45 research outputs found
Finitness of the basic intersection cohomology of a Killing foliation
We prove that the basic intersection cohomology where is the singular
foliation determined by an isometric action of a Lie group on the compact
manifold , is finite dimensional
On the localized phase of a copolymer in an emulsion: supercritical percolation regime
In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, and , each occurring with
density 1/2. The emulsion is a random mixture of liquids of two types, and
, organised in large square blocks occurring with density and ,
respectively, where . The copolymer in the emulsion has an energy
that is minus times the number of -matches minus times the
number of -matches, where without loss of generality the interaction
parameters can be taken from the cone . To make the model mathematically tractable, we assume that the
copolymer is directed and can only enter and exit a pair of neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical percolation regime , with the critical probability for directed bond percolation on
the square lattice, the free energy has a phase transition along a curve in the
cone that is independent of . At this critical curve, there is a transition
from a phase where the copolymer is fully delocalized into the -blocks to a
phase where it is partially localized near the -interface. In the present
paper we prove three theorems that complete the analysis of the phase diagram :
(1) the critical curve is strictly increasing; (2) the phase transition is
second order; (3) the free energy is infinitely differentiable throughout the
partially localized phase.Comment: 43 pages and 10 figure
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
5-dimensional contact SO(3)-manifolds and Dehn twists
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to
equivariant contactomorphisms. The construction of such manifolds with singular
orbits requires the use of generalized Dehn twists.
We show as an application that all simply connected 5-manifoldswith singular
orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The
standard contact structure on such a manifold gives right-handed Dehn twists,
and a second contact structure defined in the article gives left-handed twists.Comment: 16 pages, 1 figure; simplification of arguments by restricting
classification to coorientation preserving contactomorphism
Equivariant cohomology and analytic descriptions of ring isomorphisms
In this paper we consider a class of connected closed -manifolds with a
non-empty finite fixed point set, each of which is totally non-homologous
to zero in (or -equivariantly formal), where . With the
help of the equivariant index, we give an explicit description of the
equivariant cohomology of such a -manifold in terms of algebra, so that we
can obtain analytic descriptions of ring isomorphisms among equivariant
cohomology rings of such -manifolds, and a necessary and sufficient
condition that the equivariant cohomology rings of such two -manifolds are
isomorphic. This also leads us to analyze how many there are equivariant
cohomology rings up to isomorphism for such -manifolds in 2- and
3-dimensional cases.Comment: 20 pages, updated version with two references adde
Orbit spaces of free involutions on the product of two projective spaces
Let be a finitistic space having the mod 2 cohomology algebra of the
product of two projective spaces. We study free involutions on and
determine the possible mod 2 cohomology algebra of orbit space of any free
involution, using the Leray spectral sequence associated to the Borel fibration
. We also
give an application of our result to show that if has the mod 2 cohomology
algebra of the product of two real projective spaces (respectively complex
projective spaces), then there does not exist any -equivariant
map from for (respectively ), where
is equipped with the antipodal involution.Comment: 14 pages, to appear in Results in Mathematic
M2-branes on M-folds
We argue that the moduli space for the Bagger-Lambert A_4 theory at level k
is (R^8 \times R^8)/D_{2k}, where D_{2k} is the dihedral group of order 4k. We
conjecture that the theory describes two M2-branes on a Z_{2k} ``M-fold'', in
which a geometrical action of Z_{2k} is combined with an action on the branes.
For k=1, this arises as the strong coupling limit of two D2-branes on an O2^-
orientifold, whose worldvolume theory is the maximally supersymmetric SO(4)
gauge theory. Finally, in an appropriate large-k limit we show that one
recovers compactified M-theory and the M2-branes reduce to D2-branes.Comment: 16 pages, LaTeX, v2: typos corrected, included appendices on
Chern-Simons level quantization and monopole charge quantizatio
Locally continuously perfect groups of homeomorphisms
The notion of a locally continuously perfect group is introduced and studied.
This notion generalizes locally smoothly perfect groups introduced by Haller
and Teichmann. Next, we prove that the path connected identity component of the
group of all homeomorphisms of a manifold is locally continuously perfect. The
case of equivariant homeomorphism group and other examples are also considered.Comment: 14 page
On manifolds with nonhomogeneous factors
We present simple examples of finite-dimensional connected homogeneous spaces
(they are actually topological manifolds) with nonhomogeneous and nonrigid
factors. In particular, we give an elementary solution of an old problem in
general topology concerning homogeneous spaces