• Corpus ID: 119175819

Vector bundles and cohomotopies of spin 5-manifolds

  title={Vector bundles and cohomotopies of spin 5-manifolds},
  author={Panagiotis Konstantis},
  journal={arXiv: Geometric Topology},
The purpose of this paper is two-fold: On the one side we would like to close a gap on the classification of vector bundles over $5$-manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$-manifolds which are in $1-1$ correspondence to elements in the first cohomotopy group $\pi^4(M)=[M,S^4]$ of $M$. From previous results this group fits into a short exact sequence, which splits into $H^4(M;\mathbb Z)\oplus\mathbb Z_2$ if $M$ is spin. The second intent is to… 

A counting invariant for maps into spheres and for zero loci of sections of vector bundles

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $\pi^n(M)$ of $M$. Moreover it is known that

On Modular Cohomotopy Groups

. Let p be a prime and let π n ( X ; Z /p r ) = [ X, M n ( Z /p r )] be the set of homotopy classes of based maps from CW-complexes X into the mod p r Moore spaces M n ( Z /p r ) of degree n , where



Quaternionic Line Bundles over Quaternionic Projective Spaces

the bijection is given by the first Stiefel Withney and Chern classes, respectively. So far, the complete answer for the quaternionic case is still unknown, and as we will see in Section 2, a full

The principal fibration sequence and the second cohomotopy set

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] ->

Kreck-Stolz invariants for quaternionic line bundles

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

Borsuk's Cohomotopy Groups

In 1936 Borsuk2 [2] indicated how the homotopy classes of maps of a space X into a space Y form a group under rather general conditions. In particular, if Y is an n-sphere and X is a space with dim X


To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M,

Vector Fields on Manifolds

  • M. Atiyah
  • Mathematics
    Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen
  • 1970
This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of existence of r linearly independent vector fields. For r

Obstruction theory on 8-manifolds

This paper gives a uniform, self-contained, and fairly direct approach to a variety of obstruction-theoretic problems on 8-manifolds. We give necessary and sufficient cohomological criteria for the

Contact 5-manifolds with SU(2)-structure

We consider 5-manifolds with a contact form arising from a hypo structure [9], which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a


Generalized Killing spinors in dimension 5

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, -structures on 5-manifolds defined by a generalized Killing spinor. We prove that in