1,431 research outputs found
Intersections of Quadrics, Moment-angle Manifolds and Connected Sums
The topology of the intersection of two real homogeneous coaxial quadrics was
studied by the second author who showed that its intersection with the unit
sphere is in most cases diffeomorphic to a connected sum of sphere products.
Combining that approach with a recent one (due to Antony Bahri, Martin
Bendersky, Fred Cohen and the first author) we study here the intersections of
k>2 quadrics and we identify very general families of such manifolds that are
diffeomorphic to connected sums of sphere products. These include those
moment-angle manifolds for which the result was conjectured by Frederic Bosio
and Laurent Meersseman. As a byproduct, a simpler and neater proof of the
result for the case k=2 is obtained.
Two new sections contain results not included in the first version of this
article: Section 2 describes the topological change on the manifolds after the
operations of cutting off a vertex or an edge of the associated polytope, which
can be combined in a special way with the previos results to produce new
infinite families of manifolds that are connected sums of sphere products. In
other cases we get slightly more complicated manifolds: with this we solve
another question by Bosio-Meersseman about the manifold associated to the
truncated cube.
In Section 3 we use this to show that the known rules for the cohomology
product of a moment-angle manifold have to be drastically modified in the
general situation. We state the modified rule, but leave the details of this
for another publication.
Section 0 recalls known definitions and results and in section 2.1 some
elementary topological constructions are defined and explored. In the Appendix
we state and prove some results about specific differentiable manifolds, which
are used in sections 1 and 2.Comment: We have included many clarifying suggestions and minor corrections
from some colleagues who read the manuscript carefully. The only change in
content from the previous version is the suppression a special case (item 3)
of Theorem 1.3 because we have not been able to fill in the details of any of
the known sketched proofs (including ours
Ring graphs and complete intersection toric ideals
We study the family of graphs whose number of primitive cycles equals its
cycle rank. It is shown that this family is precisely the family of ring
graphs. Then we study the complete intersection property of toric ideals of
bipartite graphs and oriented graphs. An interesting application is that
complete intersection toric ideals of bipartite graphs correspond to ring
graphs and that these ideals are minimally generated by Groebner bases. We
prove that any graph can be oriented such that its toric ideal is a complete
intersection with a universal Groebner basis determined by the cycles. It turns
out that bipartite ring graphs are exactly the bipartite graphs that have
complete intersection toric ideals for any orientation.Comment: Discrete Math., to appea
On free loop spaces of toric spaces
Growth of the Hilbert-Poincar\"e series for the rational homology of the free
loop space of a toric space is addressed. In case the toric space is a
manifold, the structure of the fan dictates whether the Hilbert-Poincar\"e
series has exponential growth. Applications are made to the existence of
infinitely many geometrically distinct periodic geodesics
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