Stable moduli spaces of high dimensional manifolds

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1 of high dimensional manifolds University of Copenhagen August 23, 2012 joint work with Søren Galatius

2 Moduli spaces of manifolds We define W g := # g S n S n, a closed 2n-dimensional smooth manifold. When n = 1 this is an orientable surface of genus g; we consider W g to be the correct high-dimensional analogue of this. The topological group D n g := Diff(W g, D 2n ), of those diffeomorphisms of W g which are the identity on a fixed disc D 2n W g. Question: What is the cohomology of the classifying space BD n g? or less ambitiously, (a) How does H (BD n g ) vary with g? (b) What is the stable cohomology lim g H (BD n g )?

3 Modular interpretation of BD g Question: Why are we interested in BD n g? Because it is a classifying space for smooth W g -bundles (with a trivialised sub-disc-bundle), i.e. for a smooth manifold X, there is a bijection [X, BD n g ] { Wg E π X smooth fibre bundle, X D 2n E trivial sub-bundle } /iso. given by pulling back the universal bundle ED n g D n g W g BD n g. So H (BD n g ) is the ring of characteristic classes of such bundles.

4 The classes κ c Given a smooth oriented fibre bundle π : E k+2n X k, we call T π := Ker(Dπ : TE TX ) the vertical tangent bundle; it is an oriented 2n-dimensional vector bundle over E. If c H i+2n (BSO(2n)) is a characteristic class of such vector bundles, we define κ c (π) := π! (c(t π )) H i (X ), the generalised MMM-class corresponding to c. Applying this to the universal bundle, we obtain classes κ c H i (BD n g ).

5 The Mumford conjecture When 2n = 2, we have the calculation H (BSO(2); Z) = Z[e] so the only characteristic classes available are powers of the Euler class e. In this case it is traditional to write κ i := κ e i+1 H 2i (BDg 1 ; Z), and call it the ith Mumford Morita Miller class. Theorem (Mumford conjecture, Madsen Weiss) The map is an isomorphism. Q[κ 1, κ 2, κ 3,...] lim g H (BD 1 g ; Q) By work of Harer, Ivanov, and Boldsen, H (BD 1 g ) is known to be independent of g as long as 3 2g 2. So we can rephrase the theorem as shown. In fact, Madsen and Weiss proved a stronger integral statement: we will return to this later.

6 A high dimensional extension The most basic form of the result I wish to present is completely analogous to Madsen and Weiss theorem. Theorem (Galatius R-W) Let 2n > 4, and B H (BSO(2n); Q) be the subset of monomials in the classes e, p n 1, p n 2,..., p n+1 of total degree greater than 2n. Then the map 4 α : Q[κ c c B] H (BD n g ; Q) is an isomorphism in degrees 2 g 4. This is really two results: firstly that H (BD n g ; Q) is independent of g in degrees 2 g 4, and secondly an identification of the cohomology in this stable range. Both parts have an integral refinement, which I will discuss in a moment. Berglund and Madsen have also proved that H (BD n g ; Q) is independent of g, in degrees 2 min(2n 6, g 6).

7 A remark on relations Call the tautological subring. R (BD n g ; Q) := Im(α ) H (BD n g ; Q) When n = 1 this is its usual name in Algebraic Geometry, and it has been studied in great detail there, by Faber, Looijenga, Pandharipande and others. By Teichmüller theory, BD 1 g is closely related to Riemann s moduli space, so has a finite-dimensional model, and the cohomology and tautological ring of BD 1 g are clearly finitely generated. It is not known if BD n g can admit a finite-dimensional model for n > 1, but Theorem (I. Grigoriev) If n is odd, then the ring R (BD n g ; Q) is finitely generated. He finds many explicit relations among the κ c in H (BD n g ; Q), generalising a method of Morita (and R-W).

8 A remark on smoothing theory By smoothing theory, the homotopy fibre of the map f : BDiff(W g, D 2n ) BHomeo(W g, D 2n ) is equivalent to the space S of tangential smoothings, i.e. lifts D 2n W g BO(2n) BTOP(2n). It is somewhat understandable. We find that π 0 (S) is a finite set, S π 0 (S) is a rational homotopy equivalence in the range (2n/6) 7, (using Farrell Hsiang). Thus in this range f looks like a finite covering space to rational cohomology, so f is injective.

9 A remark on smoothing theory (cont.) On the other hand, it is not difficult to see that the classes κ c may be defined in H (BHomeo(W g, D 2n ); Q), by the existence of rational topological Pontryagin classes, so f is surjective in degrees 2 g 4. Corollary The map f : BDiff(W g, D 2n ) BHomeo(W g, D 2n ) induces an isomorphism on rational (co)homology in degrees 6 min(2n 42, 3g 12). Question: What should one make of this?

10 The integral statement Let Gr 2n (R N ) denote the grassmannian of 2n-planes in R N, γ its canonical 2n-plane bundle, and γ its canonical (N 2n)-plane bundle. Let θ N : Gr 2n (R N ) n Gr 2n (R N ) be the map from the n-connected cover. The sequence forms a spectrum, called MT θ n. Theorem (Galatius R-W) There is a map Th ( θ N(γ ) Gr 2n (R N ) n ) α : BD n g Ω 0 MT θ n which induces a homology isomorphism in degrees 2 g 4. Implicitly, the integral cohomology H (BD n g ; Z) is independent of g in degrees 2 g 4.

11 A remark on first homology The previous theorem reduces the calculation of the integral (co)homology of BDg n in the stable range to that of Ω 0 MT θn. In general this will still be difficult, but we can make the following calculation. Let Γ n g,1 := π 0(Dg n ) be the mapping class group of W g fixing the disc D 2n W g. A construction of Wall associates a quadratic form q W n g to the manifold Wg n, and so a homomorphism f : Γ n g,1 Aut(q Wg n). Let Ω n be the bordism theory associated to the map BO n BO. For ϕ Γ n g,1, the mapping torus T ϕ represents an element of Ω n 2n+1, which defines a homomorphism t : Γ n g,1 Ω n 2n+1. Theorem (Galatius R-W) The map f t : Γ n g,1 H 1(Aut(q W n g ); Z) Ω n 2n+1 is the abelianisation, for 2n 4 and g 6. 2

12 A remark on first homology (cont.) (Z/2) 2 n even For g 6, one calculates H 1 (Aut(q W n g ); Z) = 0 n = 1, 3, 7 Z/4 else A table of these groups in low degrees is as follows. n H 1 (Γ n g,1 ; Z) 0? 0 0 (Z/2)4 Z/4 Z/3 (Z/2) 2 Z/2 In general, Ω n 2n+1 Coker(J) 2n+1.

13 What is the theorem really about? The integral version of the theorem can also be stated as a map hocolim g BDn g Ω 0 MT θ n being a homology equivalence. Let W n g,1 = W n g \ D 2n, and W n := (W n 0,1 W n 1,1 W n 2,1 ). Then hocolim g BD n g is just BDiff c (W n ), the classifying space of the group of compactly supported diffeomorphisms of the non-compact manifold W n. There is a generalisation of the theorem from W n to certain other non-compact manifolds which are constructed in a similar way: We start with a manifold W (= W0,1 n ), which is highly connected relative to its boundary. Then we glue to it a sequence of cobordisms K i (= S n S n \ 2D 2n ) which are highly connected relative to both ends (which happened in this case to all be equal).

14 Tangential structures We fix a map θ : B BO(2n). We will now consider manifolds with a θ-structure on their tangent bundles, i.e. with a bundle map l W : TW θ γ. Definition Say θ is spherical if any θ-structure on D 2n extends to a θ-structure on S 2n. Examples: Any fibration pulled back from BO(2n + 1), such as as well as e.g. BU(3) BO(6). Non-example: EO(2n) BO(2n). BSO(2n) BO(2n) BSpin(2n) BO(2n) X BO(2n) BO(2n)

15 The space of highly connected nullbordisms Let P R be a closed (2n 1)-manifold, and l P : ε 1 TP θ γ be a θ-structure on its once-stabilised tangent bundle. Define N θ (P, l P ) to be the space of 2n-dimensional θ-manifolds W (, 0] R with boundary (P, l P ), such that (W, P) is (n 1)-connected. There is a homotopy equivalence N θ (P, l P ) [W ] Bun (TW, θ γ) /Diff(W, P) where the coproduct is over all manifolds W with boundary P, which are (n 1)-connected relative to their boundary, and Bun (TW, θ γ) is the space of bundle maps which restrict to l P on the boundary. This can be taken to be a definition.

16 Universal θ-ends Definition A universal θ-end is a 2n-dimensional θ-manifold (K, l K ) equipped with a proper map x 1 : K [0, ) such that (i) The natural numbers are regular values of x 1 ; we write K [a,b] for x 1 1 ([a, b]) when a and b are natural numbers, (ii) Each cobordism K [i,i+1] is (n 1)-connected relative to either end, (iii) For any θ-cobordism W : K i P which is (n 1)-connected relative to either end, there is an embedding of θ-manifolds j : W K [i, ) relative to K i. (iv) N θ (K 0, l K 0 ).

17 The most general statement Theorem (Galatius R-W) Let 2n > 4, θ be spherical, and (K, l K ) be a universal θ-end. Then there is a homology equivalence hocolim i N θ (K i, l K i ) Ω MT θ, where θ : B B BO(2n) is the (n 1)st stage of the Moore Postnikov tower for the map l K : K B. For more details see Søren Galatius and of high dimensional manifolds, 2012, arxiv: Søren Galatius and Homological stability for moduli spaces of high dimensional manifolds, 2012, arxiv:

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