Group actions on 4-manifolds: some recent results and open questions

A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to include several results and questions which did not appear elsewhere.Comment: 21 pages, no figures, expanded version of author's talk at the Gokova conference 2009, appeared in the proceeding

Fixed-point free circle actions on 4-manifolds

This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) $S^1$-action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free $S^1$-actions under some further conditions on the fundamental group. The connection between the classification of the $S^1$-manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the $S^1$-manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free $S^1$-action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold. The paper ends with two questions concerning the topological nature of the smooth classification and the Seiberg-Witten invariants of 4-manifolds admitting a smooth fixed-point free $S^1$-action.Comment: 42 pages, no figures, Algebraic and Geometric Topolog

Orbifold Gromov-Witten Theory

In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.Comment: Late

Innovations and Experiments in Uses of Health Manpower—The Effect of Licensure Laws

Time-resolved optical spin orientation is employed to study spin dynamics of I * and I-1* excitons bound to isoelectronic centers in bulk ZnO. It is found that spin orientation at the exciton ground state can be generated using resonant excitation via a higher lying exciton state located at about 4 meV from the ground state. Based on the performed rate equation analysis of the measured spin dynamics, characteristic times of subsequent hole, electron, and direct exciton spin flips in the exciton ground state are determined as being tau(s)(h) = 0.4 ns, tau(s)(e) greater than= 15 ns, and tau(s)(eh) greater than= 15 ns, respectively. This relatively slow spin relaxation of the isoelectronic bound excitons is attributed to combined effects of (i) weak e-h exchange interaction, (ii) restriction of the exciton movement due to its binding at the isoelectronic center, and (iii) suppressed spin-orbit coupling for the tightly bound hole

Symplectic Lefschetz fibrations on S^1 x M^3

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper18.abs.htm

On the orders of periodic diffeomorphisms of 4-manifolds

This paper initiated an investigation on the following question: Suppose a smooth 4-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold support no smooth $\Z_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant $C$ in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold.Comment: 30 pages, no figures, final version, with a slightly changed title, to appear in Duke Math.