TY - JOUR

T1 - Barycentric Subdivisions of Convex Complexes are Collapsible

AU - Adiprasito, Karim

AU - Benedetti, Bruno

N1 - Funding Information:
Karim Adiprasito acknowledges the support by ISF Grant 1050/16 and ERC StG 716424-CASe. Bruno Benedetti acknowledges the support by an NSF Grant 1600741, the DFG Collaborative Research Center TRR109, and the Swedish Research Council VR 2011-980. Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. 1 This is true because any movement of star-center x normal to the geodesic span of τ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and x induces a motion of the hemisphere N ( u , τ ) 1 B u \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm {N}^1_{(u,\tau )}B_u$$\end{document} in N ( u , τ ) 1 S d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {N}^1_{(u,\tau )} S^d$$\end{document} , specifically, by moving the midpoint of the hemisphere in the same direction.
Funding Information:
Karim?Adiprasito acknowledges the support by ISF Grant 1050/16 and ERC StG 716424-CASe. Bruno?Benedetti acknowledges the support by an NSF Grant 1600741, the DFG Collaborative Research Center TRR109, and the Swedish Research Council VR 2011-980. Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in Rd is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.

AB - A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d≤ 3. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in Rd is simplicially collapsible after d- 2 derived subdivisions at most. This presents progress on an old question by Goodrick.

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U2 - 10.1007/s00454-019-00137-3

DO - 10.1007/s00454-019-00137-3

M3 - Article

AN - SCOPUS:85075358744

VL - 64

SP - 608

EP - 626

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -