arXiv subjects

Fuquan Fang

Publications and source records attributed to Fuquan Fang.

24 records · Page 2Linked to original sources

arXiv2005

Finite isometry groups of 4-manifolds with positive sectional curvature

Let M be an oriented compact positively curved 4-manifold. Let G be a finite subgroup of the isometry group of $M$. Among others, we prove that there is a universal constant C (cf. Corollary 4.3 for the approximate value of C), such that if the order of G is odd and at least C, then G is either abelian of rank at most 2, or non-abelian and isomorphic to a subgroup of PU(3) with a presentation \{A, B| A^m=B^n=1, BAB^{-1}=A^r, (n(r-1), m)=1, r\ne r^3=1(\text{mod}m) \}. Moreover, M is homeomorphic to CP^2 if G is non-abelian, and homeomorphic to S^4 or CP^2 if G is abelian of rank 2.

math.DG↗

arXiv2003

Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive sectional curvature on which a (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By a result of Wilking, these results imply a homeomorphism classification for positively curved n-manifolds (n>7) of almost maximal symmetry rank [\frac{n-1}2].

math.DG↗

arXiv2003

Positive quaternionic Kaehler manifolds and symmetry rank

A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic Kähler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic Kähler manifold with symmetry rank at least (m-2) must be either isometric to \Bbb HP^m or Gr_2(\Bbb C^{m+2}), if m\ge 10.

math.DG↗

arXiv2003

Positively curved manifolds with maximal discrete symmetry rank

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M is homeomorphic to a sphere. (3) For M in (1) or (2), if n\ge 7 and k\ge [\frac{3n}4]+2, then M is homeomorphic to a sphere or homotopic to a complex projective space.

math.DG↗

arXiv2001

Fixed points of discrete nilpotent group actions on S^2

We prove that for each integer k of at least 2, there is an open neigborhood ν_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by elements in ν_k, then the natural action on S^2 has non-empty fixed point set. Moreover, the G-action has at least two fixed points if the action has a finite non-trivial orbit.

math.GT↗

arXiv2001

Secondary Brown-Kervaire Quadratic forms and $π$-manifolds

In this paper we define a secondary Brown-Kervaire quadratic forms. Among the applications we obtain a complete classification of (n-2)-connected 2n-dimensional framed manifolds up to homeomorphism and homotopy equivalence, . In particular, we prove that the homotopy type of such manifolds determine their homeomorphism type.

math.AT↗