Fuquan Fang

SMOOTH GROUP ACTIONS ON 4-MANIFOLDS AND SEIBERG–WITTEN INVARIANTS

In this paper we study the Seiberg–Witten invariants of 4-manifold acted on by a finite group (or a compact Lie group). Among other things, we have: Let X be a smooth closed 4-dimensional ℤp-manifold, where p is a prime. Suppose H1(X,ℝ) = 0 and where . If ℤp acts trivially on the space of self dual harmonic 2-forms, then, for any ℤp-equivariant Spinc-structure on X, the Seiberg–Witten invariant satisfies provided for j = 0,1,…,p-1, where , DA: Γ(W+) → Γ (W-) is the equivariant Dirac operator determined by for an equivariant connection A on detW+.

Curvature, diameter, homotopy groups, and cohomology rings

We establish two topological results. (A) IfM is a 1-connected compact n-manifold and q ≥ 2, then the minimal number of generators for the qth homotopy group πq(M), MNG(πq(M)), is bounded above by a number depending only onMNG(H∗(M,Z)) and q, whereH∗(M,Z) is the homology group. (C) Let M (n, Y ) be the collection of compact orientable n-manifolds whose oriented frame bundles admit SO(n)-invariant fibrations over Y with fiber compact nilpotent manifolds such that the induced SO(n)-actions on Y are equivalent. Then {πq(M) finitely generated,M ∈ M (n, Y )} contains only finite isomorphism classes depending only on n, Y , q. Together with the results of [CG] and [Gr1], from (A) we conclude that (i) if M is a complete n-manifold of nonnegative curvature, then MNG(πq(M)) is bounded above by a number depending only on n and q ≥ 2. Together with the results of [Ch] and [CFG], from (C) we conclude that (ii) ifM is a compact n-manifold whose sectional curvature and diameter satisfy λ ≤ secM ≤ and diamM ≤ d, then πq(M) has a finite number of possible isomorphism classes depending on n, λ, , d, q ≥ 2, provided πq(M) is finitely generated. We also show that (B) if M is a compact n-manifold with λ ≤ secM ≤ and diam(M) ≤ d , then the cohomology ring, H ∗(M,Q), may have infinitely many isomorphism classes. In particular, (B) answers some questions raised by K. Grove [Gro]. 0. Introduction A fundamental problem in Riemannian geometry is understanding the topological DUKE MATHEMATICAL JOURNAL Vol. 107, No. 1, c

FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

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 Tk-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.

On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Let (m_,m+) be the pair of multiplicities of an isoparametric hypersurface in the unit sphere Sn+1 with four distinct principal curvatures -w.r.g., we assume that m_ m_ > 1 and m+ = 3,5,6 or 7 (mod 8); (2) m+ > 2m> 2 and m+ = 0 (mod 4). This generalizes partial results of Wang (1988) about the topology of Clifford type examples. Consequently, the hypersurface is homeomorphic to an iterated sphere bundle under the above condition.

Complex immersions in Kähler manifolds of positive holomorphic

The main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in Kahler manifolds with positive holomorphic k-Ricci curvature. In particular this generalizes the classical Lefschetz hyperplane section theorem for projective varieties. As an immediate geometric application we prove that a complex immersion of an n-dimensional closed manifold in a simply connected closed Kahler m-manifold M with positive holomorphic k-Ricci curvature is an embedding, provided that 2n ≥ m + k. This assertion for k = 1 follows from the Fulton-Hansen theorem (1979).

k

Abstract Let X be a smooth closed spin 4-manifold with the first Betti number b1(X)=0 and signature σ(X)≤0. In this paper we use Seiberg-Witten theory theory to prove that (i) If X admits an odd type Z 2 p action preserving the spin structure, then b 2 (X)≥ 5 4 |σ(X)|+2p+2, provided b 2 + (X; Z 2 )≠b 2 + − 1 8 |σ(X)| , b + 2 (X; Z 2 p )>0 and in addition, b + 2 (X; Z 2 p ) 2 + if p≥2, where Z 2 ⊂ Z 2 p . (ii) If H 1 (X, Z 2 )=0 and X admits an even type involution τ preserving the spin structure, then the number of isolated fixed points of τ, say k, is divisible by 8 and satisfies that k≤8(b 2 + (X;τ)−1)+ 1 2 σ(X)

-Ricci curvature

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

Smooth group actions on 4-manifolds and Seiberg–Witten theory

Abstract. Let M be a Dupin hypersurface in the unit sphere

Positively curved 6-manifolds with simple symmetry groups

S^{n+1}

Topology of Dupin hypersurfaces with six distinct principal curvatures

with six distinct principal curvatures. We will prove in the present paper that M is either diffeomorphic to