1,112 research outputs found
Theta-graph and diffeomorphisms of some 4-manifolds
In this article, we construct countably many mutually non-isotopic
diffeomorphisms of some closed non simply-connected 4-manifolds that are
homotopic to but not isotopic to the identity, by surgery along
-graphs. As corollaries of this, we obtain some new results on
codimension 1 embeddings and pseudo-isotopies of 4-manifolds. In the proof of
the non-triviality of the diffeomorphisms, we utilize a twisted analogue of
Kontsevich's characteristic class for smooth bundles, which is obtained by
extending a higher dimensional analogue of March\'{e}--Lescop's "equivariant
triple intersection" in configuration spaces of 3-manifolds to allow Lie
algebraic local coefficient system.Comment: 67 pages, 14 figures, v3: corrected errors and typo
On Stein fillings of contact torus bundles
We consider a large family F of torus bundles over the circle, and we use
recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact
structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first
Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein
fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic
or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and
fall into finitely many diffeomorphism classes. Moreover, for infinitely many
hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein
fillings of (Y,C).Comment: 18 pages, 10 figures. This preprint version differs from the final
version which is to appear in the Bulletin of the London Mathematical Societ
Links, two-handles, and four-manifolds
We show that only finitely many links in a closed 3-manifold share the same
complement, up to twists along discs and annuli. Using the same techniques, we
prove that by adding 2-handles on the same link we get only finitely many
smooth cobordisms between two given closed 3-manifolds.
As a consequence, there are finitely many smooth closed 4-manifolds
constructed from some Kirby diagram with bounded number of crossings, discs,
and strands, or from some Turaev special shadow with bounded number of
vertices. (These are the 4-dimensional analogues of Heegaard diagrams and
special spines for 3-manifolds.) We therefore get two filtrations on the set of
all smooth closed 4-manifolds with finite sets. The two filtrations are
equivalent after linear rescalings, and their cardinality grows at least as
n^{c*n}.Comment: 23 pages, 9 figures. Final versio
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
Fivebranes and 4-manifolds
We describe rules for building 2d theories labeled by 4-manifolds. Using the
proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2)
theories, we obtain a number of results, which include new 3d N=2 theories
T[M_3] associated with rational homology spheres and new results for
Vafa-Witten partition functions on 4-manifolds. In particular, we point out
that the gluing measure for the latter is precisely the superconformal index of
2d (0,2) vector multiplet and relate the basic building blocks with coset
branching functions. We also offer a new look at the fusion of defect lines /
walls, and a physical interpretation of the 4d and 3d Kirby calculus as
dualities of 2d N=(0,2) theories and 3d N=2 theories, respectivelyComment: 81 pages, 18 figures. v2: misprints corrected, clarifications and
references added. v3: additions and corrections about lens space theory,
4-manifold gluing, smooth structure
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