235 research outputs found
Rational surfaces and symplectic 4-manifolds with one basic class
We present constructions of simply connected symplectic 4-manifolds which
have (up to sign) one basic class and which fill up the geographical region
between the half-Noether and Noether lines.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-19.abs.htm
Invariants for Lagrangian tori
We define an simple invariant of an embedded nullhomologous Lagrangian torus
and use this invariant to show that many symplectic 4-manifolds have infinitely
many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We
further show that for a large class of examples that lambda(T) is actually a
C-infinity invariant. In addition, this invariant is used to show that many
symplectic 4-manifolds have nontrivial homology classes which are represented
by infinitely many pairwise inequivalent Lagrangian tori, a result first proved
by S Vidussi for the homotopy K3-surface obtained from knot surgery using the
trefoil knot in [Lagrangian surfaces in a fixed homology class: existence of
knotted Lagrangian tori, J. Diff. Geom. (to appear)].Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper25.abs.htm
Surgery on Nullhomologous Tori
By studying the example of smooth structures on CP^2#3(-CP^2) we illustrate
how surgery on a single embedded nullhomologous torus can be utilized to change
the symplectic structure, the Seiberg-Witten invariant, and hence the smooth
structure on a 4-manifold
Pinwheels and nullhomologous surgery on 4-manifolds with b^+ = 1
We present a method for finding embedded nullhomologous tori in standard
4-manifolds which can be utilized to change their smooth structure. As an
application, we show how to obtain infinite families of simply connected smooth
4-manifolds with b^+ = 1 and b^- = 2,...,7, via surgery on nullhomologous tori
embedded in the standard manifolds CP^2 # k (-CP^2), k=2,...,7.Comment: Final version. To appear in AG
Knots, Links, and 4-Manifolds
In this paper we investigate the relationship between isotopy classes of
knots and links in S^3 and the diffeomorphism types of homeomorphic smooth
4-manifolds. As a corollary of this initial investigation, we begin to uncover
the surprisingly rich structure of diffeomorphism types of manifolds
homeomorphic to the K3 surface.Comment: 31 page
Double node neighborhoods and families of simply connected 4-manifolds with b^+=1
We introduce a new technique that is used to show that the complex projective
plane blown up at 6, 7, or 8 points has infinitely many distinct smooth
structures. None of these smooth structures admit smoothly embedded spheres
with self-intersection -1, i.e. they are minimal. In addition, none these
smooth structures admit an underlying symplectic structure. Shortly after the
appearance of a preliminary version of this article, Park, Stipsicz, and Szabo
used the techniques described herein to show that the complex projective plane
blown up at 5 points has infinitely many distinct smooth structures. In the
final section of this paper we give a somewhat different construction of such a
family of examples.Comment: 11 pages, More typos and minor errors correcte
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