21 research outputs found
Exotic smooth structures on 4-manifolds with zero signature
For every integer , we construct infinite families of mutually
nondiffeomorphic irreducible smooth structures on the topological -manifolds
and (2k-1)(\CP#\CPb), the connected sums of
copies of and \CP#\CPb.Comment: 6 page
Constructions of generalized complex structures in dimension four
Four-manifold theory is employed to study the existence of (twisted)
generalized complex structures. It is shown that there exist (twisted)
generalized complex structures that have more than one type change loci. In an
example-driven fashion, (twisted) generalized complex structures are
constructed on a myriad of four-manifolds, both simply and non-simply
connected, which are neither complex nor symplectic
A two-cocycle on the group of symplectic diffeomorphisms
We investigate a two-cocycle on the group of symplectic diffeomorphisms of an
exact symplectic manifolds defined by Ismagilov, Losik, and Michor and
investigate its properties. We provide both vanishing and non-vanishing results
and applications to foliated symplectic bundles and to Hamiltonian actions of
finitely generated groups.Comment: 16 pages, no figure
Exotic Smooth Structures on Small 4-Manifolds
Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first
example of a simply-connected symplectic 4-manifold that is homeomorphic but
not diffeomorphic to M.Comment: 11 page
On the geometrization of matter by exotic smoothness
In this paper we discuss the question how matter may emerge from space. For
that purpose we consider the smoothness structure of spacetime as underlying
structure for a geometrical model of matter. For a large class of compact
4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of
Fintushel and Stern to change the smoothness structure. The influence of this
surgery to the Einstein-Hilbert action is discussed. Using the Weierstrass
representation, we are able to show that the knotted torus used in knot surgery
is represented by a spinor fulfilling the Dirac equation and leading to a
mass-less Dirac term in the Einstein-Hilbert action. For sufficient complicated
links and knots, there are "connecting tubes" (graph manifolds, torus bundles)
which introduce an action term of a gauge field. Both terms are genuinely
geometrical and characterized by the mean curvature of the components. We also
discuss the gauge group of the theory to be U(1)xSU(2)xSU(3).Comment: 30 pages, 3 figures, svjour style, complete reworking now using
Fintushel-Stern knot surgery of elliptic surfaces, discussion of Lorentz
metric and global hyperbolicity for exotic 4-manifolds added, final version
for publication in Gen. Rel. Grav, small typos errors fixe
Stein structures and holomorphic mappings
We prove that every continuous map from a Stein manifold X to a complex
manifold Y can be made holomorphic by a homotopic deformation of both the map
and the Stein structure on X. In the absence of topological obstructions the
holomorphic map may be chosen to have pointwise maximal rank. The analogous
result holds for any compact Hausdorff family of maps, but it fails in general
for a noncompact family. Our main results are actually proved for smooth almost
complex source manifolds (X,J) with the correct handlebody structure. The paper
contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy
characterization of Stein manifolds and a slightly different explanation of the
construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693,
1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint
math/0509419).Comment: The original publication is available at http://www.springerlink.co
On the Number of Type Change Loci of a Generalized Complex Structure
In this note, we describe a procedure to construct generalized complex structures whose type change locus has arbitrarily many path components on products of the circle with a connected sum of closed 3-manifolds. As an application, we use the procedure to exhibit such geometric structures on a myriad of simply connected 4-manifolds among many others