3,017 research outputs found
Collapses, products and LC manifolds
Durhuus and Jonsson (1995) introduced the class of "locally constructible"
(LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are
spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We
prove this result by studying how to collapse products of manifolds with
exactly one facet removed.Comment: 6 pages; added references; minor changes. Accepted for J. Comb.
Theory, Series
Collapsibility of CAT(0) spaces
Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
(1) All CAT(0) cube complexes are collapsible.
(2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
(3) All contractible d-manifolds () admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes
has been removed and forms a new paper, called "Barycentric subdivisions of
convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration
of manifolds has also been removed and forms now a third paper, called "A
Cheeger-type exponential bound for the number of triangulated manifolds"
(arXiv:1710.00130
Knots in collapsible and non-collapsible balls
We construct the first explicit example of a simplicial 3-ball B_{15,66} that
is not collapsible. It has only 15 vertices. We exhibit a second 3-ball
B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable.
Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125}
(with only 18 vertices) that is not locally constructible. All these examples
are based on knotted subcomplexes with only three edges; the knots are the
trefoil, the double trefoil, and the triple trefoil, respectively. The more
complicated the knot is, the more distant the triangulation is from being
polytopal, collapsible, etc. Further consequences of our work are:
(1) Unshellable 3-spheres may have vertex-decomposable barycentric
subdivisions.
(This shows the strictness of an implication proven by Billera and Provan.)
(2) For d-balls, vertex-decomposable implies non-evasive implies collapsible,
and for d=3 all implications are strict.
(This answers a question by Barmak.)
(3) Locally constructible 3-balls may contain a double trefoil knot as a
3-edge subcomplex.
(This improves a result of Benedetti and Ziegler.)
(4) Rudin's ball is non-evasive.Comment: 25 pages, 5 figures, 11 tables, references update
On the dual graph of Cohen-Macaulay algebras
Given a projective algebraic set X, its dual graph G(X) is the graph whose
vertices are the irreducible components of X and whose edges connect components
that intersect in codimension one. Hartshorne's connectedness theorem says that
if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We
present two quantitative variants of Hartshorne's result:
1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where
r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for
coordinate arrangements, it yields an algebraic extension of Balinski's theorem
for simplicial polytopes.)
2) If X is a canonically embedded arrangement of lines no three of which meet
in the same point, then the diameter of the graph G(X) is not larger than the
codimension of X. (The bound is sharp; for coordinate arrangements, it yields
an algebraic expansion on the recent combinatorial result that the Hirsch
conjecture holds for flag normal simplicial complexes.)Comment: Minor changes throughout, Remark 4.1 expanded, to appear in IMR
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