{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T08:23:00Z","timestamp":1782289380513,"version":"3.54.5"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,6,9]],"date-time":"2016-06-09T00:00:00Z","timestamp":1465430400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2017,2]]},"DOI":"10.1007\/s00493-016-3149-8","type":"journal-article","created":{"date-parts":[[2016,6,9]],"date-time":"2016-06-09T09:33:57Z","timestamp":1465464837000},"page":"1-30","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Subdivisions, shellability, and collapsibility of products"],"prefix":"10.1007","volume":"37","author":[{"given":"Karim A.","family":"Adiprasito","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruno","family":"Benedetti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,6,9]]},"reference":[{"key":"3149_CR1","volume-title":"Methods from Differential Geometry in Polytope Theory","author":"K. A. Adiprasito","year":"2013","unstructured":"K. A. Adiprasito: Methods from Differential Geometry in Polytope Theory, Berlin, DE, May 2013, published on diss.fu-berlin.de."},{"key":"3149_CR2","unstructured":"K. A. Adiprasito and B. Benedetti: Metric geometry, convexity and collapsibility, preprint, available at arXiv:1107.5789."},{"key":"3149_CR3","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.ejc.2014.10.002","volume":"45","author":"K. A. Adiprasito","year":"2015","unstructured":"K. A. Adiprasito and B. Benedetti: Tight complexes in 3-space admit perfect discrete Morse functions, European J. Combinatorics 45 (2015), 71\u201384.","journal-title":"European J. Combinatorics"},{"key":"3149_CR4","volume-title":"Isr. J. Math.","author":"K. A. Adiprasito","year":"2014","unstructured":"K. A. Adiprasito and I. Izmestiev: Derived subdivisions make every PL sphere polytopal, Isr. J. Math. (2014), to appear."},{"key":"3149_CR5","unstructured":"K. A. Adiprasito and R. Sanyal: Relative Stanley-Reisner Theory and Upper Bound Theorems for Minkowski sums, preprint, arXiv:1405.7368."},{"issue":"2","key":"3149_CR6","doi-asserted-by":"publisher","first-page":"292","DOI":"10.2307\/1968099","volume":"31","author":"J. W. Alexander","year":"1930","unstructured":"J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. 31 (2) (1930), 292\u2013320.","journal-title":"Ann. of Math."},{"key":"3149_CR7","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s00454-011-9357-5","volume":"47","author":"J. A. Barmak","year":"2012","unstructured":"J. A. Barmak and E. G. Minian: Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47 (2012), 301\u2013328.","journal-title":"Discrete Comput. Geom."},{"key":"3149_CR8","doi-asserted-by":"publisher","first-page":"6631","DOI":"10.1090\/S0002-9947-2012-05614-5","volume":"364","author":"B. Benedetti","year":"2012","unstructured":"B. Benedetti: Discrete Morse theory for manifolds with boundary, Trans. Amer. Math. Soc. 364 (2012), 6631\u20136670.","journal-title":"Trans. Amer. Math. Soc."},{"key":"3149_CR9","first-page":"93","volume-title":"Lectures on modern mathematics","author":"R. H. Bing","year":"1964","unstructured":"R. H. Bing: Some aspects of the topology of 3-manifolds related to the Poincar\u00e9 conjecture, Lectures on modern mathematics, Vol. II, Wiley, New York, 1964, 93\u2013128."},{"key":"3149_CR10","first-page":"1819","volume-title":"Handbook of Combinatorics","author":"A. Bj\u00f6rner","year":"1995","unstructured":"A. Bj\u00f6rner: Topological methods, Handbook of Combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, 1819\u20131872."},{"key":"3149_CR11","doi-asserted-by":"publisher","first-page":"1299","DOI":"10.1090\/S0002-9947-96-01534-6","volume":"348","author":"A. Bj\u00f6rner","year":"1996","unstructured":"A. Bj\u00f6rner and M. L. Wachs: Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), 1299\u20131327.","journal-title":"Trans. Amer. Math. Soc."},{"issue":"1971","key":"3149_CR12","first-page":"197","volume":"29","author":"H. Bruggesser","year":"1972","unstructured":"H. Bruggesser and P. Mani: Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197\u2013205 (1972).","journal-title":"Math. Scand."},{"key":"3149_CR13","doi-asserted-by":"crossref","DOI":"10.1090\/ulect\/024","volume-title":"Torus actions and their applications in topology and combinatorics","author":"V. M. Buchstaber","year":"2002","unstructured":"V. M. Buchstaber and T. E Panov: Torus actions and their applications in topology and combinatorics, vol. 24, American Mathematical Society, 2002."},{"key":"3149_CR14","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1017\/S0305004100041268","volume":"63","author":"D. R. J. Chillingworth","year":"1967","unstructured":"D. R. J. Chillingworth: Collapsing three-dimensional convex polyhedra, Proc. Cambridge Philos. Soc. 63 (1967), 353\u2013357, Correction in Math. Proc. Cambridge Philos. Soc. 88 (2) (1980), 307\u2013310.","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"3149_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9372-6","volume-title":"A course in Simple-Homotopy Theory","author":"M. M. Cohen","year":"1973","unstructured":"M. M. Cohen: A course in Simple-Homotopy Theory, Springer-Verlag, New York, 1973, Graduate Texts in Mathematics, Vol. 10."},{"key":"3149_CR16","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1016\/0040-9383(75)90006-3","volume":"14","author":"M. M. Cohen","year":"1975","unstructured":"M. M. Cohen: Dimension estimates in collapsing X\u00d7I q, Topology 14 (1975), 253\u2013256.","journal-title":"Topology"},{"key":"3149_CR17","doi-asserted-by":"publisher","first-page":"692","DOI":"10.1016\/j.jcta.2005.05.001","volume":"113","author":"M. Courdurier","year":"2006","unstructured":"M. Courdurier: On stars and links of shellable polytopal complexes, Journal of Combinatorial Theory, Series A 113 (2006), 692\u2013697.","journal-title":"Journal of Combinatorial Theory, Series A"},{"key":"3149_CR18","doi-asserted-by":"publisher","first-page":"2403","DOI":"10.1090\/S0002-9939-07-08768-0","volume":"135","author":"S. L. \u010cuki\u0107","year":"2007","unstructured":"S. L. \u010cuki\u0107 and E. Delucchi: Simplicial shellable spheres via combinatorial blowups, Proc. Amer. Math. Soc. 135 (2007), 2403\u20132414 (electronic).","journal-title":"Proc. Amer. Math. Soc."},{"key":"3149_CR19","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1215\/S0012-7094-74-04150-7","volume":"41","author":"G. Danaraj","year":"1974","unstructured":"G. Danaraj and V. Klee: Shellings of spheres and polytopes, Duke Mathematical Journal 41 (1974), 443\u2013451.","journal-title":"Duke Mathematical Journal"},{"key":"3149_CR20","volume-title":"Low-dimensional topology","author":"M. W. Davis","year":"1998","unstructured":"M. W. Davis and G. Moussong: Notes on nonpositively curved polyhedra, in: Low-dimensional topology (Eger 1996\/Budapest 1998), Bolyai Soc. Math. Stud., vol. 8, J\u00e1nos Bolyai Math. Soc., Budapest, 1999, 11-94."},{"key":"3149_CR21","first-page":"425","volume":"19","author":"P. Dierker","year":"1968","unstructured":"P. Dierker: Note on collapsing K \u00d7 I where K is a contractible polyhedron, Proc. Amer. Math. Soc. 19 (1968), 425\u2013428.","journal-title":"Proc. Amer. Math. Soc."},{"key":"3149_CR22","doi-asserted-by":"publisher","first-page":"43","DOI":"10.4153\/CJM-1983-003-x","volume":"35","author":"R. Edwards","year":"1983","unstructured":"R. Edwards and D. Gillman: Any spine of the cube is 2-collapsible, Canad. J. Math. 35 (1983), 43\u201348.","journal-title":"Canad. J. Math."},{"key":"3149_CR23","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1016\/0040-9383(83)90017-4","volume":"22","author":"D. Gillman","year":"1983","unstructured":"D. Gillman and D. Rolfsen: The Zeeman conjecture for standard spines is equivalent to the Poincar\u00e9 conjecture, Topology 22 (1983), 315\u2013323.","journal-title":"Topology"},{"key":"3149_CR24","volume-title":"Van Nostrand Reinhold Mathematical Studies","author":"L. C. Glaser","year":"1970","unstructured":"L. C. Glaser: Geometrical combinatorial topology, i., Van Nostrand Reinhold Mathematical Studies Vol. 27. New York, 1970."},{"key":"3149_CR25","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1017\/S0305004100042511","volume":"64","author":"R. E. Goodrick","year":"1968","unstructured":"R. E. Goodrick: Non-simplicially collapsible triangulations of I n, Proc. Cambridge Philos. Soc. 64 (1968), 31\u201336.","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"3149_CR26","volume-title":"Graduate Texts in Mathematics","author":"B. Gr\u00fcnbaum","year":"2003","unstructured":"B. Gr\u00fcnbaum: Convex polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003."},{"key":"3149_CR27","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s002090000129","volume":"235","author":"M. Hachimori","year":"2000","unstructured":"M. Hachimori and G. M. Ziegler: Decompositons of simplicial balls and spheres with knots consisting of few edges, Math. Z. 235 (2000), 159\u2013171.","journal-title":"Math. Z."},{"key":"3149_CR28","first-page":"35","volume":"20","author":"G. Hetyei","year":"1996","unstructured":"G. Hetyei: Invariants des complexes cubiques, Ann. Sci. Math. Qu\u00e9. 20 (1996), 35\u201352 (French).","journal-title":"Ann. Sci. Math. Qu\u00e9."},{"key":"3149_CR29","volume-title":"Piecewise Linear Topology","author":"J. F. P. Hudson","year":"1969","unstructured":"J. F. P. Hudson: Piecewise Linear Topology, University of Chicago Lecture Notes, W. A. Benjamin, Inc., New York-Amsterdam, 1969."},{"key":"3149_CR30","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/BF02579140","volume":"4","author":"J. Kahn","year":"1984","unstructured":"J. Kahn, M. Saks and D. Sturtevant: A topological approach to evasiveness, Combinatorica 4 (1984), 297\u2013306.","journal-title":"Combinatorica"},{"key":"3149_CR31","first-page":"157","volume-title":"Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969)","author":"W. B. R. Lickorish","year":"1970","unstructured":"W. B. R. Lickorish: On collapsing X 2\u00d7I, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, 157\u2013160."},{"key":"3149_CR32","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1016\/S0195-6698(13)80103-5","volume":"12","author":"W. B. R. Lickorish","year":"1991","unstructured":"W. B. R. Lickorish: Unshellable triangulations of spheres, European J. Combin. 12 (1991), 527\u2013530.","journal-title":"European J. Combin."},{"key":"3149_CR33","first-page":"299","volume-title":"Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr.","author":"W. B. R. Lickorish","year":"1999","unstructured":"W. B. R. Lickorish: Simplicial moves on complexes and manifolds, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, 299\u2013320 (electronic)."},{"key":"3149_CR34","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1017\/CBO9780511629358.013","volume-title":"Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser.","author":"S. Matveev","year":"1993","unstructured":"S. Matveev and D. Rolfsen: Zeeman's collapsing conjecture, Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser., vol. 197, Cambridge Univ. Press, Cambridge, 1993, 335\u2013364."},{"key":"3149_CR35","first-page":"610","volume":"29","author":"M. H. A. Newman","year":"1926","unstructured":"M. H. A. Newman: On the foundations of combinatorial analysis situs, Proc. Royal Acad. Amsterdam, vol. 29, 1926, 610\u2013641.","journal-title":"Proc. Royal Acad. Amsterdam"},{"key":"3149_CR36","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF02941601","volume":"57","author":"U. Pachner","year":"1987","unstructured":"U. Pachner: Konstruktionsmethoden und das kombinatorische Hom\u00f6omorphieproblem f\u00fcr Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 57 (1987), 69\u201386.","journal-title":"Abh. Math. Sem. Univ. Hamburg"},{"key":"3149_CR37","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1287\/moor.5.4.576","volume":"5","author":"J. S. Provan","year":"1980","unstructured":"J. S. Provan and L. J. Billera: Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576\u2013594.","journal-title":"Math. Oper. Res."},{"key":"3149_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-81735-9","volume-title":"Introduction to Piecewise-Linear Topology","author":"C. P. Rourke","year":"1972","unstructured":"C. P. Rourke and B. J. Sanderson: Introduction to Piecewise-Linear Topology, Springer, New York, 1972, Ergebnisse Series vol. 69."},{"key":"3149_CR39","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1090\/S0002-9904-1958-10168-8","volume":"64","author":"M. E. Rudin","year":"1958","unstructured":"M. E. Rudin: An unshellable triangulation of a tetrahedron, Bull. Amer. Math. Soc. 64 (1958), 90\u201391.","journal-title":"Bull. Amer. Math. Soc."},{"issue":"1","key":"3149_CR40","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/BF01218376","volume":"178","author":"P. Schenzel","year":"1981","unstructured":"P. Schenzel: On the number of faces of simplicial complexes and the purity of Frobe-nius, Math. Z. 178 (1) (1981), 125\u2013142.","journal-title":"Math. Z."},{"key":"3149_CR41","volume-title":"Progress in Mathematics","author":"R. P. Stanley","year":"1983","unstructured":"R. P. Stanley: Combinatorics and Commutative Algebra, Progress in Mathematics, vol. 41, Birkh\u00e4user Boston Inc., Boston, MA, 1983."},{"key":"3149_CR42","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1007\/s00208-006-0756-5","volume":"335","author":"G. C. Verchota","year":"2006","unstructured":"G. C. Verchota and A. L. Vogel: The multidirectional Neumann problem in R4, Math. Ann. 335 (2006), 571\u2013644.","journal-title":"Math. Ann."},{"key":"3149_CR43","first-page":"71","volume":"29","author":"I. A. Volodin","year":"1974","unstructured":"I. A. Volodin, V. E. Kuznecov and A. T. Fomenko: The problem of the algorithmic discrimination of the standard three-dimensional sphere, Uspehi Mat. Nauk 29 (1974), 71\u2013168, Appendix by S. P. Novikov.","journal-title":"Uspehi Mat. Nauk"},{"key":"3149_CR44","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/S0012-365X(99)00049-7","volume":"207","author":"V. Welker","year":"1999","unstructured":"V. Welker: Constructions preserving evasiveness and collapsibility, Discrete Math. 207 (1999), 243\u2013255.","journal-title":"Discrete Math."},{"key":"3149_CR45","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1112\/plms\/s2-45.1.243","volume":"S2-45","author":"J. H. C. Whitehead","year":"1939","unstructured":"J. H. C. Whitehead: Simplicial Spaces, Nuclei and m-Groups, Proc. London Math. Soc. S2-45 (1939), 243.","journal-title":"Proc. London Math. Soc."},{"key":"3149_CR46","volume-title":"Seminar on Combinatorial Topology","author":"E. C. Zeeman","year":"1966","unstructured":"E. C. Zeeman Seminar on Combinatorial Topology, Institut des Hautes Etudes Scientifiques, Paris, 1966 (English)."},{"key":"3149_CR47","volume-title":"Graduate Texts in Mathematics","author":"G. M. Ziegler","year":"1995","unstructured":"G. M. Ziegler: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995."}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-016-3149-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00493-016-3149-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-016-3149-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-016-3149-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,24]],"date-time":"2017-06-24T16:01:15Z","timestamp":1498320075000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00493-016-3149-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,9]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,2]]}},"alternative-id":["3149"],"URL":"https:\/\/doi.org\/10.1007\/s00493-016-3149-8","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"value":"0209-9683","type":"print"},{"value":"1439-6912","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,6,9]]}}}