12,516 research outputs found
Positive Alexander Duality for Pursuit and Evasion
Considered is a class of pursuit-evasion games, in which an evader tries to
avoid detection. Such games can be formulated as the search for sections to the
complement of a coverage region in a Euclidean space over a timeline. Prior
results give homological criteria for evasion in the general case that are not
necessary and sufficient. This paper provides a necessary and sufficient
positive cohomological criterion for evasion in a general case. The principal
tools are (1) a refinement of the Cech cohomology of a coverage region with a
positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore
homology of the coverage gaps with a positive cone encoding time orientation,
and (3) a positive variant of Alexander Duality. Positive cohomology decomposes
as the global sections of a sheaf of local positive cohomology over the time
axis; we show how this decomposition makes positive cohomology computable as a
linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive
(co)homology generalized to arbitrary degrees; Positive Alexander Duality
generalized from homological degrees 0,1; Morse and smoothness conditions
generalized; illustrations of positive homology added. minor corrections in
proofs, notation, organization, and language made throughout. variant of
Borel-Moore homology now use
A Duality Exact Sequence for Legendrian Contact Homology
We establish a long exact sequence for Legendrian submanifolds L in P x R,
where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that
displaces the projection of L off of itself. In this sequence, the singular
homology H_* maps to linearized contact cohomology CH^* which maps to
linearized contact homology CH_* which maps to singular homology. In
particular, the sequence implies a duality between the kernel of the map
(CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this
duality is compatible with Poincare duality in L in the following sense: the
Poincare dual of a singular class which is the image of a in CH_* maps to a
class \alpha in CH^* such that \alpha(a)=1.
The exact sequence generalizes the duality for Legendrian knots in Euclidean
3-space [24] and leads to a refinement of the Arnold Conjecture for double
points of an exact Lagrangian admitting a Legendrian lift with linearizable
contact homology, first proved in [6].Comment: 57 pages, 10 figures. Improved exposition and expanded analytic
detai
Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups
In this paper, we investigate a relation between finite graphs, simplicial
flag complexes and right-angled Coxeter groups, and we provide a class of
reconstructible finite graphs. We show that if is a finite graph which
is the 1-skeleton of some simplicial flag complex which is a homology
manifold of dimension , then the graph is reconstructible.Comment: 7 page
- …