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## Discrete Morse Theory for Manifolds with Boundary (2012)

Citations: | 14 - 8 self |

### Citations

2041 |
Commutative algebra, with a view towards algebraic geometry
- Eisenbud
- 1995
(Show Context)
Citation Context ...≤ dim M; wecall a pseudo-manifold M endo-collapsible if cdepth M =dimM. 2.6. Stanley–Reisner rings, algebraic depth, Cohen–Macaulayness. We sketch a few definitions and results, referring to Eisenbud =-=[23]-=- or Miller–Sturmfels [52] for details. A Noetherian ring is a commutative ring all of whose ideals are finitely generated. The quotient of a Noetherian ring (modulo some ideal) is still a Noetherian r... |

1813 | Algebraic Topology - Hatcher - 2001 |

939 | The entropy formula for the Ricci flow and its geometric applications. ArXiv Mathematics e-prints,
- Perelman
- 2002
(Show Context)
Citation Context ...t is an outstanding result in algebraic topology that all manifolds with the same homotopy as a sphere are spheres. This was proven by Smale [69] for d ≥ 5, by Freedman [27] for d = 4 and by Perelman =-=[60, 61]-=- for d =3. In contrast, manifolds that are homology pseudo-spheres but not spheres exist in all dimensions d ≥ 3. None of them is simply connected: Every simply connected manifold with the same homolo... |

448 | Ricci flow with surgery on three-manifolds
- Perelman
- 2003
(Show Context)
Citation Context ...t is an outstanding result in algebraic topology that all manifolds with the same homotopy as a sphere are spheres. This was proven by Smale [69] for d ≥ 5, by Freedman [27] for d = 4 and by Perelman =-=[60, 61]-=- for d =3. In contrast, manifolds that are homology pseudo-spheres but not spheres exist in all dimensions d ≥ 3. None of them is simply connected: Every simply connected manifold with the same homolo... |

379 |
The topology of four-dimensional manifolds,
- Freedman
- 1982
(Show Context)
Citation Context ... is a homology pseudo-sphere. It is an outstanding result in algebraic topology that all manifolds with the same homotopy as a sphere are spheres. This was proven by Smale [69] for d ≥ 5, by Freedman =-=[27]-=- for d = 4 and by Perelman [60, 61] for d =3. In contrast, manifolds that are homology pseudo-spheres but not spheres exist in all dimensions d ≥ 3. None of them is simply connected: Every simply conn... |

265 |
Stratified Morse theory.
- Goresky, MacPherson
- 1988
(Show Context)
Citation Context ...ion. Bott weakened Morse’s original definition in order to also allow functions whose critical cells form submanifolds rather than isolated points (e.g. constant functions); see [31, pp. 159–162] and =-=[30]-=-. Any Morse function provides information on the homology of a manifold. For example, the number of critical points of index i is not less than the i-th Betti numbers of the manifold. These bounds are... |

235 | Elements of homotopy theory - Whitehead |

233 | A discrete Morse theory for cell complexes.
- Forman
- 1995
(Show Context)
Citation Context ... are precisely the critical cells of f. Each complex K endowed with a discrete Morse function is homotopy equivalent to a cell complex with exactly one k-cell for each critical simplex of dimension k =-=[25]-=-. For example, the complex in Figure 2 has one critical cell, which is 0-dimensional. Therefore, it is homotopy equivalent to a point. Let M be a d-pseudo-manifold and let f be a discrete Morse functi... |

203 | Munkres. Elements of Algebraic Topology - R |

198 |
A Survey of Knot Theory,
- Kawauchi
- 1996
(Show Context)
Citation Context ...the fundamental group of the knot complement inside the 3-sphere. For further definitions, e.g., of connected sums of knots, of 2-bridge knots, of spanning arcs and so on, we refer either to Kawauchi =-=[42]-=- or to our paper with Ziegler [8]. “LC 3-balls” and “endo-collapsible 3-balls” are the same. (The proof is elementary; see e.g. Corollary 4.32.) In this section, we prove that endo-collapsible 3-balls... |

165 | Topological methods - Björner - 1995 |

141 |
Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes,
- Hochster
- 1972
(Show Context)
Citation Context ...he barycentric subdivision of a constructible simplicial d-ball is collapsible (Theorem 4.13). Constructibility is a strengthening of the Cohen-Macaulay notion introduced by Hochster in the seventies =-=[38]-=-. Recursively, a pure d-complex C is constructible if and only if either C has only one facet, or C = C1 ∪ C2, where (i) C1 and C2 are constructible d-complexes and (ii) C1 ∩ C2 is a constructible (d ... |

125 | Combinatorial commutative algebra,
- Miller, Sturmfels
- 2005
(Show Context)
Citation Context ...manifold M endo-collapsible if cdepth M =dimM. 2.6. Stanley–Reisner rings, algebraic depth, Cohen–Macaulayness. We sketch a few definitions and results, referring to Eisenbud [23] or Miller–Sturmfels =-=[52]-=- for details. A Noetherian ring is a commutative ring all of whose ideals are finitely generated. The quotient of a Noetherian ring (modulo some ideal) is still a Noetherian ring. A zero-divisor is an... |

104 |
On manifolds homeomorphic to the 7-sphere.
- Milnor
- 1956
(Show Context)
Citation Context ...on on a smooth manifold without boundary has only two critical points, then the manifold is a d-sphere. The Reeb theorem is a crucial ingredient in Milnor’s proof of the existence of “exotic spheres” =-=[53]-=- and in Smale’s proof of the generalized Poincaré conjecture [69]. The idea of duality is at the core of Morse theory. If f is a smooth Morse function on a topological manifold without boundary M, soi... |

91 | Shellable decompositions of cells and spheres - Bruggersser, Mani - 1971 |

83 |
On the structure of manifolds,
- Smale
- 1962
(Show Context)
Citation Context ...ints, then the manifold is a d-sphere. The Reeb theorem is a crucial ingredient in Milnor’s proof of the existence of “exotic spheres” [53] and in Smale’s proof of the generalized Poincaré conjecture =-=[69]-=-. The idea of duality is at the core of Morse theory. If f is a smooth Morse function on a topological manifold without boundary M, sois−f. The critical points of f and −f are the same, except that th... |

76 |
Decompositions of simplicial complexes related to diameters of convex polyhedra,
- Provan, Billera
- 1980
(Show Context)
Citation Context ...ngworth [16] showed that any geometric subdivision of a convex 3-polytope collapses onto its boundary minus a facet. In particular, Rudin’s ball is endo-collapsible. In fact, it is also constructible =-=[63]-=-. If a d-sphere S splits into two endo-collapsible balls that intersect in an endocollapsible (d − 1)-sphere, by Theorem 3.18 S is endo-collapsible. The next lemma shows that the requirement on the in... |

68 |
Simplicial spaces, nuclei and m-groups,
- Whitehead
- 1939
(Show Context)
Citation Context ... 39, 41, 44] and to commutative algebra [3, 40, 68]. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties =-=[74]-=-. A complex C is collapsible if it admits a discrete Morse function with only one critical cell. All collapsible complexes are contractible, but the converse does not hold, as shown by Zeeman’s dunce ... |

67 |
On discrete Morse functions and combinatorial decompositions, Discrete Math
- Chari
(Show Context)
Citation Context ...e cannot be locally constructible if the knot is complicated enough [8]. In Section 3.2, we extend this result to manifolds with arbitrary discrete Morse functions, thus answering a question of Chari =-=[15]-=-. Main Theorem 6 (Theorem 4.18). For each integer d ≥ 3 and for each m ≥ 0, there is a PL d-sphere on which any discrete Morse function has more than m critical (d − 1)-cells. The same can be said of ... |

56 |
Shrinking cell-like decompositions of manifolds. Codimension three
- Cannon
- 1979
(Show Context)
Citation Context ...ds are PL if d ≤ 4 [13, p. 10]. However, some 5-spheres and some 5balls are not PL [20]. If a manifold H is a homology d-pseudo-sphere but not a sphere, the double suspension of H is a (d + 2)-sphere =-=[14]-=- which is not PL. (The single suspension of H is a (d + 1)-pseudo-sphere but not a manifold.) If d ≥ 5, every PL manifold that is a homology d-pseudo-sphere but not a sphere bounds a contractible PL (... |

50 |
Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer
- Pachner
- 1986
(Show Context)
Citation Context ...manifold that is a homology d-pseudo-sphere but not a sphere bounds a contractible PL (d + 1)-manifold with boundary that is not a ball [2, 62]. Every PL d-sphere bounds a collapsible PL (d + 1)-ball =-=[59, 74]-=-; see below for the definition of collapsibility. Licensed to Penn St Univ, University Park. Prepared on Fri Jul 5 10:00:06 EDT 2013 for download from IP 130.203.136.75. License or copyright restricti... |

49 |
Some aspects of the topology of 3–manifolds related to the Poincaré conjecture, Lectures on modern mathematics
- Bing
- 1964
(Show Context)
Citation Context ...alls are collapsible is open for each d ≥ 4[8]. Newman showed that all 2-balls are shellable [57]; in particular, they are all constructible and collapsible. However, some 3-balls are not collapsible =-=[9]-=-. Every constructible 3-ball B with more than one facet is splittable, that is, it contains an Licensed to Penn St Univ, University Park. Prepared on Fri Jul 5 10:00:06 EDT 2013 for download from IP 1... |

46 | Combinatorial Algebraic Topology - Kozlov - 2007 |

34 | Discrete Morse theory for cellular resolutions,
- Batzies, Welker
- 2002
(Show Context)
Citation Context ... This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry [24, 39, 41, 44] and to commutative algebra =-=[3, 40, 68]-=-. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A complex C is collapsible if it admits a dis... |

34 |
Relations between the critical points of a real function of n independent variables,
- Morse
- 1925
(Show Context)
Citation Context ...ntroduction Morse theory analyzes the topology of a manifold M by looking at smooth functions f : M −→ R whose critical points are non-degenerate. This was introduced by Marston Morse in the twenties =-=[55]-=-; it was developed in the second half of the 20th century by Bott, Thom, Milnor, Smale, Witten, Goresky and MacPherson (among others); and it was used by Smale and Freedman to prove the Poincaré conje... |

32 |
A note on some contractible 4-manifolds’,
- Mazur
- 1961
(Show Context)
Citation Context ...er a ball or a sphere. Whether the PL assumption is necessary or not in Whitehead’s result still represents an open problem. However, the collapsibility assumption is necessary: Newman [58] and Mazur =-=[50]-=- showed that the boundary of a contractible PL manifold is a homology sphere that need not be simply connected. Within the PL framework, discrete Morse functions dualize. With any triangulation of M w... |

30 |
An unshellable triangulation of a tetrahedron
- Rudin
- 1958
(Show Context)
Citation Context ...answer is “yes” for boundaries of polytopes; see [78, p. 243].) Corollary 4.5 could be viewed as a step towards a positive answer, since all shellable balls are collapsible. Example 4.7. Rudin’s ball =-=[65]-=- is a non-shellable geometric subdivision of a tetrahedron. Chillingworth [16] showed that any geometric subdivision of a convex 3-polytope collapses onto its boundary minus a facet. In particular, Ru... |

26 | Lectures on polytopes, Vol. 152 of Graduate Texts in Mathematics - Ziegler - 1995 |

24 | Optimal discrete Morse functions for 2-manifolds
- Lewiner, Lopes, et al.
(Show Context)
Citation Context ...al cells are, in fact, triangulated spheres. This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry =-=[24, 39, 41, 44]-=- and to commutative algebra [3, 40, 68]. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A comp... |

23 | The double suspension of a certain homology 3–sphere is - Edwards - 1975 |

21 |
Minimal Resolutions via Algebraic Discrete Morse Theory.
- Jollenbeck, Welker
- 2009
(Show Context)
Citation Context ... This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry [24, 39, 41, 44] and to commutative algebra =-=[3, 40, 68]-=-. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A complex C is collapsible if it admits a dis... |

21 |
Constructions preserving evasiveness and collapsibility,
- Welker
- 1999
(Show Context)
Citation Context ...ii) If B is an endo-collapsible (d−1)-ball contained in an endo-collapsible (d−1)sphere A, thed-ball v ∗ A, wherevis a new vertex, collapses onto A − B. Proof. Item (i) is well known; see e.g. Welker =-=[72]-=-. Item (ii) follows directly from Theorem 3.19. Item (iii) follows from item (ii) and Theorem 3.18 (or alternatively, from item (ii), Lemma 4.8 and the fact that all cones are collapsible). Item (iv) ... |

20 | Computing optimal Morse matchings
- Joswig, Pfetsch
- 408
(Show Context)
Citation Context ...al cells are, in fact, triangulated spheres. This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry =-=[24, 39, 41, 44]-=- and to commutative algebra [3, 40, 68]. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A comp... |

19 | Decompositions of simplicial balls and spheres with knots consisting of few edges
- Hachimori, Ziegler
(Show Context)
Citation Context ... discovered in the sixties by Bing [9, 29], is sharp for all r [47]. In contrast, if the knot is the sum of 2r−1 + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not constructible =-=[22, 34, 35]-=-. (For r = 1 this bound is “off by one” [35], [Lemma 1]; we do not know whether it is sharp for some r.) Is there an analogous theorem for endo-collapsibility with an intermediate bound between 2 r−1 ... |

17 |
Morse theory indomitable
- Bott
- 1988
(Show Context)
Citation Context ...s not less than the i-th Betti numbers of the manifold. These bounds are usually not sharp, but in some cases they can be made sharp using Smale’s cancellation theorem; see e.g. [31, pp. 196– 199] or =-=[11, 64, 66]-=-. In one case Morse theory even reveals the homeomorphism type: Reeb showed that if a Morse function on a smooth manifold without boundary has only two critical points, then the manifold is a d-sphere... |

17 |
Morse theory from an algebraic viewpoint.
- Skoldberg
- 2006
(Show Context)
Citation Context ... This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry [24, 39, 41, 44] and to commutative algebra =-=[3, 40, 68]-=-. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A complex C is collapsible if it admits a dis... |

15 |
On the dunce hat
- Zeeman
(Show Context)
Citation Context ...complex C is collapsible if it admits a discrete Morse function with only one critical cell. All collapsible complexes are contractible, but the converse does not hold, as shown by Zeeman’s dunce hat =-=[7, 75]-=-. All 2-balls are collapsible, but for each d ≥ 3somePLd-balls are not collapsible. However, Whitehead proved that every collapsible PL d-manifold is a ball [74]. Here PL stands for “piecewise-linear”... |

14 | On locally constructible spheres and balls
- Benedetti, Ziegler
(Show Context)
Citation Context ...ll of these implications are strict. The local constructibility notion mentioned above was introduced in connection with discrete quantum gravity in [19] and later linked to combinatorial topology in =-=[8]-=-. For any fixed d ≥ 2, arbitrary simplicial d-manifolds are much more than exponentially many, when counted with respect to the number of facets [4, Chapter 2]. However, there are only exponentially m... |

14 |
Non-simplicially collapsible triangulations of In
- Goodrick
- 1968
(Show Context)
Citation Context ...ry path forms a knot. If the knot is the sum of 2r + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not collapsible. This surprising obstruction, discovered in the sixties by Bing =-=[9, 29]-=-, is sharp for all r [47]. In contrast, if the knot is the sum of 2r−1 + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not constructible [22, 34, 35]. (For r = 1 this bound is “of... |

14 | Small examples of nonconstructible simplicial balls and spheres
- Lutz
(Show Context)
Citation Context ...nd 3-spheres contain 3-edge knots in their 1-skeleton. Proof. Thefirstpartisthecasee =1,k ≥ 1 of Theorem 4.19. For the second part, consider Lutz’s sphere S 3 13,56 with 13 vertices and 56 tetrahedra =-=[49]-=-. This simplicial 3-sphere contains a trefoil knot on 3 edges. Let B13,55 be the 3-ball obtained by removing from S 3 13,56 the tetrahedron Δ = {1, 2, 6, 9}. With the help of a Licensed to Penn St Uni... |

14 |
Discrete Morse Theory for Complexes of 2-Connected Graphs
- Shareshian
(Show Context)
Citation Context ...h maps are strictly increasing [26]. Moving downwards along some gradient flow line is rendered in the discrete setting by an acyclic matching of the face poset (see Figure 2). The work of Shareshian =-=[67]-=- yields a partial discrete analogue to Bott’s relative Morse theory: Given a discrete Morse function on a simplicial complex, the subcomplexes consisting only of critical cells can be “divided out” [6... |

13 |
Remarks on the entropy of 3-manifolds
- Durhuus, Jónsson
- 1995
(Show Context)
Citation Context ... constructible =⇒ simply connected, and for d ≥ 3 all of these implications are strict. The local constructibility notion mentioned above was introduced in connection with discrete quantum gravity in =-=[19]-=- and later linked to combinatorial topology in [8]. For any fixed d ≥ 2, arbitrary simplicial d-manifolds are much more than exponentially many, when counted with respect to the number of facets [4, C... |

13 |
Functions on Manifolds: Algebraic and Topological Aspects. Trans. Mathematical Monographs 506
- Sharko
- 1990
(Show Context)
Citation Context ...s not less than the i-th Betti numbers of the manifold. These bounds are usually not sharp, but in some cases they can be made sharp using Smale’s cancellation theorem; see e.g. [31, pp. 196– 199] or =-=[11, 64, 66]-=-. In one case Morse theory even reveals the homeomorphism type: Reeb showed that if a Morse function on a smooth manifold without boundary has only two critical points, then the manifold is a d-sphere... |

12 | Non-constructible complexes and the bridge index
- Ehrenborg, Hachimori
(Show Context)
Citation Context ... discovered in the sixties by Bing [9, 29], is sharp for all r [47]. In contrast, if the knot is the sum of 2r−1 + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not constructible =-=[22, 34, 35]-=-. (For r = 1 this bound is “off by one” [35], [Lemma 1]; we do not know whether it is sharp for some r.) Is there an analogous theorem for endo-collapsibility with an intermediate bound between 2 r−1 ... |

11 |
Collapsing three-dimensional convex polyhedra
- Chillingworth
- 1967
(Show Context)
Citation Context ...could be viewed as a step towards a positive answer, since all shellable balls are collapsible. Example 4.7. Rudin’s ball [65] is a non-shellable geometric subdivision of a tetrahedron. Chillingworth =-=[16]-=- showed that any geometric subdivision of a convex 3-polytope collapses onto its boundary minus a facet. In particular, Rudin’s ball is endo-collapsible. In fact, it is also constructible [63]. If a d... |

11 |
Zur Grundlegung der kombinatorischen Topologie. Abhandlungen aus dem Mathematischen Seminar der Universitt Hamburg 3
- Furch
- 1924
(Show Context)
Citation Context ...t by Bing: A ball with a knotted spanning edge cannot be collapsible if the knot is the double trefoil. In fact, its barycentric subdivision is not endo-collapsible by Theorem 4.19. Lemma 4.16 (Furch =-=[28]-=-, Bing [9]). Any (tame) knot can be realized as a 3-edge subcomplex of some (simplicial) 3-sphere and as a knotted spanning edge of some (simplicial) 3-ball. Proof. Let K be an arbitrary knot. Take a ... |

11 | Combinatorics of constructible complexes
- Hachimori
(Show Context)
Citation Context ...lls are collapsible. Since constructibility is a slight weakening of the shellability property, it is natural to ask whether all constructible balls are collapsible. (This was also asked by Hachimori =-=[32]-=- and in our work with Ziegler [8].) The problem is open for all dimensions greater than three. Here we present a partial positive answer for all dimensions. Lemma 4.12. The link of any non-empty face ... |

10 |
Geometrical connectivity
- Wall
(Show Context)
Citation Context ...depth B = 1. However, hdepth B =dimB= 3 because B is collapsible. □ In the paper [6] we show how the collapse depth of a manifold is deeply related to its geometrical connectivity, introduced by Wall =-=[71]-=-. We conclude the present work with a hierarchy of some of the properties we studied so far: Theorem 4.34. For d-manifolds (d ≥ 2), shellable =⇒ constructible =⇒ endo-collapsible =⇒ LC =⇒ simply conne... |

10 |
Unknotting combinatorial balls
- ZEEMAN
- 1978
(Show Context)
Citation Context ...boundary disk of the second one. The resulting complex is still a 3-ball. More generally, for any positive integer d one can “patch” together two d-balls alongside a (d−1)-ball in their boundary; cf. =-=[77]-=-. Unfortunately, discrete Morse theory does not relate well to this topological operation. For example, if we patch together two collapsible 3-balls, do we get a collapsible 3-ball? As far as we know,... |

9 |
A vertex-minimal non-shellable simplicial 3-ball with 9 vertices and 18 facets
- LUTZ
(Show Context)
Citation Context ...s attached to the previous ones alongside some (or maybe all) of its boundary facets. All shellable complexes are constructible, but some constructible 2-complexes [33] and some constructible 3-balls =-=[48]-=- are not shellable. Also, it is easy to show that all shellable balls are collapsible [18], but the question of whether all constructible d-balls are collapsible is open for each d ≥ 4[8]. Newman show... |

8 | Fragments of geometric topology from the sixties, - Buoncristiano, Rourke - 2003 |

8 |
Witten-Morse theory for cell complexes. Topology 37
- FORMAN
- 1998
(Show Context)
Citation Context ...ular CW complexes and smooth functions are replaced by special poset maps between the face poset of the complex and (R, ≤). ‘Critical points’ are just cells at which such maps are strictly increasing =-=[26]-=-. Moving downwards along some gradient flow line is rendered in the discrete setting by an acyclic matching of the face poset (see Figure 2). The work of Shareshian [67] yields a partial discrete anal... |

8 |
Decomposition of two dimensional simplicial complexes
- Hachimori
(Show Context)
Citation Context ...at a time so that each new simplex is attached to the previous ones alongside some (or maybe all) of its boundary facets. All shellable complexes are constructible, but some constructible 2-complexes =-=[33]-=- and some constructible 3-balls [48] are not shellable. Also, it is easy to show that all shellable balls are collapsible [18], but the question of whether all constructible d-balls are collapsible is... |

8 |
On the Cohen-Macaulay property in commutative algebra and simplicial topology
- Smith
- 1990
(Show Context)
Citation Context ...Krull-dim F[C]. A complex that is Cohen– Macaulay over any field is simply called Cohen-Macaulay. The algebraic depth of C equals the maximal integer k for which the k-skeleton of C is Cohen–Macaulay =-=[70]-=-. In particular, for each d ≥ 1, a d-complex C is connected if and only if adepth F C ≥ 1 for all fields F. In general, adepth F C depends on the field: For example, (any triangulation of) the project... |

8 |
Seminar on Combinatorial Topology. Institut des Hautes ´Etudes Scientifiques
- ZEEMAN
- 1966
(Show Context)
Citation Context ...cet of C1 and in exactly one facet of C2. Therefore, the Ci are constructible pseudo-manifolds with non-empty boundary (because C1 ∩ C2 is contained in the boundary of each Ci). By a result of Zeeman =-=[76]-=-, both the Ci are d-balls. C1 ∩C2 is either a ball or a sphere, depending on whether M is a ball or a sphere. By induction, we can assume that C1, C2 and C1 ∩ C2 are endo-collapsible. Using Theorem 3.... |

7 |
The dunce hat and a minimal non-extendably collapsible 3-ball
- BENEDETTI, LUTZ
(Show Context)
Citation Context ...complex C is collapsible if it admits a discrete Morse function with only one critical cell. All collapsible complexes are contractible, but the converse does not hold, as shown by Zeeman’s dunce hat =-=[7, 75]-=-. All 2-balls are collapsible, but for each d ≥ 3somePLd-balls are not collapsible. However, Whitehead proved that every collapsible PL d-manifold is a ball [74]. Here PL stands for “piecewise-linear”... |

7 |
Triangulations of the 3-ball with knotted spanning 1simplexes and collapsible r-th derived subdivisions
- LICKORISH, MARTIN
- 1972
(Show Context)
Citation Context ... knot is the sum of 2r + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not collapsible. This surprising obstruction, discovered in the sixties by Bing [9, 29], is sharp for all r =-=[47]-=-. In contrast, if the knot is the sum of 2r−1 + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not constructible [22, 34, 35]. (For r = 1 this bound is “off by one” [35], [Lemma 1]... |

6 |
Kosta, Ascending and descending regions of a discrete Morse function
- Jers̆e, Mramor
(Show Context)
Citation Context ...al cells are, in fact, triangulated spheres. This was first proven by Forman in [25, p. 112]; an alternative proof is given in Theorem 3.8. Discrete Morse theory opens doors to computational geometry =-=[24, 39, 41, 44]-=- and to commutative algebra [3, 40, 68]. At the same time, it builds on a classical notion in combinatorial topology, the notion of collapse introduced by J.H.C. Whitehead in the thirties [74]. A comp... |

5 |
Compact contractible n-manifolds have arc spines (n ≥ 5
- ANCEL, GUILBAULT
- 1995
(Show Context)
Citation Context ...ollary 7.8, p. 180]. In contrast, a manifold with the same homotopy as a ball need not be a ball: The boundary of a contractible manifold is a homology pseudo-sphere, not necessarily simply connected =-=[58, 50, 2]-=-. If M is a contractible d-manifold and ∂M is a (d − 1)sphere, then M is a d-ball; see e.g. [2]. 2.3. PL manifolds and Poincaré–Lefschetz duality. The barycentric subdivision sd C of a complex C is th... |

5 |
Locally Constructible Manifolds
- Benedetti
- 2010
(Show Context)
Citation Context ...tially many, when counted with respect to the number of facets [4, Chapter 2]. However, there are only exponentially many different combinatorial types of locally constructible simplicial d-manifolds =-=[4, 8, 19]-=-. Thus we essentially control the number of endo-collapsible manifolds. A knotted triangulation of the 3-sphere cannot be locally constructible if the knot is complicated enough [8]. In Section 3.2, w... |

5 | Collapses, products and LC manifolds - Benedetti |

5 |
Collapsing a triangulation of a “knotted” cell
- Hamstrom, Jerrard
- 1969
(Show Context)
Citation Context ...ut rather an application of Theorem 3.15. Corollary 4.21 establishes a concrete difference between collapsibility and endocollapsibility. In fact, some collapsible 3-balls have knotted spanning edges =-=[8, 36, 47]-=-. We also show that the barycentric subdivision of a collapsible 3-ball is endocollapsible (Proposition 4.23). In particular, we re-cover a result by Bing: A ball with a knotted spanning edge cannot b... |

5 |
Boundaries of ULC sets in Euclidean n-space
- Newman
- 1948
(Show Context)
Citation Context ...ny face is either a ball or a sphere. Whether the PL assumption is necessary or not in Whitehead’s result still represents an open problem. However, the collapsibility assumption is necessary: Newman =-=[58]-=- and Mazur [50] showed that the boundary of a contractible PL manifold is a homology sphere that need not be simply connected. Within the PL framework, discrete Morse functions dualize. With any trian... |

5 |
Introduction to picewise-linear topology
- ROURKE, SANDERSON
- 1972
(Show Context)
Citation Context ...s not less than the i-th Betti numbers of the manifold. These bounds are usually not sharp, but in some cases they can be made sharp using Smale’s cancellation theorem; see e.g. [31, pp. 196– 199] or =-=[11, 64, 66]-=-. In one case Morse theory even reveals the homeomorphism type: Reeb showed that if a Morse function on a smooth manifold without boundary has only two critical points, then the manifold is a d-sphere... |

4 |
On stars and links of shellable complexes
- COURDURIER
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Citation Context ...d B is collapsible. However, sd link∂B v = sd linkS ′ v =sdS is not LC. Remark 4.15. Is every link in a shellable complex shellable? We know the answer is positive for simplicial or cubical complexes =-=[17]-=-, but in general this still seems to be an open problem. 4.2. Knots inside 3-manifolds. A spanning edge in a 3-ball B is an interior edge that has both endpoints on ∂B. A K-knotted spanning edge of a ... |

4 |
duality for projections of polytopes
- Alexander
(Show Context)
Citation Context ...shellable complexes are constructible, but some constructible 2-complexes [33] and some constructible 3-balls [48] are not shellable. Also, it is easy to show that all shellable balls are collapsible =-=[18]-=-, but the question of whether all constructible d-balls are collapsible is open for each d ≥ 4[8]. Newman showed that all 2-balls are shellable [57]; in particular, they are all constructible and coll... |

4 | Hamiltonian Submanifolds of Regular Polytopes, Discrete Comput. Geometry 43
- Effenberger, Kühnel
- 2010
(Show Context)
Citation Context ...epeat the definition here. We called the other notion “hamiltonian depth”. The name comes from the fact that a subcomplex H of a complex C is called k-hamiltonian if the k-skeleta of H and C coincide =-=[21]-=-. Definition 4.1 (Hochster, Reisner [52, p. 101]). Given a regular CW complex C and a field F, we define { adepthF C =max m : ˜ Hi (linkC σ; F) = 0 for all faces σ } and for all integers i<m−dim σ − 1... |

4 | Morse theory in the 1990s - GUEST - 2002 |

4 |
Tangle sum and constructible spheres
- Hachimori, Shimokawa
(Show Context)
Citation Context ... discovered in the sixties by Bing [9, 29], is sharp for all r [47]. In contrast, if the knot is the sum of 2r−1 + 1 trefoils, then the r-th barycentric subdivision of the 3-ball is not constructible =-=[22, 34, 35]-=-. (For r = 1 this bound is “off by one” [35], [Lemma 1]; we do not know whether it is sharp for some r.) Is there an analogous theorem for endo-collapsibility with an intermediate bound between 2 r−1 ... |

4 | Reciprocal domains and Cohen-Macaulay d-complexes in Rd
- Miller, Reiner
(Show Context)
Citation Context ...t manifolds with different topologies may be Cohen–Macaulay as well. For example, Miller and Reiner found a PL Cohen– Macaulay non-simply-connected 4-manifold that embeds in R4 without being a 4-ball =-=[51]-=-. Here we present two similar notions (dual to one another) that seem equally interesting from the combinatorial point of view. The goal is to assign to each manifold an integer that (1) tells us abou... |

3 | Discrete Morse theory is at least as perfect as Morse Theory. Preprint at arXiv
- Benedetti
(Show Context)
Citation Context ...e) Morse functions on sd M with the same number of critical cells [25]. Sometimes a triangulation “improves” by taking subdivisions, so sd M might admit Morse functions with fewer critical cells; see =-=[6]-=-. The last result of this section shows how to get from a boundary-critical discrete Morse function on a manifold M with boundary to a discrete Morse function on sd M. Then we show how to obtain a bou... |

3 |
An unsplittable triangulation
- LICKORISH
- 1971
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Citation Context ...r copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-useDISCRETE MORSE THEORY FOR MANIFOLDS WITH BOUNDARY 6657 embedded 2-disk D such that ∂D ⊂ ∂B. Lickorish =-=[45]-=- showed that some 3-balls do not split. By taking cones over Lickorish’s example, one obtains non-constructible d-balls for each d ≥ 3. The reason for the existence of non-constructible spheres and no... |

3 | Fifty years ago: topology of manifolds in the 50’s and 60’s
- MILNOR
- 2009
(Show Context)
Citation Context ...plex. A PL + d-sphere is a sphere piecewise-linearly homeomorphic to the boundary of a (d +1)-simplex. (All PL + d-spheres and d-balls are PL. The converse is true for all d = 4and unknown for d = 4 =-=[54]-=-: A priori some 4-spheres or 4-balls might not be PL + , whereas all 4-manifolds are PL.) Theorem 3.14 (Forman [25, Theorem 5.2]). Let M be any PL + d-ball or d-sphere. By performing a finite sequence... |

3 |
A property of 2-dimensional elements
- NEWMAN
- 1926
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Citation Context ...sy to show that all shellable balls are collapsible [18], but the question of whether all constructible d-balls are collapsible is open for each d ≥ 4[8]. Newman showed that all 2-balls are shellable =-=[57]-=-; in particular, they are all constructible and collapsible. However, some 3-balls are not collapsible [9]. Every constructible 3-ball B with more than one facet is splittable, that is, it contains an... |

3 |
contractible n-manifolds and their boundaries
- PRICE, Compact
- 1971
(Show Context)
Citation Context ...a (d + 1)-pseudo-sphere but not a manifold.) If d ≥ 5, every PL manifold that is a homology d-pseudo-sphere but not a sphere bounds a contractible PL (d + 1)-manifold with boundary that is not a ball =-=[2, 62]-=-. Every PL d-sphere bounds a collapsible PL (d + 1)-ball [59, 74]; see below for the definition of collapsibility. Licensed to Penn St Univ, University Park. Prepared on Fri Jul 5 10:00:06 EDT 2013 fo... |

2 |
ENGSTRÖM, Discrete Morse functions from Fourier transforms
- B
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Citation Context |

1 |
Unshellable triangulations of spheres, Europ
- Lickorish
- 1991
(Show Context)
Citation Context ...iscrete Morse function g on B we have cint 2 (g)+1≥ t, whence cint 2 (g) ≥ t − 1. Notice the gap with Example 3.16. The crucial idea for the proof of Theorem 3.15 comes from a 1991 paper by Lickorish =-=[46]-=-, who treated the special case when M is a 3-sphere, Lint = L is a 1-sphere, c1(g) =c2(g) =0andf0(L) =f1(L) = 3. A generalization by the author and Ziegler to the case ∂M = ∅, Lint = L and cd−1(g) = 0... |