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.ChessboardComplex
simplicial_complexes.ComplexProjectivePlane
simplicial_complexes.KleinBottle
simplicial_complexes.MatchingComplex
simplicial_complexes.MooreSpace
simplicial_complexes.NotIConnectedGraphs
simplicial_complexes.PoincareHomologyThreeSphere
simplicial_complexes.RandomComplex
simplicial_complexes.RealProjectivePlane
simplicial_complexes.RealProjectiveSpace
simplicial_complexes.Simplex
simplicial_complexes.Sphere
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:

ChessboardComplex
ComplexProjectivePlane
KleinBottle
MatchingComplex
MooreSpace
NotIConnectedGraphs
PoincareHomologyThreeSphere
RandomComplex
RealProjectivePlane
RealProjectiveSpace
Simplex
Sphere
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}
ChessboardComplex(n, i)

The chessboard complex for an n by 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. for information on computing its homology by computer, and see Wachs 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}

REFERENCES:

  • Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.
  • 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)
ComplexProjectivePlane()

A minimal triangulation of the complex projective plane.

This was constructed by Kühnel and Banchoff.

REFERENCES:

  • 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
KleinBottle()

A triangulation of the Klein bottle, formed by taking the connected sum of a real projective plane with itself. (This is not a minimal triangulation.)

EXAMPLES:

sage: simplicial_complexes.KleinBottle()
Simplicial complex with 9 vertices and 18 facets
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. for information on computing its homology by computer, and see Wachs 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 (a few seconds)
Z^132

REFERENCES:

  • Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.
  • 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 - 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(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. for information on computing its homology by computer, and see Babson et al. 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:

  • Babson, Bjorner, Linusson, Shareshian, and Welker, “Complexes of not i-connected graphs,” Topology 38 (1999), 271-299
  • 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. The triangulation given here has 16 vertices and is due to Björner and Lutz.

REFERENCES:

  • 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
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
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, 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: simplicial_complexes.RandomComplex(6, 2)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6) and 15 facets

If d is too large (if d > n+1, 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, 6, 7) and facets {(0, 1, 2, 3, 4, 5, 6, 7)}

REFERENCES:

  • 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; this function returns a triangulation with 11 vertices, as given by Lutz.
  • \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, 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 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.

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:

  • Basudeb Datta, “Minimal triangulations of manifolds”, J. Indian Inst. Sci. 87 (2007), no. 4, 429-449.
  • W. Kühnel, “Minimal triangulations of Kummer varieties”, Abh. Math. Sem. Univ. Hamburg 57 (1987), 7-20.
  • Frank H. Lutz, “Triangulated Manifolds with Few Vertices: Combinatorial Manifolds”, preprint (2005), arXiv:math/0506372.
  • 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) # long time: 2 seconds
sage: P4.f_vector() # long time
[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: 45 seconds
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]]
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.

EXAMPLES:

sage: simplicial_complexes.Torus().homology(1)
Z x Z
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 x B where each a, b appears in at most one pair. A maximal matching is one which is maximal with respect to inclusion of subsets of A x 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])
[set([(1, 3), (2, 4)]), set([(2, 3), (1, 4)])]
sage: matching([0,2], [0])
[set([(0, 0)]), set([(2, 0)])]

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