## Abstract

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any T^{k}-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

Original language | English (US) |
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Pages (from-to) | 75-86 |

Number of pages | 12 |

Journal | Communications in Contemporary Mathematics |

Volume | 2 |

Issue number | 1 |

State | Published - Feb 2000 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics