367 research outputs found
Topological classification of torus manifolds which have codimension one extended actions
A toric manifold is a compact non-singular toric variety equipped with a
natural half-dimensional compact torus action. A torus manifold is an oriented,
closed, smooth manifold of dimension with an effective action of a compact
torus having a non-empty fixed point set. Hence, a torus manifold can
be thought of as a generalization of a toric manifold. In the present paper, we
focus on a certain class \mM in the family of torus manifolds with
codimension one extended actions, and we give a topological classification of
\mM. As a result, their topological types are completely determined by their
cohomology rings and real characteristic classes.
The problem whether the cohomology ring determines the topological type of a
toric manifold or not is one of the most interesting open problems in toric
topology. One can also ask this problem for the class of torus manifolds even
if its orbit spaces are highly structured. Our results provide a negative
answer to this problem for torus manifolds. However, we find a sub-class of
torus manifolds with codimension one extended actions which is not in the class
of toric manifolds but which is classified by their cohomology rings.Comment: 20 page
On the microlocal properties of the range of systems of principal type
The purpose of this paper is to study microlocal conditions for inclusion
relations between the ranges of square systems of pseudodifferential operators
which fail to be locally solvable. The work is an extension of earlier results
for the scalar case in this direction, where analogues of results by L.
H\"ormander about inclusion relations between the ranges of first order
differential operators with coefficients in which fail to be locally
solvable were obtained. We shall study the properties of the range of systems
of principal type with constant characteristics for which condition (\Psi) is
known to be equivalent to microlocal solvability.Comment: Added Theorem 4.7, Corollary 4.8 and Lemma A.4, corrected misprints.
The paper has 40 page
On the cohomology algebra of a fiber
Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that
there is a natural isomorphism of vector spaces between H^*(F;F_p) and
Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan,
Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453]
proved that if X is a finite r-connected CW-complex of dimension < rp+1 then
the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative
differential graded algebra A(X) with the same cohomology. Therefore if we
suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of
dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra
H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of
algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP
Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29
(1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France
9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of
algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th
powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-36.abs.htm
Geometric formality of homogeneous spaces and of biquotients
We provide examples of homogeneous spaces which are neither symmetric spaces
nor real cohomology spheres, yet have the property that every invariant metric
is geometrically formal. We also extend the known obstructions to geometric
formality to some new classes of homogeneous spaces and of biquotients, and to
certain sphere bundles.Comment: 15 page
Cohomological non-rigidity of generalized real Bott manifolds of height 2
We investigate when two generalized real Bott manifolds of height 2 have
isomorphic cohomology rings with Z/2 coefficients and also when they are
diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not
distinguish those manifolds up to diffeomorphism in general. This gives a
counterexample to the cohomological rigidity problem for real toric manifolds
posed in \cite{ka-ma08}. We also prove that generalized real Bott manifolds of
height 2 are diffeomorphic if they are homotopy equivalent
On the existence of branched coverings between surfaces with prescribed branch data, II
For a given branched covering between closed connected surfaces, there are
several easy relations one can establish between the Euler characteristics of
the surfaces, their orientability, the total degree, and the local degrees at
the branching points, including the classical Riemann-Hurwitz formula. These
necessary relations have been khown to be also sufficient for the existence of
the covering except when the base surface is the sphere (and when it is the
projective plane, but this case reduces to the case of the sphere). If the base
surface is the sphere, many exceptions are known to occur and the problem is
widely open. Generalizing methods of Baranski, we prove in this paper that the
necessary relations are actually sufficient in a specific but rather
interesting situation. Namely under the assumption that the base surface is the
sphere, that there are three branching points, that one of these branching
points has only two preimages with one being a double point, and either that
the covering surface is the sphere and that the degree is odd, or that the
covering surface has genus at least one, with a single specific exception. For
the case of the covering surface the sphere we also show that for each even
degree there are precisely two exceptions.Comment: 38 pages, 21 figures. This is a sequel of math.GT/050843
Universal circles for quasigeodesic flows
We show that if M is a hyperbolic 3-manifold which admits a quasigeodesic
flow, then pi_1(M) acts faithfully on a universal circle by homeomorphisms, and
preserves a pair of invariant laminations of this circle. As a corollary, we
show that the Thurston norm can be characterized by quasigeodesic flows,
thereby generalizing a theorem of Mosher, and we give the first example of a
closed hyperbolic 3-manifold without a quasigeodesic flow, answering a
long-standing question of Thurston.Comment: This is the version published by Geometry & Topology on 29 November
2006. V4: typsetting correction
Relative Gerbes
This paper introduces the notion of ``relative gerbes'' for smooth maps of
manifolds, and discusses their differential geometry. The equivalence classes
of relative gerbes are further classified by the relative integral cohomology
in degree three
- …