87 research outputs found
The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature
No full textThe derivative map for diffeomorphism of disks: an example
No full textWe prove that the derivative map , defined by taking the derivative of a diffeomorphism, can induce
a nontrivial map on homotopy groups. Specifically, for we prove that
the following homomorphism is non-zero: As a consequence we give a counter-example to a conjecture
of Burghelea and Lashof and so give an example of a non-trivial vector bundle
over a sphere which is trivial as a topological -bundle (the
rank of is and the base sphere is .)
The proof relies on a recent result of Burklund and Senger which determines
those homotopy 17-spheres bounding -connected manifolds, the plumbing
approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an
explicit construction of low-codimension embeddings of certain homotopy
spheres
Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets
No full textPrincipal bundles and the Dixmier Douady class
No full textA 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, Ures bundles and other examples motivated by considerations from quantum field theory.Alan L. Carey, Diarmuid Crowley, Michael K. Murra
On the mapping class groups of #(r) (S(p) x S(p)) for p=3, 7
No full textFor Mr := _r (Sp×Sp), p = 3, 7, we calculate π0Diff(Mr)/_2p+1 and E(Mr), respectively the group of isotopy classes of orientation preserving diffeomorphisms of Mr modulo isotopy classes with representatives which are the identity outside a 2p-disc and the group of homotopy classes of orientation preserving homotopy equivalences of Mr .Diarmuid J. Crowle
The topology of Stein fillable manifolds in high dimensions II
No full textWe continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling (W, \omega) with \omega(\pi _2(M))=0. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not.
Concerning obstructions to Stein fillings, we show that the (8k-1)-dimensional sphere has an almost contact structure which is not Stein fillable once k > 1. As a consequence we deduce that any highly connected almost contact (8k-1)-manifold (with k > 1) admits an almost contact structure which is not Stein fillable
Connected sum decompositions of high-dimensional manifolds
No full textThe classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions
Quaternionic line bundles on spin 7-manifolds
No full textWe study quaternionic line bundles over closed, connected, spin manifolds of dimensions 6 and 7. In dimension 6, the Puppe sequence of the pair (N, N^3) gives a complete classification of the set Bun(N) of quaternionic line bundles in terms of the cohomology of the base manifold N.In dimension 7, we consider two ways to decompose the base manifold M. First, we study the cell structure on M/M^2. Based on this, we show that the second Chern class c_2 : Bun(M) -> H^4(M;Z) is surjective. If M is simply connected and TH^3(M;Z) = 0, we obtain a partial description of the fibers over c_2 in terms of topological invariants of M. As second decomposition we consider Heegard type splittings of M. We show that the set Bun(M) can be described as a biquotient. Based on this description we see that the fibers over c_2 may differ in size
The smooth structure set of Sp x Sq
No full textAppears in this Special issue: Algebraic and Geometric Topology, in honor of Bruce Williams / Guest Edited by Bill Dwyer, John Klein and Shmuel WeinbergerDiarmuid Crowle
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