Homotopy type of non-Cohen-Macaulay complexesWhy do wedges of spheres often appear in combinatorics?Simplicial Representations of (Hyper)Graph ComplexesDiscrete Morse theory and existence of minimal complexHomotopy type of TOP(4)/PL(4)Testing simplicial complexes for shellabilityCohen-Macaulay versus shellable simplicial complexesHow much of homotopy theory can be done using only finite topological spaces?How a “sequentially Cohen–Macaulay” simplicial complex relates to “Cohen–Macaulay” simplicial complex?On a Robin Forman's remark on combinatorial simplicial complexesSimplicial set are to cubical sets what simplicial complexes are to …?
Homotopy type of non-Cohen-Macaulay complexes
Why do wedges of spheres often appear in combinatorics?Simplicial Representations of (Hyper)Graph ComplexesDiscrete Morse theory and existence of minimal complexHomotopy type of TOP(4)/PL(4)Testing simplicial complexes for shellabilityCohen-Macaulay versus shellable simplicial complexesHow much of homotopy theory can be done using only finite topological spaces?How a “sequentially Cohen–Macaulay” simplicial complex relates to “Cohen–Macaulay” simplicial complex?On a Robin Forman's remark on combinatorial simplicial complexesSimplicial set are to cubical sets what simplicial complexes are to …?
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Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension).
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable.
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?
co.combinatorics at.algebraic-topology
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
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Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension).
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable.
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?
co.combinatorics at.algebraic-topology
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
$endgroup$
add a comment |
$begingroup$
Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension).
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable.
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?
co.combinatorics at.algebraic-topology
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
$endgroup$
Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension).
I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable.
Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?
co.combinatorics at.algebraic-topology
co.combinatorics at.algebraic-topology
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
5045 silver badges12 bronze badges
asked 13 hours ago
Priyavrat DeshpandePriyavrat Deshpande
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3 Answers
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In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by there definition) are listed in Section 5. For example, included in the list are triangulations of $mathbbRP^2n$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."
Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the complex and not just the space. So, all spheres and balls are CM, but there are non-shellable examples of each.
answered 11 hours ago
John MachacekJohn Machacek
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An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.
answered 5 hours ago
Richard StanleyRichard Stanley
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The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.
One typical kind of bad example is by subword or pattern containment. A particular instance of this is:
Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.
In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.
Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
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3 Answers
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3 Answers
3
active
oldest
votes
active
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active
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votes
$begingroup$
In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by there definition) are listed in Section 5. For example, included in the list are triangulations of $mathbbRP^2n$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."
Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the complex and not just the space. So, all spheres and balls are CM, but there are non-shellable examples of each.
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
$endgroup$
add a comment |
$begingroup$
In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by there definition) are listed in Section 5. For example, included in the list are triangulations of $mathbbRP^2n$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."
Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the complex and not just the space. So, all spheres and balls are CM, but there are non-shellable examples of each.
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
$endgroup$
add a comment |
$begingroup$
In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by there definition) are listed in Section 5. For example, included in the list are triangulations of $mathbbRP^2n$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."
Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the complex and not just the space. So, all spheres and balls are CM, but there are non-shellable examples of each.
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
$endgroup$
In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by there definition) are listed in Section 5. For example, included in the list are triangulations of $mathbbRP^2n$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."
Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the complex and not just the space. So, all spheres and balls are CM, but there are non-shellable examples of each.
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
edited 11 hours ago
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
answered 11 hours ago
John MachacekJohn Machacek
4,8471 gold badge11 silver badges31 bronze badges
answered 11 hours ago
John MachacekJohn Machacek
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4,8471 gold badge11 silver badges31 bronze badges
add a comment |
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$begingroup$
An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.
answered 5 hours ago
Richard StanleyRichard Stanley
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$begingroup$
An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
$endgroup$
add a comment |
$begingroup$
An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
$endgroup$
An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $h$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
answered 5 hours ago
Richard StanleyRichard Stanley
29.8k9 gold badges118 silver badges194 bronze badges
29.8k9 gold badges118 silver badges194 bronze badges
add a comment |
add a comment |
$begingroup$
The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.
One typical kind of bad example is by subword or pattern containment. A particular instance of this is:
Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.
In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.
Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
$endgroup$
add a comment |
$begingroup$
The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.
One typical kind of bad example is by subword or pattern containment. A particular instance of this is:
Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.
In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.
Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
$endgroup$
add a comment |
$begingroup$
The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.
One typical kind of bad example is by subword or pattern containment. A particular instance of this is:
Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.
In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.
Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
$endgroup$
The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.
One typical kind of bad example is by subword or pattern containment. A particular instance of this is:
Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.
In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.
Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
2,7261 gold badge16 silver badges18 bronze badges
answered 2 hours ago
Russ WoodroofeRuss Woodroofe
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