2,660 research outputs found
Fixed-point free circle actions on 4-manifolds
This paper is concerned with fixed-point free -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)
-action and realize the given group as the fundamental group. A similar
statement holds for the number of equivalence classes of fixed-point free
-actions under some further conditions on the fundamental group. The
connection between the classification of the -manifolds and the
fundamental group is given by a certain decomposition, called fiber-sum
decomposition, of the -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 -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 -action.Comment: 42 pages, no figures, Algebraic and Geometric Topolog
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 such that the manifold support no smooth -actions of prime
order for ? We gave affirmative results to this question for the case of
holomorphic and symplectic actions, with an interesting finding that the
constant 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.
A Homotopy Theory of Orbispaces
In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on
Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was
the prediction that a certain physicist's Euler number of the orbifold must be
equal to the Euler number of any of its crepant resolutions. This was soon
related to the so called McKay correspondence in mathematics (cf. [McK]). Later
developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of
Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants
of symplectic orbifolds (cf. [CR1-2]). One common feature of these studies is
that certain contributions from singularities, which are called ``twisted
sectors'' in physics, have to be properly incorporated. This is called the
``stringy aspect'' of an orbifold (cf. [R]).
This paper makes an effort to understand the stringy aspect of orbifolds in
the realm of ``traditional mathematics''.Comment: latex, 59 pages, minor mistakes corrected, more references adde
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
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