Showing papers on "Geometry and topology published in 1977"

Closed 2-forms and an embedding theorem for symplectic manifolds

Abstract: The existence of universal connections was shown by Narasimhan and Ramanan [5], and Kostant [3] showed that any integral closed 2-form is the curvature form of a connection on some circle bundle. These results can be combined to show the existence of a universal closed 2-form with integral periods. In this paper we will use the symplectic structure of a complex projective space to give an elementary proof of this result the precise statement is given in Theorem A. The result of Kostant is in fact a corollary of the existence of a universal closed 2-form, as is indicated below. Another immediate corollary of Theorem A is the result of Gromov [3] that closed symplectic manifolds can be symplecticalΓy immersed in CP, for large enough n see Theorem B. First we indicate why the proof which we are going to give here is a simple and natural generalization of an elementary fact about exact 2-forms. Consider the standard symplectic form Ω = Σιl=i dXidyt on R . Any exact 2-form on a manifold M can be induced from Ω by a mapping to R for some n, since any exact 2-form on M can be written in the form 2*=i dft A dgt, where /*, gi are real valued functions on M. CP has a symplectic structure Ωo which is locally given by Ωo = 2?=i dxi A dyt. Furthermore, CP n is the 2π-skeleton of an Eilenberg-MacLane space of type K(Z, 2). It is thus natural to expect that any closed 2-form with integral periods can be induced from Ωo by a map to CP, because there is some map to CP, for large n, which pulls back Ωo to within an exact 2-form of the given closed 2-form. The only complication that is met in CP to adjusting the map to account for the exact 2-form is that, unlike in R, the symplectic charts on CP have finite radius, so the fi9 g/s utilized would have to be bounded. The proof we give of Theorem A depends only on estimating the bounds on fi9 gt as n becomes large. A closed &-form on a manifold M will be said to be integral if its de Rham cohomology class is in the image of the canonical coefficient map H(M Z) -*H(M;R). Complex projective space CP has a Kahlerian structure, and we will denote its Kahler form by flj. The 2-form Ω% can be chosen to represent a generator in the image of H\\CP Z) -> H\\CP R), and we can assume that /*(βJ0 = Ωl where i is the standard inclusion of CP in CP,

Transformation groups : proceedings of the conference in the University of Newcastle upon Tyne, August 1976

Abstract: Part I: 1. Generators and relations for groups of homeomorphisms Herbert Abels 2. Affine embeddings of real Lie groups Nguiffo B. Boyom 3. Equivariant regular neighbourhoods Allan L. Edmonds 4. Characteristic numbers and equivariant spin cobordism V. Giambalvo 5. Equivariant K-theory and cyclic subgroups Stefan Jakowski 6. Z/p manifolds with low dimensional fixed point set Czes Kosniowski 7. Gaps in the relative degree of symmetry Hsu-Tung Ku and Mei-Chin Ku 8. Characters do not lie Arunas Liulevicius 9. Actions of Z/2n on S3 Gerhard X. Ritter 10. Periodic homeomorphisms on non-compact 3 manifolds Gerhard X. Ritter and Bradd E. Clark 11. Equivariant function spaces and equivariant stable homotopy theory Reinhard Schulz 12. A property of a characteristic class of an orbit foliation Haruo Suzuki 13. Orbit structure for Lie group actions on higher cohomology projective spaces Per Tomter 14. On the existence of group actions on certain manifolds Steven H. Weintraub Part II. Summaries and Surveys: 15. Proper transformation groups H. Abels 16. Problems on group actions on Q manifolds R. D. Anderson 17. A non-abelian view of abelian varieties L. Auslander, B. Kolb and R. Tolimieri 18. Non compact Lie groups of transformation and invariant operator measures on homogenous spaces in Hilbert space M. P. Heble 19. Approximation of simplicial G-maps by equivariantly non degenerate maps Soren Illman 20. Equivariant Riemann-Roch type theorems and related topics Jatsuo Kawakubo 21. Knots and diffeomorphisms M. Kreck 22. Some remarks on free differentiable involuetions on homotopy spheres Peter Loffler 23. Compact transitive isometry spaces Gordon Lukesh 24. A problem of Breson concerning homology manifolds W. J. R. Mitchell.