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 [KMM2004], 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: len(S1.maximal_cells())
4
sage: sorted(S1.maximal_cells())
[[0,0] x [2,3], [0,1] x [2,2], [0,1] x [3,3], [1,1] x [2,3]]
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() # needs sage.modules
{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() # needs sage.modules
{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.
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.topology.cubical_complex.Cube(data)#
Bases:
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 intervalI
,I[1]
will produce a value, not anIndexError
.)EXAMPLES:
sage: from sage.topology.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
- alexander_whitney(dim)#
Subdivide this cube into pairs of cubes.
This provides a cubical approximation for the diagonal map \(K \to K \times K\).
INPUT:
dim
– integer between 0 and one more than the dimension of this cube
OUTPUT:
a list containing triples
(coeff, left, right)
This uses the algorithm described by Pilarczyk and Réal [PR2015] on p. 267; the formula is originally due to Serre. Calling this method
alexander_whitney
is an abuse of notation, since the actual Alexander-Whitney map goes from \(C(K \times L) \to C(K) \otimes C(L)\), where \(C(-)\) denotes the associated chain complex, but this subdivision of cubes is at the heart of it.EXAMPLES:
sage: from sage.topology.cubical_complex import Cube sage: C1 = Cube([[0,1], [3,4]]) sage: C1.alexander_whitney(0) [(1, [0,0] x [3,3], [0,1] x [3,4])] sage: C1.alexander_whitney(1) [(1, [0,1] x [3,3], [1,1] x [3,4]), (-1, [0,0] x [3,4], [0,1] x [4,4])] sage: C1.alexander_whitney(2) [(1, [0,1] x [3,4], [1,1] x [4,4])]
- dimension()#
The dimension of this cube: the number of its nondegenerate intervals.
EXAMPLES:
sage: from sage.topology.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-degenerate interval with either its upper or lower endpoint.
EXAMPLES:
sage: from sage.topology.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.topology.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.topology.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.topology.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.topology.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
andother
, as a Cube
EXAMPLES:
sage: from sage.topology.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.topology.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.topology.cubical_complex.CubicalComplex(maximal_faces=None, maximality_check=True)#
Bases:
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 classCube
, 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])]); S1 Cubical complex with 4 vertices and 8 cubes sage: S1.homology() # needs sage.modules {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() # needs sage.modules {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() # needs sage.modules {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)) # needs sage.modules True
You can get the set of maximal cells or a dictionary of all cells:
sage: X.maximal_cells() # random: order may depend on the version of Python {[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: sorted(X.maximal_cells()) [[0,0] x [2,3] x [-12,-12], [0,0] x [2,3] x [0,0], [0,0] x [2,3] x [3,3], [0,0] x [2,3] x [5,5], [0,0] x [2,3] x [6,6], [0,1] x [2,2] x [-12,-12], [0,1] x [2,2] x [0,0], [0,1] x [2,2] x [3,3], [0,1] x [2,2] x [5,5], [0,1] x [2,2] x [6,6], [0,1] x [3,3] x [-12,-12], [0,1] x [3,3] x [0,0], [0,1] x [3,3] x [3,3], [0,1] x [3,3] x [5,5], [0,1] x [3,3] x [6,6], [1,1] x [2,3] x [-12,-12], [1,1] x [2,3] x [0,0], [1,1] x [2,3] x [3,3], [1,1] x [2,3] x [5,5], [1,1] x [2,3] x [6,6]] sage: S1.cells() {-1: set(), 0: {[0,0] x [2,2], [0,0] x [3,3], [1,1] x [2,2], [1,1] x [3,3]}, 1: {[0,0] x [2,3], [0,1] x [2,2], [0,1] x [3,3], [1,1] x [2,3]}}
Chain complexes, homology, and cohomology:
sage: T = S1.product(S1); T Cubical complex with 16 vertices and 64 cubes sage: T.chain_complex() # needs sage.modules Chain complex with at most 3 nonzero terms over Integer Ring sage: T.homology(base_ring=QQ) # needs sage.modules {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)) # needs sage.modules {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.
- alexander_whitney(cube, dim_left)#
Subdivide
cube
in this cubical complex into pairs of cubes.See
Cube.alexander_whitney()
for more details. This method just calls that one.INPUT:
cube
– a cube in this cubical complexdim
– integer between 0 and one more than the dimension of this cube
OUTPUT: a list containing triples
(coeff, left, right)
EXAMPLES:
sage: C = cubical_complexes.Cube(3) sage: c = list(C.n_cubes(3))[0]; c [0,1] x [0,1] x [0,1] sage: C.alexander_whitney(c, 1) [(1, [0,1] x [0,0] x [0,0], [1,1] x [0,1] x [0,1]), (-1, [0,0] x [0,1] x [0,0], [0,1] x [1,1] x [0,1]), (1, [0,0] x [0,0] x [0,1], [0,1] x [0,1] x [1,1])]
- algebraic_topological_model(base_ring=None)#
Algebraic topological model for this cubical complex with coefficients in
base_ring
.The term “algebraic topological model” is defined by Pilarczyk and Réal [PR2015].
INPUT:
base_ring
- coefficient ring (optional, defaultQQ
). Must be a field.
Denote by \(C\) the chain complex associated to this cubical complex. The algebraic topological model is a chain complex \(M\) with zero differential, with the same homology as \(C\), along with chain maps \(\pi: C \to M\) and \(\iota: M \to C\) satisfying \(\iota \pi = 1_M\) and \(\pi \iota\) chain homotopic to \(1_C\). The chain homotopy \(\phi\) must satisfy
\(\phi \phi = 0\),
\(\pi \phi = 0\),
\(\phi \iota = 0\).
Such a chain homotopy is called a chain contraction.
OUTPUT: a pair consisting of
chain contraction
phi
associated to \(C\), \(M\), \(\pi\), and \(\iota\)the chain complex \(M\)
Note that from the chain contraction
phi
, one can recover the chain maps \(\pi\) and \(\iota\) viaphi.pi()
andphi.iota()
. Then one can recover \(C\) and \(M\) from, for example,phi.pi().domain()
andphi.pi().codomain()
, respectively.EXAMPLES:
sage: # needs sage.modules sage: RP2 = cubical_complexes.RealProjectivePlane() sage: phi, M = RP2.algebraic_topological_model(GF(2)) sage: M.homology() {0: Vector space of dimension 1 over Finite Field of size 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} sage: T = cubical_complexes.Torus() sage: phi, M = T.algebraic_topological_model(QQ) sage: M.homology() {0: Vector space of dimension 1 over Rational Field, 1: Vector space of dimension 2 over Rational Field, 2: Vector space of dimension 1 over Rational Field}
- 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: sorted(S2.cells()[2]) [[0,0] x [0,1] x [0,1], [0,1] x [0,0] x [0,1], [0,1] x [0,1] x [0,0], [0,1] x [0,1] x [1,1], [0,1] x [1,1] x [0,1], [1,1] x [0,1] x [0,1]]
- chain_complex(subcomplex=None, augmented=False, verbose=False, check=False, dimensions=None, base_ring=Integer Ring, cochain=False)#
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 (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: # needs sage.modules 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}
Check that github issue #32203 has been fixed:
sage: # needs sage.modules sage: Square = CubicalComplex([([0,1],[0,1])]) sage: EdgesLTR = CubicalComplex([([0,0],[0,1]),([0,1],[1,1]),([1,1],[0,1])]) sage: EdgesLBR = CubicalComplex([([0,0],[0,1]),([0,1],[0,0]),([1,1],[0,1])]) sage: Square.homology(subcomplex=EdgesLTR)[2] == Square.homology(subcomplex=EdgesLBR)[2] True
- 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() # needs sage.modules True sage: RP2 = cubical_complexes.RealProjectivePlane() sage: T.connected_sum(RP2).homology(1) # needs sage.modules Z x Z x C2 sage: RP2.connected_sum(RP2).connected_sum(RP2).homology(1) # needs sage.modules 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() # needs sage.modules {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 ofother
.- Parameters:
other – a cubical complex
Each maximal cube of
self
must be a face of a maximal cube ofother
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 ofC2
because it’s not embedded in \(\RR^2\). On the other hand,C1 x C0
is a face ofC2
. 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 then
-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: sorted(C.n_cubes(2)) [[0,0] x [0,1] x [0,1], [0,1] x [0,0] x [0,1], [0,1] x [0,1] x [0,0], [0,1] x [0,1] x [1,1], [0,1] x [1,1] 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] # long time: 5 seconds 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 andother
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
orother
is not path-connected.EXAMPLES:
sage: S1 = cubical_complexes.Sphere(1) sage: S2 = cubical_complexes.Sphere(2) sage: S1.wedge(S2).homology() # needs sage.modules {0: 0, 1: Z, 2: Z}
- class sage.topology.cubical_complex.CubicalComplexExamples#
Bases:
object
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