### Navigation

- index
- modules |
- next |
- previous |
- Cell complexes and their homology »

# Finite cubical complexes

AUTHORS:

- John H. Palmieri (2009-08)

This module implements the basic structure of finite cubical complexes. For full mathematical details, see Kaczynski, Mischaikow, and Mrozek [KMM], for example.

Cubical complexes are topological spaces built from gluing together cubes of various dimensions; the collection of cubes must be closed under taking faces, just as with a simplicial complex. In this context, a “cube” means a product of intervals of length 1 or length 0 (degenerate intervals), with integer endpoints, and its faces are obtained by using the nondegenerate intervals: if \(C\) is a cube – a product of degenerate and nondegenerate intervals – and if \([i,i+1]\) is the \(k\)-th nondegenerate factor, then \(C\) has two faces indexed by \(k\): the cubes obtained by replacing \([i, i+1]\) with \([i, i]\) or \([i+1, i+1]\).

So to construct a space homeomorphic to a circle as a cubical complex, we could take for example the four line segments in the plane from \((0,2)\) to \((0,3)\) to \((1,3)\) to \((1,2)\) to \((0,2)\). In Sage, this is done with the following command:

    sage: S1 = CubicalComplex([([0,0], [2,3]), ([0,1], [3,3]), ([0,1], [2,2]), ([1,1], [2,3])]); S1
    Cubical complex with 4 vertices and 8 cubes

The argument to `CubicalComplex` is a list of the maximal “cubes” in the complex. Each “cube” can be an instance of the class `Cube` or a list (or tuple) of “intervals”, and an “interval” is a pair of integers, of one of the two forms \([i, i]\) or \([i, i+1]\). So the cubical complex `S1` above has four maximal cubes:

    sage: S1.maximal_cells()
    {[0,0] x [2,3], [1,1] x [2,3], [0,1] x [3,3], [0,1] x [2,2]}

The first of these, for instance, is the product of the degenerate interval \([0,0]\) with the unit interval \([2,3]\): this is the line segment in the plane from \((0,2)\) to \((0,3)\). We could form a topologically equivalent space by inserting some degenerate simplices:

    sage: S1.homology()
    {0: 0, 1: Z}
    sage: X = CubicalComplex([([0,0], [2,3], [2]), ([0,1], [3,3], [2]), ([0,1], [2,2], [2]), ([1,1], [2,3], [2])])
    sage: X.homology()
    {0: 0, 1: Z}

Topologically, the cubical complex `X` consists of four edges of a square in \(\RR^3\): the same unit square as `S1`, but embedded in \(\RR^3\) with \(z\)-coordinate equal to 2. Thus `X` is homeomorphic to `S1` (in fact, they’re “cubically equivalent”), and this is reflected in the fact that they have isomorphic homology groups.

REFERENCES:

[KMM] | Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, “Computational Homology”, Springer-Verlag (2004).

Note

This class derives from `GenericCellComplex`, and so inherits its methods. Some of those methods are not listed here; see the `Generic Cell Complex` page instead.

class `sage.homology.cubical_complex.``Cube`(data)

  Bases: `sage.structure.sage_object.SageObject`

  Define a cube for use in constructing a cubical complex.

  “Elementary cubes” are products of intervals with integer endpoints, each of which is either a unit interval or a degenerate (length 0) interval; for example,

  \[[0,1] \times [3,4] \times [2,2] \times [1,2]\]

  is a 3-dimensional cube (since one of the intervals is degenerate) embedded in \(\RR^4\).

  Parameters: | data – list or tuple of terms of the form `(i,i+1)` or `(i,i)` or `(i,)` – the last two are degenerate intervals.
  Returns: | an elementary cube

  Each cube is stored in a standard form: a tuple of tuples, with a nondegenerate interval `[j,j]` represented by `(j,j)`, not `(j,)`. (This is so that for any interval `I`, `I[1]` will produce a value, not an `IndexError`.)

  EXAMPLES:

      sage: from sage.homology.cubical_complex import Cube
      sage: C = Cube([[1,2], [5,], [6,7], [-1, 0]]); C
      [1,2] x [5,5] x [6,7] x [-1,0]
      sage: C.dimension() # number of nondegenerate intervals
      3
      sage: C.nondegenerate_intervals()  # indices of these intervals
      [0, 2, 3]
      sage: C.face(1, upper=False)
      [1,2] x [5,5] x [6,6] x [-1,0]
      sage: C.face(1, upper=True)
      [1,2] x [5,5] x [7,7] x [-1,0]
      sage: Cube(()).dimension()  # empty cube has dimension -1
      -1

  `dimension`()

    The dimension of this cube: the number of its nondegenerate intervals.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [5,], [6,7], [-1, 0]])
        sage: C.dimension()
        3
        sage: C = Cube([[1,], [5,], [6,], [-1,]])
        sage: C.dimension()
        0
        sage: Cube([]).dimension()  # empty cube has dimension -1
        -1

  `face`(n, upper=True)

    The nth primary face of this cube.

    Parameters: | n – an integer between 0 and one less than the dimension of this cube upper (boolean; optional, default=True) – if True, return the “upper” nth primary face; otherwise, return the “lower” nth primary face.
    Returns: | the cube obtained by replacing the nth non-degenrate interval with either its upper or lower endpoint.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [5,], [6,7], [-1, 0]]); C
        [1,2] x [5,5] x [6,7] x [-1,0]
        sage: C.face(0)
        [2,2] x [5,5] x [6,7] x [-1,0]
        sage: C.face(0, upper=False)
        [1,1] x [5,5] x [6,7] x [-1,0]
        sage: C.face(1)
        [1,2] x [5,5] x [7,7] x [-1,0]
        sage: C.face(2, upper=False)
        [1,2] x [5,5] x [6,7] x [-1,-1]
        sage: C.face(3)
        Traceback (most recent call last):
        ...
        ValueError: Can only compute the nth face if 0 <= n < dim.

  `faces`()

    The list of faces (of codimension 1) of this cube.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [3,4]])
        sage: C.faces()
        [[2,2] x [3,4], [1,2] x [4,4], [1,1] x [3,4], [1,2] x [3,3]]

  `faces_as_pairs`()

    The list of faces (of codimension 1) of this cube, as pairs (upper, lower).

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [3,4]])
        sage: C.faces_as_pairs()
        [([2,2] x [3,4], [1,1] x [3,4]), ([1,2] x [4,4], [1,2] x [3,3])]

  `is_face`(other)

    Return True iff this cube is a face of other.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C1 = Cube([[1,2], [5,], [6,7], [-1, 0]])
        sage: C2 = Cube([[1,2], [5,], [6,], [-1, 0]])
        sage: C1.is_face(C2)
        False
        sage: C1.is_face(C1)
        True
        sage: C2.is_face(C1)
        True

  `nondegenerate_intervals`()

    The list of indices of nondegenerate intervals of this cube.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [5,], [6,7], [-1, 0]])
        sage: C.nondegenerate_intervals()
        [0, 2, 3]
        sage: C = Cube([[1,], [5,], [6,], [-1,]])
        sage: C.nondegenerate_intervals()
        []

  `product`(other)

    Cube obtained by concatenating the underlying tuples of the two arguments.

    Parameters: | other – another cube
    Returns: | the product of `self` and `other`, as a Cube

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [3,]])
        sage: D = Cube([[4], [0,1]])
        sage: C.product(D)
        [1,2] x [3,3] x [4,4] x [0,1]

    You can also use `__add__` or `+` or `__mul__` or `*`:

        sage: D * C
        [4,4] x [0,1] x [1,2] x [3,3]
        sage: D + C * C
        [4,4] x [0,1] x [1,2] x [3,3] x [1,2] x [3,3]

  `tuple`()

    The tuple attached to this cube.

    EXAMPLES:

        sage: from sage.homology.cubical_complex import Cube
        sage: C = Cube([[1,2], [5,], [6,7], [-1, 0]])
        sage: C.tuple()
        ((1, 2), (5, 5), (6, 7), (-1, 0))

class `sage.homology.cubical_complex.``CubicalComplex`(maximal_faces=[], **kwds)

  Bases: `sage.homology.cell_complex.GenericCellComplex`

  Define a cubical complex.

  Parameters: | maximal_faces – set of maximal faces maximality_check (boolean; optional, default True) – see below
  Returns: | a cubical complex

  `maximal_faces` should be a list or tuple or set (or anything which may be converted to a set) of “cubes”: instances of the class `Cube`, or lists or tuples suitable for conversion to cubes. These cubes are the maximal cubes in the complex.

  In addition, `maximal_faces` may be a cubical complex, in which case that complex is returned. Also, `maximal_faces` may instead be any object which has a `_cubical_` method (e.g., a simplicial complex); then that method is used to convert the object to a cubical complex.

  If `maximality_check` is True, check that each maximal face is, in fact, maximal. In this case, when producing the internal representation of the cubical complex, omit those that are not. It is highly recommended that this be True; various methods for this class may fail if faces which are claimed to be maximal are in fact not.

  EXAMPLES:

  The empty complex, consisting of one cube, the empty cube:

      sage: CubicalComplex()
      Cubical complex with 0 vertices and 1 cube

  A “circle” (four edges connecting the vertices (0,2), (0,3), (1,2), and (1,3)):

      sage: S1 = CubicalComplex([([0,0], [2,3]), ([0,1], [3,3]), ([0,1], [2,2]), ([1,1], [2,3])])
      sage: S1
      Cubical complex with 4 vertices and 8 cubes
      sage: S1.homology()
      {0: 0, 1: Z}

  A set of five points and its product with `S1`:

      sage: pts = CubicalComplex([([0],), ([3],), ([6],), ([-12],), ([5],)])
      sage: pts
      Cubical complex with 5 vertices and 5 cubes
      sage: pts.homology()
      {0: Z x Z x Z x Z}
      sage: X = S1.product(pts); X
      Cubical complex with 20 vertices and 40 cubes
      sage: X.homology()
      {0: Z x Z x Z x Z, 1: Z^5}

  Converting a simplicial complex to a cubical complex:

      sage: S2 = simplicial_complexes.Sphere(2)
      sage: C2 = CubicalComplex(S2)
      sage: all([C2.homology(n) == S2.homology(n) for n in range(3)])
      True

  You can get the set of maximal cells or a dictionary of all cells:

      sage: X.maximal_cells()
      {[0,0] x [2,3] x [-12,-12], [0,1] x [3,3] x [5,5], [0,1] x [2,2] x [3,3], [0,1] x [2,2] x [0,0], [0,1] x [3,3] x [6,6], [1,1] x [2,3] x [0,0], [0,1] x [2,2] x [-12,-12], [0,0] x [2,3] x [6,6], [1,1] x [2,3] x [-12,-12], [1,1] x [2,3] x [5,5], [0,1] x [2,2] x [5,5], [0,1] x [3,3] x [3,3], [1,1] x [2,3] x [3,3], [0,0] x [2,3] x [5,5], [0,1] x [3,3] x [0,0], [1,1] x [2,3] x [6,6], [0,1] x [2,2] x [6,6], [0,0] x [2,3] x [0,0], [0,0] x [2,3] x [3,3], [0,1] x [3,3] x [-12,-12]}
      sage: S1.cells()
      {-1: set(),
       0: {[0,0] x [3,3], [1,1] x [3,3], [0,0] x [2,2], [1,1] x [2,2]},
       1: {[0,1] x [2,2], [0,0] x [2,3], [1,1] x [2,3], [0,1] x [3,3]}}

  Chain complexes, homology, and cohomology:

      sage: T = S1.product(S1); T
      Cubical complex with 16 vertices and 64 cubes
      sage: T.chain_complex()
      Chain complex with at most 3 nonzero terms over Integer Ring
      sage: T.homology(base_ring=QQ)
      {0: Vector space of dimension 0 over Rational Field,
       1: Vector space of dimension 2 over Rational Field,
       2: Vector space of dimension 1 over Rational Field}
      sage: RP2 = cubical_complexes.RealProjectivePlane()
      sage: RP2.cohomology(dim=[1, 2], base_ring=GF(2))
      {1: Vector space of dimension 1 over Finite Field of size 2,
       2: Vector space of dimension 1 over Finite Field of size 2}

  Joins are not implemented:

      sage: S1.join(S1)
      Traceback (most recent call last):
      ...
      NotImplementedError: Joins are not implemented for cubical complexes.

  Therefore, neither are cones or suspensions.

  `cells`(subcomplex=None)

    The cells of this cubical complex, in the form of a dictionary: the keys are integers, representing dimension, and the value associated to an integer d is the list of d-cells.

    If the optional argument `subcomplex` is present, then return only the faces which are not in the subcomplex.

    Parameters: | subcomplex (a cubical complex; optional, default None) – a subcomplex of this cubical complex
    Returns: | cells of this complex not contained in `subcomplex`
    Return type: | dictionary

    EXAMPLES:

        sage: S2 = cubical_complexes.Sphere(2)
        sage: S2.cells()[2]
        {[0,1] x [0,1] x [0,0],
         [0,1] x [0,1] x [1,1],
         [0,0] x [0,1] x [0,1],
         [0,1] x [1,1] x [0,1],
         [0,1] x [0,0] x [0,1],
         [1,1] x [0,1] x [0,1]}

  `chain_complex`(**kwds)

    The chain complex associated to this cubical complex.

    Parameters: | dimensions – if None, compute the chain complex in all dimensions. If a list or tuple of integers, compute the chain complex in those dimensions, setting the chain groups in all other dimensions to zero. NOT IMPLEMENTED YET: this function always returns the entire chain complex base_ring (optional, default ZZ) – commutative ring subcomplex (optional, default empty) – a subcomplex of this cubical complex. Compute the chain complex relative to this subcomplex. augmented (boolean; optional, default False) – If True, return the augmented chain complex (that is, include a class in dimension \(-1\) corresponding to the empty cell). This is ignored if `dimensions` is specified. cochain (boolean; optional, default False) – If True, return the cochain complex (that is, the dual of the chain complex). verbose (boolean; optional, default False) – If True, print some messages as the chain complex is computed. check_diffs (boolean; optional, default False) – If True, make sure that the chain complex is actually a chain complex: the differentials are composable and their product is zero.

    Note

    If subcomplex is nonempty, then the argument `augmented` has no effect: the chain complex relative to a nonempty subcomplex is zero in dimension \(-1\).

    EXAMPLES:

        sage: S2 = cubical_complexes.Sphere(2)
        sage: S2.chain_complex()
        Chain complex with at most 3 nonzero terms over Integer Ring
        sage: Prod = S2.product(S2); Prod
        Cubical complex with 64 vertices and 676 cubes
        sage: Prod.chain_complex()
        Chain complex with at most 5 nonzero terms over Integer Ring
        sage: Prod.chain_complex(base_ring=QQ)
        Chain complex with at most 5 nonzero terms over Rational Field
        sage: C1 = cubical_complexes.Cube(1)
        sage: S0 = cubical_complexes.Sphere(0)
        sage: C1.chain_complex(subcomplex=S0)
        Chain complex with at most 1 nonzero terms over Integer Ring
        sage: C1.homology(subcomplex=S0)
        {0: 0, 1: Z}

  `cone`()

    The cone on this cubical complex.

    NOT IMPLEMENTED

    The cone is the complex formed by taking the join of the original complex with a one-point complex (that is, a 0-dimensional cube). Since joins are not implemented for cubical complexes, neither are cones.

    EXAMPLES:

        sage: C1 = cubical_complexes.Cube(1)
        sage: C1.cone()
        Traceback (most recent call last):
        ...
        NotImplementedError: Cones are not implemented for cubical complexes.

  `connected_sum`(other)

    Return the connected sum of self with other.

    Parameters: | other – another cubical complex
    Returns: | the connected sum `self # other`

    Warning

    This does not check that self and other are manifolds, only that their facets all have the same dimension. Since a (more or less) random facet is chosen from each complex and then glued together, this method may return random results if applied to non-manifolds, depending on which facet is chosen.

    EXAMPLES:

        sage: T = cubical_complexes.Torus()
        sage: S2 = cubical_complexes.Sphere(2)
        sage: T.connected_sum(S2).cohomology() == T.cohomology()
        True
        sage: RP2 = cubical_complexes.RealProjectivePlane()
        sage: T.connected_sum(RP2).homology(1)
        Z x Z x C2
        sage: RP2.connected_sum(RP2).connected_sum(RP2).homology(1)
        Z x Z x C2

  `disjoint_union`(other)

    The disjoint union of this cubical complex with another one.

    Parameters: | right – the other cubical complex (the right-hand factor)

    Algorithm: first embed both complexes in d-dimensional Euclidean space. Then embed in (1+d)-dimensional space, calling the new axis \(x\), and putting the first complex at \(x=0\), the second at \(x=1\).

    EXAMPLES:

        sage: S1 = cubical_complexes.Sphere(1)
        sage: S2 = cubical_complexes.Sphere(2)
        sage: S1.disjoint_union(S2).homology()
        {0: Z, 1: Z, 2: Z}

  `graph`()

    The 1-skeleton of this cubical complex, as a graph.

    EXAMPLES:

        sage: cubical_complexes.Sphere(2).graph()
        Graph on 8 vertices

  `is_pure`()

    True iff this cubical complex is pure: that is, all of its maximal faces have the same dimension.

    Warning

    This may give the wrong answer if the cubical complex was constructed with `maximality_check` set to False.

    EXAMPLES:

        sage: S4 = cubical_complexes.Sphere(4)
        sage: S4.is_pure()
        True
        sage: C = CubicalComplex([([0,0], [3,3]), ([1,2], [4,5])])
        sage: C.is_pure()
        False

  `is_subcomplex`(other)

    Return True if `self` is a subcomplex of `other`.

    Parameters: | other – a cubical complex

    Each maximal cube of `self` must be a face of a maximal cube of `other` for this to be True.

    EXAMPLES:

        sage: S1 = cubical_complexes.Sphere(1)
        sage: C0 = cubical_complexes.Cube(0)
        sage: C1 = cubical_complexes.Cube(1)
        sage: cyl = S1.product(C1)
        sage: end = S1.product(C0)
        sage: end.is_subcomplex(cyl)
        True
        sage: cyl.is_subcomplex(end)
        False

    The embedding of the cubical complex is important here:

        sage: C2 = cubical_complexes.Cube(2)
        sage: C1.is_subcomplex(C2)
        False
        sage: C1.product(C0).is_subcomplex(C2)
        True

    `C1` is not a subcomplex of `C2` because it’s not embedded in \(\RR^2\). On the other hand, `C1 x C0` is a face of `C2`. Look at their maximal cells:

        sage: C1.maximal_cells()
        {[0,1]}
        sage: C2.maximal_cells()
        {[0,1] x [0,1]}
        sage: C1.product(C0).maximal_cells()
        {[0,1] x [0,0]}

  `join`(other)

    The join of this cubical complex with another one.

    NOT IMPLEMENTED.

    Parameters: | other – another cubical complex

    EXAMPLES:

        sage: C1 = cubical_complexes.Cube(1)
        sage: C1.join(C1)
        Traceback (most recent call last):
        ...
        NotImplementedError: Joins are not implemented for cubical complexes.

  `maximal_cells`()

    The set of maximal cells (with respect to inclusion) of this cubical complex.

    Returns: | Set of maximal cells

    This just returns the set of cubes used in defining the cubical complex, so if the complex was defined with no maximality checking, none is done here, either.

    EXAMPLES:

        sage: interval = cubical_complexes.Cube(1)
        sage: interval
        Cubical complex with 2 vertices and 3 cubes
        sage: interval.maximal_cells()
        {[0,1]}
        sage: interval.product(interval).maximal_cells()
        {[0,1] x [0,1]}

  `n_cubes`(n, subcomplex=None)

    The set of cubes of dimension n of this cubical complex. If the optional argument `subcomplex` is present, then return the `n`-dimensional cubes which are not in the subcomplex.

    Parameters: | n (integer) – dimension subcomplex (a cubical complex; optional, default None) – a subcomplex of this cubical complex
    Returns: | cells in dimension `n`
    Return type: | set

    EXAMPLES:

        sage: C = cubical_complexes.Cube(3)
        sage: C.n_cubes(3)
        {[0,1] x [0,1] x [0,1]}
        sage: C.n_cubes(2)
        {[0,1] x [0,1] x [0,0],
         [0,1] x [0,1] x [1,1],
         [0,0] x [0,1] x [0,1],
         [0,1] x [1,1] x [0,1],
         [0,1] x [0,0] x [0,1],
         [1,1] x [0,1] x [0,1]}

  `n_skeleton`(n)

    The n-skeleton of this cubical complex.

    Parameters: | n (non-negative integer) – dimension
    Returns: | cubical complex

    EXAMPLES:

        sage: S2 = cubical_complexes.Sphere(2)
        sage: C3 = cubical_complexes.Cube(3)
        sage: S2 == C3.n_skeleton(2)
        True

  `product`(other)

    The product of this cubical complex with another one.

    Parameters: | other – another cubical complex

    EXAMPLES:

        sage: RP2 = cubical_complexes.RealProjectivePlane()
        sage: S1 = cubical_complexes.Sphere(1)
        sage: RP2.product(S1).homology()[1]
        Z x C2

  `suspension`(n=1)

    The suspension of this cubical complex.

    NOT IMPLEMENTED

    Parameters: | n (positive integer; optional, default 1) – suspend this many times

    The suspension is the complex formed by taking the join of the original complex with a two-point complex (the 0-sphere). Since joins are not implemented for cubical complexes, neither are suspensions.

    EXAMPLES:

        sage: C1 = cubical_complexes.Cube(1)
        sage: C1.suspension()
        Traceback (most recent call last):
        ...
        NotImplementedError: Suspensions are not implemented for cubical complexes.

  `wedge`(other)

    The wedge (one-point union) of this cubical complex with another one.

    Parameters: | right – the other cubical complex (the right-hand factor)

    Algorithm: if `self` is embedded in \(d\) dimensions and `other` in \(n\) dimensions, embed them in \(d+n\) dimensions: `self` using the first \(d\) coordinates, `other` using the last \(n\), translating them so that they have the origin as a common vertex.

    Note

    This operation is not well-defined if `self` or `other` is not path-connected.

    EXAMPLES:

        sage: S1 = cubical_complexes.Sphere(1)
        sage: S2 = cubical_complexes.Sphere(2)
        sage: S1.wedge(S2).homology()
        {0: 0, 1: Z, 2: Z}

class `sage.homology.cubical_complex.``CubicalComplexExamples`

  Some examples of cubical complexes.

  Here are the available examples; you can also type “cubical_complexes.” and hit TAB to get a list:

      Sphere
      Torus
      RealProjectivePlane
      KleinBottle
      SurfaceOfGenus
      Cube

  EXAMPLES:

      sage: cubical_complexes.Torus()  # indirect doctest
      Cubical complex with 16 vertices and 64 cubes
      sage: cubical_complexes.Cube(7)
      Cubical complex with 128 vertices and 2187 cubes
      sage: cubical_complexes.Sphere(7)
      Cubical complex with 256 vertices and 6560 cubes

  `Cube`(n)

    A cubical complex representation of an \(n\)-dimensional cube.

    Parameters: | n (non-negative integer) – the dimension

    EXAMPLES:

        sage: cubical_complexes.Cube(0)
        Cubical complex with 1 vertex and 1 cube
        sage: cubical_complexes.Cube(3)
        Cubical complex with 8 vertices and 27 cubes

  `KleinBottle`()

    A cubical complex representation of the Klein bottle, formed by taking the connected sum of the real projective plane with itself.

    EXAMPLES:

        sage: cubical_complexes.KleinBottle()
        Cubical complex with 42 vertices and 168 cubes

  `RealProjectivePlane`()

    A cubical complex representation of the real projective plane. This is taken from the examples from CHomP, the Computational Homology Project: http://chomp.rutgers.edu/.

    EXAMPLES:

        sage: cubical_complexes.RealProjectivePlane()
        Cubical complex with 21 vertices and 81 cubes

  `Sphere`(n)

    A cubical complex representation of the \(n\)-dimensional sphere, formed by taking the boundary of an \((n+1)\)-dimensional cube.

    Parameters: | n (non-negative integer) – the dimension of the sphere

    EXAMPLES:

        sage: cubical_complexes.Sphere(7)
        Cubical complex with 256 vertices and 6560 cubes

  `SurfaceOfGenus`(g, orientable=True)

    A surface of genus g as a cubical complex.

    Parameters: | g (non-negative integer) – the genus orientable (bool, optional, default True) – whether the surface should be orientable

    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: cubical_complexes.SurfaceOfGenus(2)
        Cubical complex with 32 vertices and 134 cubes
        sage: cubical_complexes.SurfaceOfGenus(1, orientable=False)
        Cubical complex with 21 vertices and 81 cubes

  `Torus`()

    A cubical complex representation of the torus, obtained by taking the product of the circle with itself.

    EXAMPLES:

        sage: cubical_complexes.Torus()
        Cubical complex with 16 vertices and 64 cubes

### Navigation

- index
- modules |
- next |
- previous |
- Cell complexes and their homology »