SimplicialComplexes -- exploring abstract simplicial complexes within commutative algebra

Description

An abstract simplicial complex is a family of finite sets closed under the operation of taking subsets. In this package, the finite set consists of variables in a polynomial ring and each subset is represented as a product of the corresponding variables. In other words, we exploit the natural bijection between abstract simplicial complexes and Stanley-Reisner ideals.

This package is designed to explore applications of abstract simplicial complexes within combinatorial commutative algebra. Introductions to this theory can be found in the following textbooks:

Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39,Cambridge University Press, Cambridge, 1993. ISBN: 0-521-41068-1
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer-Verlag, New York, 2005. ISBN: 0-387-22356-8
Irena Peeva, Graded Syzygies, Algebra and Applications 14, Springer-Verlag, London, 2011. ISBN: 978-0-85729-176-9
Richard Stanley, Combinatorics and commutative algebra, Progress in Mathematics 41, Birkhäuser Boston, Inc., Boston, MA, 1983. ISBN: 0-8176-3112-7

Contributors

The following people have generously contributed code, improved existing code, or enhanced the documentation: Janko Böhm, Sorin Popescu, Mike Stillman, and Lorenzo Venturello.

Caveat

This package is not intended to handle abstract simplicial complexes with a very large number of vertices, because computations in the corresponding polynomial ring are typically prohibitive.

Authors

Certification

Version 2.0 of this package was accepted for publication in volume 13 of Journal of Software for Algebra and Geometry on 2023-03-21, in the article Simplicial complexes in Macaulay2. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 2.0 of SimplicialComplexes.

Source code

The source code from which this documentation is derived is in the file SimplicialComplexes.m2. The auxiliary files accompanying it are in the directory SimplicialComplexes/.

Exports

Types
- SimplicialComplex -- the class of all abstract simplicial complexes
- SimplicialMap -- the class of all maps between simplicial complexes
Functions and commands
- algebraicShifting -- the algebraic shifting of a simplicial complex
- bartnetteSphereComplex -- see bartnetteSphereComplex(PolynomialRing) -- make a non-polytopal 3-sphere with 8 vertices
- barycentricSubdivision -- see barycentricSubdivision(SimplicialComplex,Ring) -- create the barycentric subdivision of a simplicial complex
- bjornerComplex -- see bjornerComplex(PolynomialRing) -- make a shellable 2-polyhedron with 6 vertices
- boundaryMap -- see boundaryMap(ZZ,SimplicialComplex) -- make a boundary map between the oriented faces of an abstract simplicial complex
- buchbergerResolution -- make a Buchberger resolution of a monomial ideal
- buchbergerSimplicialComplex -- make the Buchberger complex of a monomial ideal
- connectedComponents -- see connectedComponents(SimplicialComplex) -- find the connected components of an abstract simplicial complex
- dunceHatComplex -- see dunceHatComplex(PolynomialRing) -- make a realization of the dunce hat with 8 vertices
- elementaryCollapse -- see elementaryCollapse(SimplicialComplex,RingElement) -- construct the elementary collapse of a free face in a simplicial complex
- flagfVector -- see flagfVector(SimplicialComplex) -- compute the flag $f$-vector of an colored simplicial complex
- grunbaumBallComplex -- see grunbaumBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 29 facets
- inducedSubcomplex -- see inducedSubcomplex(SimplicialComplex,List) -- make the induced simplicial complex on a subset of vertices
- isProper -- see isProper(SimplicialComplex) -- whether an abstract simplicial complex is properly colored
- kleinBottleComplex -- see kleinBottleComplex(PolynomialRing) -- make a minimal triangulation of the Klein bottle
- link -- see link(SimplicialComplex,RingElement) -- make the link of a face in an abstract simplicial complex
- lyubeznikResolution -- create the Lyubeznik resolution of an ordered set of monomials.
- lyubeznikSimplicialComplex -- create a simplicial complex supporting a Lyubeznik resolution of a monomial ideal
- nonPiecewiseLinearSphereComplex -- see nonPiecewiseLinearSphereComplex(PolynomialRing) -- make a non-piecewise-linear 5-sphere with 18 vertices
- poincareSphereComplex -- see poincareSphereComplex(PolynomialRing) -- make a homology 3-sphere with 16 vertices
- realProjectiveSpaceComplex -- see realProjectiveSpaceComplex(ZZ,PolynomialRing) -- make a small triangulation of real projective space
- rudinBallComplex -- see rudinBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 41 facets
- scarfChainComplex -- create the Scarf chain complex for a list of monomials.
- scarfSimplicialComplex -- create the Scarf simplicial complex for a list of monomials
- simplexComplex -- see simplexComplex(ZZ,PolynomialRing) -- make the simplex as an abstract simplicial complex
- simplicialComplex -- see simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
- smallManifold -- see smallManifold(ZZ,ZZ,ZZ,PolynomialRing) -- get a small manifold from the Frank Lutz database
- star -- see star(SimplicialComplex,RingElement) -- make the star of a face
- taylorResolution -- create the Taylor resolution of a monomial ideal
- wedge -- see wedge(SimplicialComplex,SimplicialComplex,RingElement,RingElement) -- make the wedge sum of two abstract simplicial complexes
- zieglerBallComplex -- see zieglerBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 10 vertices and 21 facets
Methods
- algebraicShifting(SimplicialComplex) -- see algebraicShifting -- the algebraic shifting of a simplicial complex
- bartnetteSphereComplex(PolynomialRing) -- make a non-polytopal 3-sphere with 8 vertices
- barycentricSubdivision(SimplicialComplex,Ring) -- create the barycentric subdivision of a simplicial complex
- barycentricSubdivision(SimplicialMap,Ring,Ring) -- create the map between barycentric subdivisions corresponding to a simplicial map
- bjornerComplex(PolynomialRing) -- make a shellable 2-polyhedron with 6 vertices
- boundaryMap(ZZ,SimplicialComplex) -- make a boundary map between the oriented faces of an abstract simplicial complex
- buchbergerResolution(List) -- see buchbergerResolution -- make a Buchberger resolution of a monomial ideal
- buchbergerResolution(MonomialIdeal) -- see buchbergerResolution -- make a Buchberger resolution of a monomial ideal
- buchbergerSimplicialComplex(List,Ring) -- see buchbergerSimplicialComplex -- make the Buchberger complex of a monomial ideal
- buchbergerSimplicialComplex(MonomialIdeal,Ring) -- see buchbergerSimplicialComplex -- make the Buchberger complex of a monomial ideal
- chainComplex(SimplicialComplex) -- create the chain complex associated to a simplicial complex.
- chainComplex(SimplicialMap) -- constructs the associated map between chain complexes
- coefficientRing(SimplicialComplex) -- get the coefficient ring of the underlying polynomial ring
- connectedComponents(SimplicialComplex) -- find the connected components of an abstract simplicial complex
- dim(SimplicialComplex) -- find the dimension of an abstract simplicial complex
- dual(SimplicialComplex) -- make the Alexander dual of an abstract simplicial complex
- dunceHatComplex(PolynomialRing) -- make a realization of the dunce hat with 8 vertices
- elementaryCollapse(SimplicialComplex,RingElement) -- construct the elementary collapse of a free face in a simplicial complex
- faces(SimplicialComplex) -- get the list of faces for an abstract simplicial complex
- faces(ZZ,SimplicialComplex) -- get the $i$-faces of an abstract simplicial complex
- facets(SimplicialComplex) -- get the list of maximal faces
- flagfVector(List,SimplicialComplex) -- compute a flag $f$-number of a colored simplicial complex
- flagfVector(SimplicialComplex) -- compute the flag $f$-vector of an colored simplicial complex
- fVector(SimplicialComplex) -- compute the f-vector of an abstract simplicial complex
- grunbaumBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 29 facets
- HH SimplicialMap -- Compute the induced map on homology of a simplicial map.
- homology(Nothing,SimplicialMap) -- see HH SimplicialMap -- Compute the induced map on homology of a simplicial map.
- HH^ZZ SimplicialMap -- Compute the induced map on cohomology of a simplicial map.
- HH^ZZ SimplicialComplex -- see HH^ZZ(SimplicialComplex,Ring) -- compute the reduced cohomology of an abstract simplicial complex
- HH^ZZ(SimplicialComplex,Ring) -- compute the reduced cohomology of an abstract simplicial complex
- HH^ZZ(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- HH_ZZ SimplicialMap -- Compute the induced map on homology of a simplicial map.
- HH_ZZ SimplicialComplex -- see HH_ZZ(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH_ZZ(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH SimplicialComplex -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(Nothing,SimplicialComplex) -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(Nothing,SimplicialComplex,Ring) -- see homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- homology(SimplicialComplex,Ring) -- compute the reduced homology of an abstract simplicial complex
- HH_ZZ(SimplicialComplex,SimplicialComplex) -- see homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- homology(Nothing,SimplicialComplex,SimplicialComplex) -- see homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- homology(SimplicialComplex,SimplicialComplex) -- compute the relative homology of two simplicial complexes
- ideal(SimplicialComplex) -- get the Stanley–Reisner ideal
- image(SimplicialMap) -- construct the image of a simplicial map
- inducedSubcomplex(SimplicialComplex,List) -- make the induced simplicial complex on a subset of vertices
- isInjective(SimplicialMap) -- checks if a simplicial map is injective
- isProper(SimplicialComplex) -- whether an abstract simplicial complex is properly colored
- isPure(SimplicialComplex) -- whether the facets are equidimensional
- isSurjective(SimplicialMap) -- checks if a simplicial map is surjective
- isWellDefined(SimplicialComplex) -- whether underlying data is uncontradictory
- isWellDefined(SimplicialMap) -- whether underlying data is uncontradictory
- kleinBottleComplex(PolynomialRing) -- make a minimal triangulation of the Klein bottle
- link(SimplicialComplex,RingElement) -- make the link of a face in an abstract simplicial complex
- lyubeznikResolution(List) -- see lyubeznikResolution -- create the Lyubeznik resolution of an ordered set of monomials.
- lyubeznikResolution(MonomialIdeal) -- see lyubeznikResolution -- create the Lyubeznik resolution of an ordered set of monomials.
- lyubeznikSimplicialComplex(List,Ring) -- see lyubeznikSimplicialComplex -- create a simplicial complex supporting a Lyubeznik resolution of a monomial ideal
- lyubeznikSimplicialComplex(MonomialIdeal,Ring) -- see lyubeznikSimplicialComplex -- create a simplicial complex supporting a Lyubeznik resolution of a monomial ideal
- map(SimplicialComplex,List) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,Matrix) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,RingMap) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,List) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialComplex,SimplicialComplex,RingMap) -- see map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes
- map(SimplicialMap) -- the underlying ring map associated to a simplicial map
- matrix(SimplicialMap) -- get the underlying map of rings
- monomialIdeal(SimplicialComplex) -- get the Stanley–Reisner monomial ideal
- net(SimplicialComplex) -- make a symbolic representation of an abstract simplicial complex
- net(SimplicialMap) -- make a symbolic representation for a map of abstract simplicial complexes
- nonPiecewiseLinearSphereComplex(PolynomialRing) -- make a non-piecewise-linear 5-sphere with 18 vertices
- poincareSphereComplex(PolynomialRing) -- make a homology 3-sphere with 16 vertices
- prune(SimplicialComplex) -- make a minimal presentation of an abstract simplicial complex
- realProjectiveSpaceComplex(ZZ,PolynomialRing) -- make a small triangulation of real projective space
- ring(SimplicialComplex) -- get the polynomial ring of its Stanley–Reisner ideal
- rudinBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 41 facets
- scarfChainComplex(List) -- see scarfChainComplex -- create the Scarf chain complex for a list of monomials.
- scarfChainComplex(MonomialIdeal) -- see scarfChainComplex -- create the Scarf chain complex for a list of monomials.
- scarfSimplicialComplex(List,Ring) -- see scarfSimplicialComplex -- create the Scarf simplicial complex for a list of monomials
- scarfSimplicialComplex(MonomialIdeal,Ring) -- see scarfSimplicialComplex -- create the Scarf simplicial complex for a list of monomials
- simplexComplex(ZZ,PolynomialRing) -- make the simplex as an abstract simplicial complex
- SimplicialComplex * SimplicialComplex -- make the join for two abstract simplicial complexes
- simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
- simplicialComplex(Matrix) -- see simplicialComplex(List) -- make an abstract simplicial complex from a list of faces
- simplicialComplex(Ideal) -- see simplicialComplex(MonomialIdeal) -- make a simplicial complex from its Stanley–Reisner ideal
- simplicialComplex(MonomialIdeal) -- make a simplicial complex from its Stanley–Reisner ideal
- skeleton(ZZ,SimplicialComplex) -- make a new simplicial complex generated by all faces of a bounded dimension
- smallManifold(ZZ,ZZ,ZZ,PolynomialRing) -- get a small manifold from the Frank Lutz database
- source(SimplicialMap) -- get the source of the map
- star(SimplicialComplex,RingElement) -- make the star of a face
- substitute(SimplicialComplex,PolynomialRing) -- change the ring of a simplicial complex
- target(SimplicialMap) -- get the target of the map
- taylorResolution(List) -- see taylorResolution -- create the Taylor resolution of a monomial ideal
- taylorResolution(MonomialIdeal) -- see taylorResolution -- create the Taylor resolution of a monomial ideal
- vertices(SimplicialComplex) -- get the list of the vertices for an abstract simplicial complex
- wedge(SimplicialComplex,SimplicialComplex,RingElement,RingElement) -- make the wedge sum of two abstract simplicial complexes
- zieglerBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 10 vertices and 21 facets
Symbols
- Labels -- see chainComplex(SimplicialComplex) -- create the chain complex associated to a simplicial complex.

For the programmer

The object SimplicialComplexes is a package.