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# Examples of simplicial complexes

AUTHORS:

- John H. Palmieri (2009-04)

This file constructs some examples of simplicial complexes. There are two main types: manifolds and examples related to graph theory.

For manifolds, there are functions defining the \(n\)-sphere for any \(n\), the torus, \(n\)-dimensional real projective space for any \(n\), the complex projective plane, surfaces of arbitrary genus, and some other manifolds, all as simplicial complexes.

Aside from surfaces, this file also provides some functions for constructing some other simplicial complexes: the simplicial complex of not-\(i\)-connected graphs on \(n\) vertices, the matching complex on n vertices, and the chessboard complex for an \(n\) by \(i\) chessboard. These provide examples of large simplicial complexes; for example, `simplicial_complexes.NotIConnectedGraphs(7,2)` has over a million simplices.

All of these examples are accessible by typing `simplicial_complexes.NAME`, where `NAME` is the name of the example. You can get a list by typing `simplicial_complexes.` and hitting the TAB key:

    simplicial_complexes.BarnetteSphere
    simplicial_complexes.BrucknerGrunbaumSphere
    simplicial_complexes.ChessboardComplex
    simplicial_complexes.ComplexProjectivePlane
    simplicial_complexes.K3Surface
    simplicial_complexes.KleinBottle
    simplicial_complexes.MatchingComplex
    simplicial_complexes.MooreSpace
    simplicial_complexes.NotIConnectedGraphs
    simplicial_complexes.PoincareHomologyThreeSphere
    simplicial_complexes.PseudoQuaternionicProjectivePlane
    simplicial_complexes.RandomComplex
    simplicial_complexes.RealProjectivePlane
    simplicial_complexes.RealProjectiveSpace
    simplicial_complexes.Simplex
    simplicial_complexes.Sphere
    simplicial_complexes.SumComplex
    simplicial_complexes.SurfaceOfGenus
    simplicial_complexes.Torus

See the documentation for `simplicial_complexes` and for each particular type of example for full details.

class `sage.homology.examples.``SimplicialComplexExamples`

  Some examples of simplicial complexes.

  Here are the available examples; you can also type `simplicial_complexes.` and hit tab to get a list:

  - `BarnetteSphere()`
  - `BrucknerGrunbaumSphere()`
  - `ChessboardComplex()`
  - `ComplexProjectivePlane()`
  - `K3Surface()`
  - `KleinBottle()`
  - `MatchingComplex()`
  - `MooreSpace()`
  - `NotIConnectedGraphs()`
  - `PoincareHomologyThreeSphere()`
  - `PseudoQuaternionicProjectivePlane()`
  - `RandomComplex()`
  - `RealProjectivePlane()`
  - `RealProjectiveSpace()`
  - `Simplex()`
  - `Sphere()`
  - `SumComplex()`
  - `SurfaceOfGenus()`
  - `Torus()`

  EXAMPLES:

      sage: S = simplicial_complexes.Sphere(2) # the 2-sphere
      sage: S.homology()
      {0: 0, 1: 0, 2: Z}
      sage: simplicial_complexes.SurfaceOfGenus(3)
      Simplicial complex with 15 vertices and 38 facets
      sage: M4 = simplicial_complexes.MooreSpace(4)
      sage: M4.homology()
      {0: 0, 1: C4, 2: 0}
      sage: simplicial_complexes.MatchingComplex(6).homology()
      {0: 0, 1: Z^16, 2: 0}

  `BarnetteSphere`()

    Returns Barnette’s triangulation of the 3-sphere.

    This is a pure simplicial complex of dimension 3 with 8 vertices and 19 facets, which is a non-polytopal triangulation of the 3-sphere. It was constructed by Barnette in [B1970]. The construction here uses the labeling from De Loera, Rambau and Santos [DLRS2010]. Another reference is chapter III.4 of Ewald [E1996].

    EXAMPLES:

        sage: BS = simplicial_complexes.BarnetteSphere() ; BS
        Simplicial complex with vertex set (1, 2, 3, 4, 5, 6, 7, 8) and 19 facets
        sage: BS.f_vector()
        [1, 8, 27, 38, 19]

    TESTS:

    Checks that this is indeed the same Barnette Sphere as the one given on page 87 of [E1996].:

        sage: BS2 = SimplicialComplex([[1,2,3,4],[3,4,5,6],[1,2,5,6],
        ...                            [1,2,4,7],[1,3,4,7],[3,4,6,7],
        ...                            [3,5,6,7],[1,2,5,7],[2,5,6,7],
        ...                            [2,4,6,7],[1,2,3,8],[2,3,4,8],
        ...                            [3,4,5,8],[4,5,6,8],[1,2,6,8],
        ...                            [1,5,6,8],[1,3,5,8],[2,4,6,8],
        ...                            [1,3,5,7]])
        sage: BS.is_isomorphic(BS2)
        True

    REFERENCES:

    [B1970] | Barnette, “Diagrams and Schlegel diagrams”, in Combinatorial Structures and Their Applications, Proc. Calgary Internat. Conference 1969, New York, 1970, Gordon and Breach.

    [DLRS2010] | De Loera, Rambau and Santos, “Triangulations: Structures for Algorithms and Applications”, Algorithms and Computation in Mathematics, Volume 25, Springer, 2011.

    [E1996] | (1, 2) Ewald, “Combinatorial Convexity and Algebraic Geometry”, vol. 168 of Graduate Texts in Mathematics, Springer, 1996

  `BrucknerGrunbaumSphere`()

    Returns Bruckner and Grunbaum’s triangulation of the 3-sphere.

    This is a pure simplicial complex of dimension 3 with 8 vertices and 20 facets, which is a non-polytopal triangulation of the 3-sphere. It appeared first in [Br1910] and was studied in [GrS1967].

    It is defined here as the link of any vertex in the unique minimal triangulation of the complex projective plane, see chapter 4 of [Ku1995].

    EXAMPLES:

        sage: BGS = simplicial_complexes.BrucknerGrunbaumSphere() ; BGS
        Simplicial complex with vertex set (1, 2, 3, 4, 5, 6, 7, 8) and 20 facets
        sage: BGS.f_vector()
        [1, 8, 28, 40, 20]

    REFERENCES:

    [Br1910] | Bruckner, “Uber die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope)”, Verhand. Konik. Akad. Wetenschap, Erste Sectie, 10 (1910)

    [GrS1967] | Grunbaum and Sreedharan, “An enumeration of simplicial 4-polytopes with 8 vertices”, J. Comb. Th. 2, 437-465 (1967)

    [Ku1995] | Kuhnel, “Tight Polyhedral Submanifolds and Tight Triangulations” Lecture Notes in Mathematics Volume 1612, 1995

  `ChessboardComplex`(n, i)

    The chessboard complex for an \(n \times i\) chessboard.

    Fix integers \(n, i > 0\) and consider sets \(V\) of \(n\) vertices and \(W\) of \(i\) vertices. A ‘partial matching’ between \(V\) and \(W\) is a graph formed by edges \((v,w)\) with \(v \in V\) and \(w \in W\) so that each vertex is in at most one edge. If \(G\) is a partial matching, then so is any graph obtained by deleting edges from \(G\). Thus the set of all partial matchings on \(V\) and \(W\), viewed as a set of subsets of the \(n+i\) choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex called the ‘chessboard complex’. This function produces that simplicial complex. (It is called the chessboard complex because such graphs also correspond to ways of placing rooks on an \(n\) by \(i\) chessboard so that none of them are attacking each other.)

    INPUT:

    - `n, i` – positive integers.

    See Dumas et al. [DHSW2003] for information on computing its homology by computer, and see Wachs [Wa2003] for an expository article about the theory.

    EXAMPLES:

        sage: C = simplicial_complexes.ChessboardComplex(5,5)
        sage: C.f_vector()
        [1, 25, 200, 600, 600, 120]
        sage: simplicial_complexes.ChessboardComplex(3,3).homology()
        {0: 0, 1: Z x Z x Z x Z, 2: 0}

  `ComplexProjectivePlane`()

    A minimal triangulation of the complex projective plane.

    This was constructed by Kühnel and Banchoff [KB1983].

    REFERENCES:

    [KB1983] | W. Kühnel and T. F. Banchoff, “The 9-vertex complex projective plane”, Math. Intelligencer 5 (1983), no. 3, 11-22.

    EXAMPLES:

        sage: C = simplicial_complexes.ComplexProjectivePlane()
        sage: C.f_vector()
        [1, 9, 36, 84, 90, 36]
        sage: C.homology(2)
        Z
        sage: C.homology(4)
        Z

  `K3Surface`()

    Returns a minimal triangulation of the K3 surface.

    This is a pure simplicial complex of dimension 4 with 16 vertices and 288 facets. It was constructed by Casella and Kühnel in [CK2001]. The construction here uses the labeling from Spreer and Kühnel [SK2011].

    REFERENCES:

    [CK2001] | (1, 2) M. Casella and W. Kühnel, “A triangulated K3 surface with the minimum number of vertices”, Topology 40 (2001), 753–772.

    [SK2011] | (1, 2) J. Spreer and W. Kühnel, “Combinatorial properties of the K3 surface: Simplicial blowups and slicings”, Experimental Mathematics, Volume 20, Issue 2, 2011.

    EXAMPLES:

        sage: K3=simplicial_complexes.K3Surface() ; K3
        Simplicial complex with 16 vertices and 288 facets
        sage: K3.f_vector()
        [1, 16, 120, 560, 720, 288]

    This simplicial complex is implemented just by listing all 288 facets. The list of facets can be computed by the function `facets_for_K3()`, but running the function takes a few seconds.

  `KleinBottle`()

    A minimal triangulation of the Klein bottle, as presented for example in Davide Cervone’s thesis [Ce1994].

    EXAMPLES:

        sage: simplicial_complexes.KleinBottle()
        Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 16 facets

    REFERENCES:

    [Ce1994] | D. P. Cervone, “Vertex-minimal simplicial immersions of the Klein bottle in three-space”, Geom. Ded. 50 (1994) 117-141, http://www.math.union.edu/~dpvc/papers/1993-03.kb/vmkb.pdf.

  `MatchingComplex`(n)

    The matching complex of graphs on \(n\) vertices.

    Fix an integer \(n>0\) and consider a set \(V\) of \(n\) vertices. A ‘partial matching’ on \(V\) is a graph formed by edges so that each vertex is in at most one edge. If \(G\) is a partial matching, then so is any graph obtained by deleting edges from \(G\). Thus the set of all partial matchings on \(n\) vertices, viewed as a set of subsets of the \(n\) choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex called the ‘matching complex’. This function produces that simplicial complex.

    INPUT:

    - `n` – positive integer.

    See Dumas et al. [DHSW2003] for information on computing its homology by computer, and see Wachs [Wa2003] for an expository article about the theory. For example, the homology of these complexes seems to have only mod 3 torsion, and this has been proved for the bottom non-vanishing homology group for the matching complex \(M_n\).

    EXAMPLES:

        sage: M = simplicial_complexes.MatchingComplex(7)
        sage: H = M.homology()
        sage: H
        {0: 0, 1: C3, 2: Z^20}
        sage: H[2].ngens()
        20
        sage: simplicial_complexes.MatchingComplex(8).homology(2)  # long time (6s on sage.math, 2012)
        Z^132

    REFERENCES:

    [Wa2003] | (1, 2) Wachs, “Topology of Matching, Chessboard and General Bounded Degree Graph Complexes” (Algebra Universalis Special Issue in Memory of Gian-Carlo Rota, Algebra Universalis, 49 (2003) 345-385)

  `MooreSpace`(q)

    Triangulation of the mod \(q\) Moore space.

    INPUT:

    - `q` -0 integer, at least 2

    This is a simplicial complex with simplices of dimension 0, 1, and 2, such that its reduced homology is isomorphic to \(\ZZ/q\ZZ\) in dimension 1, zero otherwise.

    If \(q=2\), this is the real projective plane. If \(q>2\), then construct it as follows: start with a triangle with vertices 1, 2, 3. We take a \(3q\)-gon forming a \(q\)-fold cover of the triangle, and we form the resulting complex as an identification space of the \(3q\)-gon. To triangulate this identification space, put \(q\) vertices \(A_0\), ..., \(A_{q-1}\), in the interior, each of which is connected to 1, 2, 3 (two facets each: \([1, 2, A_i]\), \([2, 3, A_i]\)). Put \(q\) more vertices in the interior: \(B_0\), ..., \(B_{q-1}\), with facets \([3, 1, B_i]\), \([3, B_i, A_i]\), \([1, B_i, A_{i+1}]\), \([B_i, A_i, A_{i+1}]\). Then triangulate the interior polygon with vertices \(A_0\), \(A_1\), ..., \(A_{q-1}\).

    EXAMPLES:

        sage: simplicial_complexes.MooreSpace(2)
        Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets
        sage: simplicial_complexes.MooreSpace(3).homology()[1]
        C3
        sage: simplicial_complexes.MooreSpace(4).suspension().homology()[2]
        C4
        sage: simplicial_complexes.MooreSpace(8)
        Simplicial complex with 19 vertices and 54 facets

  `NotIConnectedGraphs`(n, i)

    The simplicial complex of all graphs on \(n\) vertices which are not \(i\)-connected.

    Fix an integer \(n>0\) and consider the set of graphs on \(n\) vertices. View each graph as its set of edges, so it is a subset of a set of size \(n\) choose 2. A graph is \(i\)-connected if, for any \(j<i\), if any \(j\) vertices are removed along with the edges emanating from them, then the graph remains connected. Now fix \(i\): it is clear that if \(G\) is not \(i\)-connected, then the same is true for any graph obtained from \(G\) by deleting edges. Thus the set of all graphs which are not \(i\)-connected, viewed as a set of subsets of the \(n\) choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex. This function produces that simplicial complex.

    INPUT:

    - `n`, `i` – non-negative integers with \(i\) at most \(n\)

    See Dumas et al. [DHSW2003] for information on computing its homology by computer, and see Babson et al. [BBLSW1999] for theory. For example, Babson et al. show that when \(i=2\), the reduced homology of this complex is nonzero only in dimension \(2n-5\), where it is free abelian of rank \((n-2)!\).

    EXAMPLES:

        sage: simplicial_complexes.NotIConnectedGraphs(5,2).f_vector()
        [1, 10, 45, 120, 210, 240, 140, 20]
        sage: simplicial_complexes.NotIConnectedGraphs(5,2).homology(5).ngens()
        6

    REFERENCES:

    [BBLSW1999] | Babson, Bjorner, Linusson, Shareshian, and Welker, “Complexes of not i-connected graphs,” Topology 38 (1999), 271-299

    [DHSW2003] | (1, 2, 3) Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.

  `PoincareHomologyThreeSphere`()

    A triangulation of the Poincare homology 3-sphere.

    This is a manifold whose integral homology is identical to the ordinary 3-sphere, but it is not simply connected. In particular, its fundamental group is the binary icosahedral group, which has order 120. The triangulation given here has 16 vertices and is due to Björner and Lutz [BL2000].

    REFERENCES:

    [BL2000] | Anders Björner and Frank H. Lutz, “Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere”, Experiment. Math. 9 (2000), no. 2, 275-289.

    EXAMPLES:

        sage: S3 = simplicial_complexes.Sphere(3)
        sage: Sigma3 = simplicial_complexes.PoincareHomologyThreeSphere()
        sage: S3.homology() == Sigma3.homology()
        True
        sage: Sigma3.fundamental_group().cardinality() # long time
        120

  `ProjectivePlane`()

    A minimal triangulation of the real projective plane.

    EXAMPLES:

        sage: P = simplicial_complexes.RealProjectivePlane()
        sage: Q = simplicial_complexes.ProjectivePlane()
        sage: P == Q
        True
        sage: P.cohomology(1)
        0
        sage: P.cohomology(2)
        C2
        sage: P.cohomology(1, base_ring=GF(2))
        Vector space of dimension 1 over Finite Field of size 2
        sage: P.cohomology(2, base_ring=GF(2))
        Vector space of dimension 1 over Finite Field of size 2

  `PseudoQuaternionicProjectivePlane`()

    Returns a pure simplicial complex of dimension 8 with 490 facets.

    Warning

    This is expected to be a triangulation of the projective plane \(HP^2\) over the ring of quaternions, but this has not been proved yet.

    This simplicial complex has the same homology as \(HP^2\). Its automorphism group is isomorphic to the alternating group \(A_5\) and acts transitively on vertices.

    This is defined here using the description in [BrK92]. This article deals with three different triangulations. This procedure returns the only one which has a transitive group of automorphisms.

    EXAMPLES:

        sage: HP2 = simplicial_complexes.PseudoQuaternionicProjectivePlane() ; HP2
        Simplicial complex with 15 vertices and 490 facets
        sage: HP2.f_vector()
        [1, 15, 105, 455, 1365, 3003, 4515, 4230, 2205, 490]

    Checking its automorphism group:

        sage: HP2.automorphism_group().is_isomorphic(AlternatingGroup(5))
        True

    REFERENCES:

    [BrK92] | Brehm U., Kuhnel W., “15-vertex triangulations of an 8-manifold”, Math. Annalen 294 (1992), no. 1, 167-193.

  `RandomComplex`(n, d, p=0.5)

    A random `d`-dimensional simplicial complex on `n` vertices.

    INPUT:

    - `n` – number of vertices
    - `d` – dimension of the complex
    - `p` – floating point number between 0 and 1 (optional, default 0.5)

    A random \(d\)-dimensional simplicial complex on \(n\) vertices, as defined for example by Meshulam and Wallach [MW2009], is constructed as follows: take \(n\) vertices and include all of the simplices of dimension strictly less than \(d\), and then for each possible simplex of dimension \(d\), include it with probability \(p\).

    EXAMPLES:

        sage: X = simplicial_complexes.RandomComplex(6, 2); X
        Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets
        sage: len(list(X.vertices()))
        6

    If \(d\) is too large (if \(d+1 > n\), so that there are no \(d\)-dimensional simplices), then return the simplicial complex with a single \((n+1)\)-dimensional simplex:

        sage: simplicial_complexes.RandomComplex(6, 12)
        Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(0, 1, 2, 3, 4, 5)}

    REFERENCES:

    [MW2009] | Meshulam and Wallach, “Homological connectivity of random \(k\)-dimensional complexes”, preprint, math.CO/0609773.

  `RealProjectivePlane`()

    A minimal triangulation of the real projective plane.

    EXAMPLES:

        sage: P = simplicial_complexes.RealProjectivePlane()
        sage: Q = simplicial_complexes.ProjectivePlane()
        sage: P == Q
        True
        sage: P.cohomology(1)
        0
        sage: P.cohomology(2)
        C2
        sage: P.cohomology(1, base_ring=GF(2))
        Vector space of dimension 1 over Finite Field of size 2
        sage: P.cohomology(2, base_ring=GF(2))
        Vector space of dimension 1 over Finite Field of size 2

  `RealProjectiveSpace`(n)

    A triangulation of \(\Bold{R}P^n\) for any \(n \geq 0\).

    INPUT:

    - `n` – integer, the dimension of the real projective space to construct

    The first few cases are pretty trivial:

    - \(\Bold{R}P^0\) is a point.
    - \(\Bold{R}P^1\) is a circle, triangulated as the boundary of a single 2-simplex.
    - \(\Bold{R}P^2\) is the real projective plane, here given its minimal triangulation with 6 vertices, 15 edges, and 10 triangles.
    - \(\Bold{R}P^3\): any triangulation has at least 11 vertices by a result of Walkup [Wa1970]; this function returns a triangulation with 11 vertices, as given by Lutz [Lu2005].
    - \(\Bold{R}P^4\): any triangulation has at least 16 vertices by a result of Walkup; this function returns a triangulation with 16 vertices as given by Lutz; see also Datta [Da2007], Example 3.12.
    - \(\Bold{R}P^n\): Lutz has found a triangulation of \(\Bold{R}P^5\) with 24 vertices, but it does not seem to have been published. Kühnel [Ku1987] has described a triangulation of \(\Bold{R}P^n\), in general, with \(2^{n+1}-1\) vertices; see also Datta, Example 3.21. This triangulation is presumably not minimal, but it seems to be the best in the published literature as of this writing. So this function returns it when \(n > 4\).

    ALGORITHM: For \(n < 4\), these are constructed explicitly by listing the facets. For \(n = 4\), this is constructed by specifying 16 vertices, two facets, and a certain subgroup \(G\) of the symmetric group \(S_{16}\). Then the set of all facets is the \(G\)-orbit of the two given facets. This is implemented here by explicitly listing all of the facets; the facets can be computed by the function `facets_for_RP4()`, but running the function takes a few seconds.

    For \(n > 4\), the construction is as follows: let \(S\) denote the simplicial complex structure on the \(n\)-sphere given by the first barycentric subdivision of the boundary of an \((n+1)\)-simplex. This has a simplicial antipodal action: if \(V\) denotes the vertices in the boundary of the simplex, then the vertices in its barycentric subdivision \(S\) correspond to nonempty proper subsets \(U\) of \(V\), and the antipodal action sends any subset \(U\) to its complement. One can show that modding out by this action results in a triangulation for \(\Bold{R}P^n\). To find the facets in this triangulation, find the facets in \(S\). These are indentified in pairs to form \(\Bold{R}P^n\), so choose a representative from each pair: for each facet in \(S\), replace any vertex in \(S\) containing 0 with its complement.

    Of course these complexes increase in size pretty quickly as \(n\) increases.

    REFERENCES:

    [Da2007] | (1, 2) Basudeb Datta, “Minimal triangulations of manifolds”, J. Indian Inst. Sci. 87 (2007), no. 4, 429-449.

    [Ku1987] | W. Kühnel, “Minimal triangulations of Kummer varieties”, Abh. Math. Sem. Univ. Hamburg 57 (1987), 7-20.

    [Lu2005] | Frank H. Lutz, “Triangulated Manifolds with Few Vertices: Combinatorial Manifolds”, preprint (2005), arXiv:math/0506372.

    [Wa1970] | David W. Walkup, “The lower bound conjecture for 3- and 4-manifolds”, Acta Math. 125 (1970), 75-107.

    EXAMPLES:

        sage: P3 = simplicial_complexes.RealProjectiveSpace(3)
        sage: P3.f_vector()
        [1, 11, 51, 80, 40]
        sage: P3.homology()
        {0: 0, 1: C2, 2: 0, 3: Z}
        sage: P4 = simplicial_complexes.RealProjectiveSpace(4)
        sage: P4.f_vector()
        [1, 16, 120, 330, 375, 150]
        sage: P4.homology() # long time
        {0: 0, 1: C2, 2: 0, 3: C2, 4: 0}
        sage: P5 = simplicial_complexes.RealProjectiveSpace(5)  # long time (44s on sage.math, 2012)
        sage: P5.f_vector()  # long time
        [1, 63, 903, 4200, 8400, 7560, 2520]

    The following computation can take a long time – over half an hour – with Sage’s default computation of homology groups, but if you have CHomP installed, Sage will use that and the computation should only take a second or two. (You can download CHomP from http://chomp.rutgers.edu/, or you can install it as a Sage package using `sage -i chomp`).

        sage: P5.homology()  # long time # optional - CHomP
        {0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: Z}
        sage: simplicial_complexes.RealProjectiveSpace(2).dimension()
        2
        sage: P3.dimension()
        3
        sage: P4.dimension() # long time
        4
        sage: P5.dimension() # long time
        5

  `Simplex`(n)

    An \(n\)-dimensional simplex, as a simplicial complex.

    INPUT:

    - `n` – a non-negative integer

    OUTPUT: the simplicial complex consisting of the \(n\)-simplex on vertices \((0, 1, ..., n)\) and all of its faces.

    EXAMPLES:

        sage: simplicial_complexes.Simplex(3)
        Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2, 3)}
        sage: simplicial_complexes.Simplex(5).euler_characteristic()
        1

  `Sphere`(n)

    A minimal triangulation of the \(n\)-dimensional sphere.

    INPUT:

    - `n` – positive integer

    EXAMPLES:

        sage: simplicial_complexes.Sphere(2)
        Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
        sage: simplicial_complexes.Sphere(5).homology()
        {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: Z}
        sage: [simplicial_complexes.Sphere(n).euler_characteristic() for n in range(6)]
        [2, 0, 2, 0, 2, 0]
        sage: [simplicial_complexes.Sphere(n).f_vector() for n in range(6)]
        [[1, 2],
         [1, 3, 3],
         [1, 4, 6, 4],
         [1, 5, 10, 10, 5],
         [1, 6, 15, 20, 15, 6],
         [1, 7, 21, 35, 35, 21, 7]]

  `SumComplex`(n, A)

    The sum complexes of Linial, Meshulam, and Rosenthal [LMR2010].

    If \(k+1\) is the cardinality of \(A\), then this returns a \(k\)-dimensional simplicial complex \(X_A\) with vertices \(\ZZ/(n)\), and facets given by all \(k+1\)-tuples \((x_0, x_1, ..., x_k)\) such that the sum \(\sum x_i\) is in \(A\). See the paper by Linial, Meshulam, and Rosenthal [LMR2010], in which they prove various results about these complexes; for example, if \(n\) is prime, then \(X_A\) is rationally acyclic, and if in addition \(A\) forms an arithmetic progression in \(\ZZ/(n)\), then \(X_A\) is \(\ZZ\)-acyclic. Throughout their paper, they assume that \(n\) and \(k\) are relatively prime, but the construction makes sense in general.

    In addition to the results from the cited paper, these complexes can have large torsion, given the number of vertices; for example, if \(n=10\), and \(A=\{0,1,2,3,6\}\), then \(H_3(X_A)\) is cyclic of order 2728, and there is a 4-dimensional complex on 13 vertices with \(H_3\) having a cyclic summand of order

    \[706565607945 = 3 \cdot 5 \cdot 53 \cdot 79 \cdot 131 \cdot 157 \cdot 547.\]

    See the examples.

    INPUT:

    - `n` – a positive integer
    - `A` – a subset of \(\ZZ/(n)\)

    REFERENCES:

    [LMR2010] | (1, 2) N. Linial, R. Meshulam and M. Rosenthal, “Sum complexes – a new family of hypertrees”, Discrete & Computational Geometry, 2010, Volume 44, Number 3, Pages 622-636

    EXAMPLES:

        sage: S = simplicial_complexes.SumComplex(10, [0,1,2,3,6]); S
        Simplicial complex with 10 vertices and 126 facets
        sage: S.homology()
        {0: 0, 1: 0, 2: 0, 3: C2728, 4: 0}
        sage: factor(2728)
        2^3 * 11 * 31

        sage: S = simplicial_complexes.SumComplex(11, [0, 1, 3]); S
        Simplicial complex with 11 vertices and 45 facets
        sage: S.homology(1)
        C23
        sage: S = simplicial_complexes.SumComplex(11, [0,1,2,3,4,7]); S
        Simplicial complex with 11 vertices and 252 facets
        sage: S.homology() # long time
        {0: 0, 1: 0, 2: 0, 3: 0, 4: C645679, 5: 0}
        sage: factor(645679)
        23 * 67 * 419

        sage: S = simplicial_complexes.SumComplex(13, [0, 1, 3]); S
        Simplicial complex with 13 vertices and 66 facets
        sage: S.homology(1)
        C159
        sage: factor(159)
        3 * 53
        sage: S = simplicial_complexes.SumComplex(13, [0,1,2,5]); S
        Simplicial complex with 13 vertices and 220 facets
        sage: S.homology() # long time
        {0: 0, 1: 0, 2: C146989209, 3: 0}
        sage: factor(1648910295)
        3^2 * 5 * 53 * 521 * 1327
        sage: S = simplicial_complexes.SumComplex(13, [0,1,2,3,5]); S
        Simplicial complex with 13 vertices and 495 facets
        sage: S.homology() # long time
        {0: 0, 1: 0, 2: 0, 3: C3 x C237 x C706565607945, 4: 0}
        sage: factor(706565607945)
        3 * 5 * 53 * 79 * 131 * 157 * 547

        sage: S = simplicial_complexes.SumComplex(17, [0, 1, 4]); S
        Simplicial complex with 17 vertices and 120 facets
        sage: S.homology(1)
        C140183
        sage: factor(140183)
        103 * 1361
        sage: S = simplicial_complexes.SumComplex(19, [0, 1, 4]); S
        Simplicial complex with 19 vertices and 153 facets
        sage: S.homology(1)
        C5670599
        sage: factor(5670599)
        11 * 191 * 2699
        sage: S = simplicial_complexes.SumComplex(31, [0, 1, 4]); S
        Simplicial complex with 31 vertices and 435 facets
        sage: S.homology(1) # long time
        C5 x C5 x C5 x C5 x C26951480558170926865
        sage: factor(26951480558170926865)
        5 * 311 * 683 * 1117 * 11657 * 1948909

  `SurfaceOfGenus`(g, orientable=True)

    A surface of genus \(g\).

    INPUT:

    - `g` – a non-negative integer. The desired genus
    - `orientable` – boolean (optional, default `True`). If `True`, return an orientable surface, and if `False`, return a non-orientable surface.

    In the orientable case, return a sphere if \(g\) is zero, and otherwise return a \(g\)-fold connected sum of a torus with itself.

    In the non-orientable case, raise an error if \(g\) is zero. If \(g\) is positive, return a \(g\)-fold connected sum of a real projective plane with itself.

    EXAMPLES:

        sage: simplicial_complexes.SurfaceOfGenus(2)
        Simplicial complex with 11 vertices and 26 facets
        sage: simplicial_complexes.SurfaceOfGenus(1, orientable=False)
        Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets

  `Torus`()

    A minimal triangulation of the torus.

    This is a simplicial complex with 7 vertices, 21 edges and 14 faces. It is the unique triangulation of the torus with 7 vertices, and has been found by Möbius in 1861.

    This is also the combinatorial structure of the Császár polyhedron (see Wikipedia article Császár_polyhedron).

    EXAMPLES:

        sage: T = simplicial_complexes.Torus(); T.homology(1)
        Z x Z
        sage: T.f_vector()
        [1, 7, 21, 14]

    TESTS:

        sage: T.flip_graph().is_isomorphic(graphs.HeawoodGraph())
        True

    REFERENCES:

    [LutzCsas] | Császár’s Torus

`sage.homology.examples.``facets_for_K3`()

  Returns the facets for a minimal triangulation of the K3 surface.

  This is a pure simplicial complex of dimension 4 with 16 vertices and 288 facets. The facets are obtained by constructing a few facets and a permutation group \(G\), and then computing the \(G\)-orbit of those facets.

  See Casella and Kühnel in [CK2001] and Spreer and Kühnel [SK2011]; the construction here uses the labeling from Spreer and Kühnel.

  EXAMPLES:

      sage: from sage.homology.examples import facets_for_K3
      sage: A = facets_for_K3()   # long time (a few seconds)
      sage: SimplicialComplex(A) == simplicial_complexes.K3Surface()  # long time
      True

`sage.homology.examples.``facets_for_RP4`()

  Return the list of facets for a minimal triangulation of 4-dimensional real projective space.

  We use vertices numbered 1 through 16, define two facets, and define a certain subgroup \(G\) of the symmetric group \(S_{16}\). Then the set of all facets is the \(G\)-orbit of the two given facets.

  See the description in Example 3.12 in Datta [Da2007].

  EXAMPLES:

      sage: from sage.homology.examples import facets_for_RP4
      sage: A = facets_for_RP4()   # long time (1 or 2 seconds)
      sage: SimplicialComplex(A) == simplicial_complexes.RealProjectiveSpace(4) # long time
      True

`sage.homology.examples.``matching`(A, B)

  List of maximal matchings between the sets `A` and `B`.

  A matching is a set of pairs \((a,b) \in A \times B\) where each \(a\) and \(b\) appears in at most one pair. A maximal matching is one which is maximal with respect to inclusion of subsets of \(A \times B\).

  INPUT:

  - `A`, `B` – list, tuple, or indeed anything which can be converted to a set.

  EXAMPLES:

      sage: from sage.homology.examples import matching
      sage: matching([1,2], [3,4])
      [{(1, 3), (2, 4)}, {(1, 4), (2, 3)}]
      sage: matching([0,2], [0])
      [{(0, 0)}, {(2, 0)}]

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