Diarmuid Crowley

PRINCIPAL BUNDLES AND THE DIXMIER DOUADY CLASS

Abstract:A systematic consideration of the problem of the reduction and “lifting” of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the Dixmier-Douady class in various contexts including string structures, bundles and other examples motivated by considerations from quantum field theory.

The Gromoll filtration, KO–characteristic classes and metrics of positive scalar curvature

Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let Pos(X) be the space of all such metrics. For any g in Pos(X), Hitchin used the KO-valued alpha-invariant to define a homomorphism A_{n-1} from \pi_{n-1}(Pos(X) to KO_{m+n}. He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0 if m = 8k-1 or 8

New invariants of G2–structures

. In this paper we use Hitchins methods and extend these results by proving that A_{8j+1-m} is not 0 whenever m>6 and 8j - m >= 0. The new input are elements with non-trivial alpha-invariant deep down in the Gromoll filtration of the group \Gamma^{n+1} = \pi_0(\Diff(D^n, \del)). We show that \alpha(\Gamma^{8j+2}_{8j-5}) is not 0 for j>0. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

A classification of S3-bundles over S4

We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu. We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and diffeomorphism. The class of a G_2-structure is determined by nu on its own when the greatest divisor of p_1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G_2-manifolds. We also prove that the parametric h-principle holds for coclosed G_2-structures.

Kreck-Stolz invariants for quaternionic line bundles

We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller that each of these total spaces admits metrics with nonnegative sectional curvature.

The topology of Stein fillable manifolds in high dimensions I

We generalise the Kreck-Stolz invariants s2 and s3 by defining a new invariant, the t-invariant, for quaternionic line bundles over closed spin- manifolds M of dimension 4k − 1 with H 3 (M;Q) = H 4 (M;Q) = 0. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism and detects exotic homeomorphisms between such manifolds. The t-invariant also provides information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HP k . The t-invariant for S 4k−1 is closely related to the Adams e-invariant on the (4k − 5)-stem.

The rational classification of links of codimension > 2

We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashbergs h-principle for Stein manifolds in the setting of Krecks modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

A CLASSIFICATION OF SMOOTH EMBEDDINGS OF FOUR-MANIFOLDS IN SEVEN-SPACE, II

Abstract. Let m and be positive integers. The set of links of codimension , , is the set of smooth isotopy classes of smooth embeddings . Haefliger showed that is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. . For and for restrictions on the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group in general. In particular we determine precisely when is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.

Stably diffeomorphic manifolds and l 2q+1(ℤ[π])

Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12. This group acts on E7(N) by an embedded connected sum. Boechat and Haefliger constructed an invariant ℵ: E7(N) → H2(N;ℤ) which is injective on the orbit space of this action; they also described im(ℵ). We determine the orbits of the action: for u ∈ im(ℵ) the number of elements in ℵ-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2. The proof is based on Krecks modified formulation of surgery.

The additivity of the ρ–invariant and periodicity in topological surgery

Abstract The Kreck monoids l 2q+1(ℤ[π]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall surgery obstruction groups, . In this paper we identify l 2q+1(ℤ[π]) as the edge set of a directed graph with vertices a set of equivalence classes of quadratic forms on finitely generated free ℤ[π] modules. Our main theorem computes the set of edges l 2q+1(υ, υ′) ⊂ l 2q+1(ℤ[π]) between the classes of the forms υ and υ′ via an exact sequence Here sbIso(υ, υ′) denotes the set of “stable boundary isomorphisms” between the algebraic boundaries of υ and υ′. As a consequence we deduce new classification results for stably diffeomorphic manifolds.