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On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes

Martin ČadekJiří Vanžura — 1998

Colloquium Mathematicae

Let ξ be an oriented 8-dimensional spin vector bundle over an 8-complex. In this paper we give necessary and sufficient conditions for ξ to have 4 linearly independent sections or to be a sum of two 4-dimensional spin vector bundles, in terms of characteristic classes and higher order cohomology operations. On closed connected spin smooth 8-manifolds these operations can be computed.

On 2-distributions in 8-dimensional vector bundles over 8-complexes

Martin ČadekJiří Vanžura — 1996

Colloquium Mathematicae

It is shown that the 2 -index of a 2-distribution in an 8-dimensional spin vector bundle over an 8-complex is independent of the 2-distribution. Necessary and sufficient conditions for the existence of 2-distributions in such vector bundles are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.

On S(2) and S(2) · S(1) structures in 8-dimensional vector bundles.

Martin CadekJirí Vanzura — 1997

Publicacions Matemàtiques

Let ξ be an oriented 8-dimensional vector bundle. We prove that the structure group SO(8) of ξ can be reduced to S(2) or S(2) · S(1) if and only if the vector bundle associated to ξ via a certain outer automorphism of the group Spin(8) has 3 linearly independent sections or contains a 3-dimensional subbundle. Necessary and sufficient conditions for the existence of an S(2)- structure in ξ over a closed connected spin manifold of dimension 8 are also given in terms of characteristic classes.

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