Circle actions on simply connected 5-manifolds

The aim of this paper is to study compact 5-manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel-Whitney class have to satisfy strong restrictions. We then show that for simply connected 5-manifolds these restrictions are necessary and sufficient.     

Circle bundles over 4-manifolds

Every 1-connected topological 4-manifold M admits a S¹-covering by # _{r − 1} S² × S³, where Received: 4 July 2004     

On 4-manifolds with singular torus actions

Mathematische Annalen -     

On the topological classification of pseudofree group actions on 4-manifolds: II

This paper is concerned with the algebraic aspects of the classification of pseudofree, locally linear group actions on a simply connected 4-manifold, particularly with the splitting and stability properties of the associated Hermitian intersection module and its isometry group. Our main result is the proof of stability of the equivariant intersection form for a large class of pseudofree actions. We also prove a topological rigidity theorem stating that two locally linear, pseudofree actions on a closed, oriented, simply connected 4-manifold, with the equivariant intersection forms indefinite and of rank at least 3 at each irreducible character, are topologically conjugate by an orientation preserving homeomorphism if and only if their oriented local representations at the corresponding fixed points are linearly equivalent.Partially supported by the N.S.F.     

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.     

Analogies between group actions on 3-manifolds and number fields

Let a cyclic group $G$ act either on a number field $\mathbb L$ or on a $3$-manifold $M$. Let $s_{\mathbb L}$ be the number of ramified primes in the extension $\mathbb L^G\subset \mathbb L$ and $s_M$ be the number of components of the branching set of the branched covering $M\to M/G$. In this paper, we prove several formulas relating $s_{\mathbb L}$ and $s_M$ to the induced $G$-action on $Cl(\mathbb L)$ and $H_1(M),$ respectively. We observe that the formulas for $3$-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between $3$-manifolds and number fields postulated in arithmetic topology.     

Fold maps on 4-manifolds

We give a complete characterization of those closed orientable 4-manifolds which admit smooth maps into R ³ with only fold singularities. We also clarify the relationship between the existence problem of fold maps and that of linearly independent vector fields on manifolds.     

Nonsmoothable involutions on spin 4-manifolds

Let X be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to n( − E ₈) ⊕ mH, where H is the hyperbolic form. In this paper, we prove that for n such that n ≡ 2  mod  4, there exists a locally linear pseudofree ℤ₂-action on X which is nonsmoothable with respect to any possible smooth structure on X.     

Finite maximal orientation reversing group actions on handlebodies and 3-manifolds

We study arbitrary (that is not necessarily orientation preserving) finite group actions on 3-dimensional orientable or nonorientable handlebodies of genus g. For g>1, the maximal possible order is 24(g−1); we characterize the corresponding groups of this order and also the occuring quotient orbifolds. Then we use this to study finite group actions of large order (with respect to the equivariant Heegaard genus g) on closed 3-manifolds, again concentrating on the maximal case of order 24(g−1). Our results extend corresponding results in the orientation preserving setting. Whereas for arbitrary finite group actions on handlebodies much more types of quotient orbifolds occur than in the orientation preserving case, it turns out that for closed 3-manifolds the situation is quite rigid, in contrast to the orientation preserving case where one has many possibilities to construct manifolds with large group actions.     

Stabilization of 4-manifolds

The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.