12 research outputs found
Higher chordality: From graphs to complexes
We generalize the fundamental graph-theoretic notion of chordality for higher
dimensional simplicial complexes by putting it into a proper context within
homology theory. We generalize some of the classical results of graph
chordality to this generality, including the fundamental relation to the Leray
property and chordality theorems of Dirac.Comment: 13 pages, revised; to appear in Proc. AM
Rigidity and volume preserving deformation on degenerate simplices
Given a degenerate -simplex in a -dimensional space
(Euclidean, spherical or hyperbolic space, and ), for each , , Radon's theorem induces a partition of the set of -faces into two
subsets. We prove that if the vertices of the simplex vary smoothly in
for , and the volumes of -faces in one subset are constrained only to
decrease while in the other subset only to increase, then any sufficiently
small motion must preserve the volumes of all -faces; and this property
still holds in for if an invariant of
the degenerate simplex has the desired sign. This answers a question posed by
the author, and the proof relies on an invariant we discovered
for any -stress on a cell complex in . We introduce a
characteristic polynomial of the degenerate simplex by defining
, and prove that the roots
of are real for the Euclidean case. Some evidence suggests the same
conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
Affine stresses, inverse systems, and reconstruction problems
A conjecture of Kalai asserts that for , the affine type of a prime
simplicial -polytope can be reconstructed from the space of affine
-stresses of . We prove this conjecture for all . We also prove
the following generalization: for all pairs with , the affine type of a simplicial -polytope that has
no missing faces of dimension can be reconstructed from the space
of affine -stresses of . A consequence of our proofs is a strengthening
of the Generalized Lower Bound Theorem: it was proved by Nagel that for any
simplicial -sphere and ,
is at least as large as the number of missing -faces of
; here we show that, for ,
equality holds if and only if is -stacked. Finally, we show that
for , any simplicial -polytope that has no missing faces of
dimension is redundantly rigid, that is, for each edge of ,
there exists an affine -stress on with a non-zero value on .Comment: 21 pages. Added a few remarks and examples (see Remark 3.6, Examples
2.5 and 3.3-3.5). To appear in IMR
-vectors of manifolds with boundary
We extend several -type theorems for connected, orientable homology
manifolds without boundary to manifolds with boundary. As applications of these
results we obtain K\"uhnel-type bounds on the Betti numbers as well as on
certain weighted sums of Betti numbers of manifolds with boundary. Our main
tool is the completion of a manifold with boundary ; it is
obtained from by coning off the boundary of with a single new
vertex. We show that despite the fact that has a singular
vertex, its Stanley--Reisner ring shares a few properties with the
Stanley--Reisner rings of homology spheres. We close with a discussion of a
connection between three lower bound theorems for manifolds, PL-handle
decompositions, and surgery