65 research outputs found
Homologous non-isotopic symplectic tori in a K3-surface
For each member of an infinite family of homology classes in the K3-surface
E(2), we construct infinitely many non-isotopic symplectic tori representing
this homology class. This family has an infinite subset of primitive classes.
We also explain how these tori can be non-isotopically embedded as homologous
symplectic submanifolds in many other symplectic 4-manifolds including the
elliptic surfaces E(n) for n>2.Comment: 15 pages, 9 figures; v2: extended the main theorem, gave a second
construction of symplectic tori, added a figure, added/updated references,
minor changes in 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
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
Fake R^4's, Einstein Spaces and Seiberg-Witten Monopole Equations
We discuss the possible relevance of some recent mathematical results and
techniques on four-manifolds to physics. We first suggest that the existence of
uncountably many R^4's with non-equivalent smooth structures, a mathematical
phenomenon unique to four dimensions, may be responsible for the observed
four-dimensionality of spacetime. We then point out the remarkable fact that
self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean
signature without affecting the metric. As a specific example, we consider
solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are
covariantly constant, the monopole Weyl spinor has only a single constant
component, and the 4-manifold M_4 is a product of two Riemann surfaces
Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric)
vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being
excluded). When the two genuses are equal, the electromagnetic fields are
self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole
condensate serving as the cosmological constant.Comment: 9 pages, Talk at the Second Gursey Memorial Conference, June 2000,
Istanbu
PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees
In this article we complete the proof---for a broad class of
four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten
series coincide, at least through terms of degree less than or equal to c-2,
where c is a linear combination of the Euler characteristic and signature of
the four-manifold. This article is a revision of sections 4--7 of an earlier
version, while a revision of sections 1--3 of that earlier version now appear
in a separate companion article (math.DG/0007190). Here, we use our
computations of Chern classes for the virtual normal bundles for the
Seiberg-Witten strata from the companion article (math.DG/0007190), a
comparison of all the orientations, and the PU(2) monopole cobordism to compute
pairings with the links of level-zero Seiberg-Witten moduli subspaces of the
moduli space of PU(2) monopoles. These calculations then allow us to compute
low-degree Donaldson invariants in terms of Seiberg-Witten invariants and
provide a partial verification of Witten's conjecture.Comment: Journal fur die Reine und Angewandte Mathematik, to appear; 65 pages.
Revision of sections 4-7 of version v1 (December 1997
On the Quantum Invariant for the Brieskorn Homology Spheres
We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev
invariant for the Brieskorn homology spheres by use of
properties of the modular form following a method proposed by Lawrence and
Zagier. Key observation is that the invariant coincides with a limiting value
of the Eichler integral of the modular form with weight 3/2. We show that the
Casson invariant is related to the number of the Eichler integrals which do not
vanish in a limit . Correspondingly there is a
one-to-one correspondence between the non-vanishing Eichler integrals and the
irreducible representation of the fundamental group, and the Chern-Simons
invariant is given from the Eichler integral in this limit. It is also shown
that the Ohtsuki invariant follows from a nearly modular property of the
Eichler integral, and we give an explicit form in terms of the L-function.Comment: 26 pages, 2 figure
Further restrictions on the topology of stationary black holes in five dimensions
We place further restriction on the possible topology of stationary
asymptotically flat vacuum black holes in 5 spacetime dimensions. We prove that
the horizon manifold can be either a connected sum of Lens spaces and "handles"
, or the quotient of by certain finite groups of
isometries (with no "handles"). The resulting horizon topologies include Prism
manifolds and quotients of the Poincare homology sphere. We also show that the
topology of the domain of outer communication is a cartesian product of the
time direction with a finite connected sum of 's
and 's, minus the black hole itself. We do not assume the existence of
any Killing vector beside the asymptotically timelike one required by
definition for stationarity.Comment: LaTex, 22 pages, 9 figure
Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds
We define relative Ruan invariants that count embedded connected symplectic
submanifolds which contact a fixed stable symplectic hypersurface V in a
symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders
(in addition to insertions on X\V) for stable V. We obtain invariants of the
deformation class of (X,V,w). Two large issues must be tackled to define such
invariants: (1) Curves lying in the hypersurface V and (2) genericity results
for almost complex structures constrained to make V pseudo-holomorphic (or
almost complex). Moreover, these invariants are refined to take into account
rim tori decompositions. In the latter part of the paper, we extend the
definition to disconnected submanifolds and construct relative Gromov-Taubes
invariants
A Large k Asymptotics of Witten's Invariant of Seifert Manifolds
We calculate a large asymptotic expansion of the exact surgery formula
for Witten's invariant of Seifert manifolds. The contributions of all
flat connections are identified. An agreement with the 1-loop formula is
checked. A contribution of the irreducible connections appears to contain only
a finite number of terms in the asymptotic series. A 2-loop correction to the
contribution of the trivial connection is found to be proportional to Casson's
invariant.Comment: 51 pages (Some changes are made to the Discussion section. A surgery
formula for perturbative corrections to the contribution of the trivial
connection is suggested.
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
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