60 research outputs found
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
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