,1 , G/O) (1) and a nullhomotopy of σ • f . For simplicity of exposition we restrict to the case n = 2k. We let B = S n × S n , write a, b ∈ H n (B; Q) for a hyperbolic basis, and write e 1 , f 1 , . . . , e g , f g ∈ H n (W g,1 , ∂W g,1 ; Q) for a hyperbolic basis. Write the nth Hirzebruch L-polynomial as L n = Ap n + Bp 2 n/2 modulo other Pontrjagin classes, for some constants A and B. It is well-known that A = 0, and less well-known but true [21, Lemma A.1] that B = 0.
As the composition has homotopy fibre with finite homotopy groups, we claim that may find a map f whose adjointf : which has p n/2 (ξ ) = C · (a ⊗ e 1 + b ⊗ f 1 ), p n (ξ ) = − 2BC 2 A · a · b ⊗ e 1 · f 1 , and all other rational Pontrjagin classes zero, for some constant C = 0. To establish this claim, let the map classify the pair of relative cohomology classes L · (a ⊗ e 1 + b ⊗ f 1 ) and − 2B L 2 A · a · b ⊗ e 1 · f 1 , for some integer L = 0 large enough that these classes are integral. For each N > 0 consider the map φ N : then admits a dotted liftf for N large enough, as the universal obstructions to finding such a lift lie in the cohomology of i K (Z, 4i) with finite coefficients, and are therefore annihilated (on each skeleton) by some φ N . The resulting mapf gives p n/2 (ξ ) = L · N n/2 · (a ⊗ e 1 + b ⊗ f 1 ), p n (ξ ) = − 2B L 2 A · N n · a · b ⊗ e 1 · f 1 , and all other Pontrjagin classes zero, as required (with C = L · N n/2 ).
We must show that the composition (1) is nullhomotopic, but we shall allow ourselves to precompose f with self-maps k N : S n × S n → S n × S n having degree N = 0 on both factors (such a precomposition preserves the form of Pontrjagin classes which has been arranged above). With this in mind, it is enough to show that This group may be identified with H 4 * (B; Q). If n ≡ 0 mod 4 then the component of degree n = 2k = 4 is identified with the Künneth factor of in H 4 (B; Q) ∼ = H 4 (B; Q) ⊗ H 8 (W g,1 , ∂W g,1 ; Q). But L 3 (ξ ) = 0 by observation, as only p 4 (ξ ) and p 2 (ξ ) are non-zero. Whatever the class of n modulo 4, the component of degree 2n = 4k is identified with the Künneth factor of in H 4k (B; Q). But by construction We therefore obtain a map f , with σ • f nullhomotopic and i •f classifying a vector bundle ξ having p n/2 (ξ )  [8,Lemma 3.3]) and sõ It is not difficult to adapt the above argument to work for n = 2k + 1. The essential point is that if we write L n = Ap n + Bp n−1

Rational connectivity of Top(2n)/O(2n)
Our goal in this section is to show how similar techniques to those we have been using imply the following.
Proof of Theorem 4.1 Let W g = # g S n × S n \int(D 2n ), with 2n ≥ 6, and choose a collar [0, 1) × ∂ W g,1 ⊂ W g,1 and a disc D 2n ⊂ (0, 1) × ∂ W g,1 . The map is (2n − 4)-connected, by Morlet's lemma of disjunction [6,Corollary 3.2]. Consider the fibration Up to isotopy any diffeomorphism ϕ representing an element of π 1 (B Diff ∂ (W g,1 )) may be supposed to be equal to the identity on the collar: the map (4.1) is then preserved by that induced by ϕ, and it then follows that ϕ acts trivially on H * Diff ∂ (W g,1 ) Diff ∂ (W g,1 ) ; Q in the range of degrees * ≤ 2n − 4 where the map (4.1) is a cohomology injection. Hence the Serre spectral sequence H p B Diff ∂ (W g,1 ); H q Diff ∂ (W g,1 ) Diff ∂ (W g,1 ) ; Q ⇒ H p+q (BDiff ∂ (W g,1 ); Q) associated to this fibration has a product structure in this range of degrees. But Berglund-Madsen have shown that the map i * : H * (B Diff ∂ (W g,1 ); Q) −→ H * (BDiff ∂ (W g,1 ); Q) is an isomorphism in degrees * ≤ 2n − 1 as long as g 0. This result will appear soon in a revision of [2]. It follows that H q Diff ∂ (W g,1 ) Diff ∂ (W g,1 ) ; Q = 0 for 1 ≤ q ≤ 2n − 4, and hence by the map (4.1) that H q Diff ∂ (D 2n ) Diff ∂ (D 2n ) ; Q = 0 for 1 ≤ q ≤ 2n − 5. On the other hand, the surgery fibration sequence shows that hAut ∂ (D 2n ) is rationally acyclic, and hAut ∂ (D 2n ) * by the Alexander trick, so B Diff ∂ (D 2n ) is rationally acyclic. Boundary connect-sum makes this into an H -space, so it has trivial rational homotopy groups. Thus Diff ∂ (D 2n ) has finitely-many components, and each one is rationally acyclic. The group Diff ∂ (D 2n ) has the same components, and so the quotient Diff ∂ (D 2n ) is rationally homotopy equivalent to BDiff ∂ (D 2n ) 0 , the classifying space of the component of the identity in BDiff ∂ (D 2n ). It follows from the above that its rational cohomology, and hence rational homotopy, vanishes in degrees 1 ≤ * ≤ 2n − 5.