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On 2-cycles of B Diff ( S 1 ) which are represented by foliated S 1 -bundles over T 2

Takashi Tsuboi (1981)

Annales de l'institut Fourier

We give several sufficients conditions for a 2-cycle of B Diff ( S 1 ) d (resp. B Diff K ( R ) d ) represented by a foliated S 1 -(resp. R -) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms f , g of S 1 (resp. R ). If f , g have fixed points, we construct decompositions: f = π f i , g = π g i , where the interiors of Supp ( f i ) Supp ( g i ) are disjoint, and f i and g i belong either to { h i n ; n Z } ( h i Diff ) or to a one-parameter subgroup generated by a C 1 -vectorfield ξ i . Under some conditions on the norms...

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

Martin Čadek, Jiří 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 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes

Martin Čadek, Jiří 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 a secondary invariant of the hyperelliptic mapping class group

Takayuki Morifuji (2009)

Banach Center Publications

We discuss relations among several invariants of 3-manifolds including Meyer's function, the η-invariant, the von Neumann ρ-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.

On algebraic sets invariant by one-dimensional foliations on 𝐂 P ( 3 )

Marcio G. Soares (1993)

Annales de l'institut Fourier

We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on C P ( 2 ) without algebraic solutions to the case of foliations by curves on C P ( 3 ) . We give an example of a foliation on C P ( 3 ) with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.

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